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Must Do “Must Do Better” Better

Tue, 03 Mar 2026 15:00:00 +0000

Abstract: Starting with the his much-discussed 2006 paper “Must Do Better,’ Timothy Williamson has offered a series of methodological reflections and recommendations for philosophers. Good philosophy, for Williamson, (1) prefers precision and rigour to depth and profundity, (2) where possible, it avails itself of formal modelling, so as to not only prescribe definitions or analyses of concepts, but make a range of predictions that can be tested, (3) embraces the constraints of connections to other disciplines and to a range of scientific norms. While there is much to appreciate in Williamson’s methodology—at least as an example of one fruitful way to pursue philosophical questions—I will argue that Williamson’s idiosyncratic understanding of logic and of semantics helps explain how he has come to such surprising conclusions in epistemology (e.g. that we know all logical and necessary truths) and metaphysics (e.g. that everything that exists exists necessarily), and once we adopt a more well-rounded—and, dare I say, orthodox—view of logic and semantics, the argument to these more surprising conclusions are undercut, and a quite different picture emerges concerning the place for logic, and formal tools more generally, in philosophy. I will end with some short, speculative suggestions about what this means for analytic philosophy, more generally.

Debating the Status of Mathematics

Wed, 18 Feb 2026 10:12:00 +0000

The St Andrews Union Debating Society is a long-standing institution around these parts. Founded in 1794, this student-run society hosts regular debates. Last week, the society collaborated with SUMS, the St Andrews University Mathematics Society, to host a public debate on the status of mathematics, featuring the eminent Emeritus Professor Peter Cameron and my colleague Dr Walter Pedriali on one side, and on the other, Dr David Waszek (an historian and philosopher of mathematics working at the ÉNS in Paris), and me.

PY4612: Advanced Logic

Tue, 27 Jan 2026 00:00:00 +0000

py4612: Advanced Logic applies the tools of formal logic to make logic itself the object of study. We will explore the power and limits of logical tools and techniques. The main goals of the module will be to come to grips with some standard ‘metatheoretical’ results about logic: (1) the Soundness and Completeness Theorems, which together show that proofs and models can be used analyse the same consequence relation in two very different ways. (2) The Compactness Theorem and the Löwenheim–Skolem Theorems, which explore some of the limits of first-order classical predicate logic for classifying infinite structures. (3) The contrast between first-order predicate logic and second-order logic. Finally (4), we will work through Gödel’s celebrated Incompleteness Theorems and come to grips with what they mean. Along the way, there will be some preparatory discussion of elementary set theory, proof theory, model theory, and recursion theory. Kurt Gödel, seated

Why Logic Matters for Philosophy, and why Philosophy Matters for Logic

Tue, 18 Nov 2025 19:30:00 +0000

This is a longer, more relaxed and interactive version of the talk “Why Logic Matters for Philosophy, and why Philosophy Matters for Logic” that I first gave as a part of the Inaugural Lecture Showcase of the Philosophy Department. The poster for the presentation. The slides for the talk are available here.

Why Logic Matters for Philosophy, and why Philosophy Matters for Logic

Wed, 12 Nov 2025 17:10:00 +0000

I gave a short, 25 minute lecture, entitled “Why Logic Matters for Philosophy, and why Philosophy Matters for Logic” as a part of the Inaugural Lecture Showcase of the Philosophy Department. The slides for the talk are available here.

2025 Wendy Huang Lectures

Tue, 21 Oct 2025 14:00:00 +0000

I will be giving the 2025 Wendy Huang Lectures at the invitation of the Taiwan Association for Logic, Methodology and Philosophy of Science, at National Taiwan University, in Taipei, from October 21 to 23. The third lecture will also be a keynote presentation at the Taiwan Philosophical Logic Colloquium. The three talks are: Inferentialism for Everyone Tuesday October 21 • (handout) I aim to give an opinionated introduction to inferentialist semantics, giving an account about what is so distinctive about logic, insofar as logical notions have a grip on whatever can be said or thought. In doing this, I aim to clarify the connections between logic and semantics, the theory of meaning. Existence & Modality (Wednesday October 22) • (handout) In the second lecture, I will show how the inferentialist semantics, introduced in Lecture 1, applies to issues of predication, quantification and modality, and thereby provides some distinctive insight into ‘possible worlds’ models for quantified modal logics. Foundations for Truth-Conditional Semantics (Thursday October 23) • (handout) Finally, I show how we might relate inferentialist semantics to truth-conditional accounts of meaning for natural languages. The poster for the 2025 Wendy Huang Lectures A draft of the text of all three lectures is available here.

The Shape of an Intermediate Logic Class

Mon, 29 Sep 2025 20:40:00 +0100

Alongside coordinating and lecturing in PY2010: Intermediate Logic this semester, I have the fun task of teaching one of the tutorial groups, together with our enthusiastic and capable graduate student tutors. My Tuesday morning cohort is a microcosm of the diverse student body at St Andrews. Of the 13 students, we have two each from England, China, and the USA, and we have one each from Botswana, Denmark, France, Greece, Hong Kong, Scotland and Spain. That’s a diverse bunch. In addition, six of the 13 are Philosophy majors, six are Mathematics majors, five major in International Relations, and we have one each from French, Computer Science and Psychology. (I haven’t miscounted: some of our students are completing double majors.) Finally, this cohort consists of eight women and five men. Given their submissions for our first tutorial, starting tomorrow—and given the engagement and enthusiasm that the whole group are showing in lectures—I’m looking forward to spending time with them.

Inferentialism: Logic and Meaning

Wed, 24 Sep 2025 14:00:00 +0000

In October, I will be giving the 2025 Wendy Huang Lectures at the invitation of the Taiwan Association for Logic, Methodology and Philosophy of Science, at National Taiwan University, in Taipei. In preparation for those lectures, I’m trying out the ideas in three sessions in the Metpahysics and Logic Group at Arché. The three talks are: Inferentialism for Everyone (handout) I aim to give an opinionated introduction to inferentialist semantics, giving an account about what is so distinctive about logic, insofar as logical notions have a grip on whatever can be said or thought. In doing this, I aim to clarify the connections between logic and semantics, the theory of meaning. Existence & Modality (handout) In the second lecture, I will show how the inferentialist semantics, introduced in Lecture 1, applies to issues of predication, quantification and modality, and thereby provides some distinctive insight into ‘possible worlds’ models for quantified modal logics. Foundations for Truth-Conditional Semantics (handout) Finally, I show how we might relate inferentialist semantics to truth-conditional accounts of meaning for natural languages.

Back in Edgecliffe

Mon, 22 Sep 2025 12:15:00 +0100

I’ve returned from a productive semester of research leave, and I’m enjoying being back in the classroom teaching Intermediate Logic. Now that my big book has a complete first draft, my writing energies are focused elsewhere. I’m gearing up to give the 2025 Wendy Huang Lectures in Taiwan next month, so from this coming Wednesday, I’ll be giving the material a test run with the home crowd. The handout for the first Wendy Huang Lecture

PY2010: Intermediate Logic

Mon, 15 Sep 2025 00:00:00 +0000

py2010: Intermediate Logic is a University of St Andrews undergraduate subject in which we cover important results in logic to philosophy students. It’s taught by Greg Restall, together with a committed crew of postgraduate students. The subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. I’m using the new textbook Logical Methods, written with my colleague Shawn Standefer.

Reflections on Brady's Logic of Meaning Containment

Wed, 10 Sep 2025 00:00:00 +0000

This paper is a series of reflections on Ross Brady’s favourite substructural logic, the logic MC of meaning containment. In the first section, I describe some of the distinctive features of MC, including depth relevance, and its principled rejection of some con- cepts that have been found useful in many substructural logics, namely intensional or multiplicative conjunction (sometimes known as ‘fusion’), the Church constants (⊤ and ⊥), and the Ackermann constants (t and f). A further distinctive feature of the axiomatic formulation of MC is its meta-rule, which is a unique feature of MC Hilbert proofs. This meta-rule gives rise to one further special property of MC, in that the logic is distributive in one sense, and non-distributive in another. The distribution of additive conjunction over disjunction (the step from p∧(q∨r) to (p∧q)∨(p∧r)) holds in MC as a rule, but not as a provable conditional, and in this way, MC is distinctive among popular substructural logics. (Anderson and Belnap’s favourite logics R and E are distributive in both senses, while Girard’s linear logic is distributive in neither.) In this paper, I aim to increase our understanding of each of these distinctive features of MC, giving an account of what it might take for a propositional logic to meet these constraints.

Proof, Rules and Meaning

Fri, 29 Aug 2025 00:00:00 +0000

This is my next book-length writing project. I am writing a book which aims to do these things: Be a useable introduction to philosophical logic, accessible to someone who's done only an introductory course in logic, covering at least some models and proofs for propositional logic, and maybe a little bit of predicate logic. Be a user-friendly, pedagogically useful and philosophically motivated presentation of cut-elimination, normalisation and conservative extension, both (a) why they're important to semantics and (b) how to actually prove them. (I don't think there are any books like this currently available, but I'd be happy to be shown wrong.) Develop an extended argument, showing how proof rules (of a certain shape) can be understood as definitions. And finally, relate these results to issues concerning meaning, epistemology and metaphysics, including issues of logical consequence and rationality, the problem of absolute generality, and the status of modality. Here is an outline of the manuscript, showing how the parts hold together. The book is in three parts. The Tools: in which core concepts from proof theory are introduced. Three chapters, one on natural deduction proofs (and normalisation), one on the sequent calculus (and cut elimination) and one on models and more, relating proofs and models, including soundness and completeness results, understood from a proof-first perspective.

Must Do “Must Do Better” Better

Wed, 18 Jun 2025 17:00:00 -0700

Type Theory and Themes in Philosophical Logic

Tue, 10 Jun 2025 13:30:00 +0100

Abstract: Contemporary type theorists share interests and concerns with philosophical logicians. This isn’t surprising, since Per Martin-Löf is (among other things) a philosopher, and type theory was born in philosophy. Although type theory has come into its own in computer science—and, more recently, in mathematics with the rise of assistants—the connections between type theory and philosophical logic go beyond Martin-Löf’s original motivations. There are many fresh points of contact with active research areas in philosophical logic. In the interest of fostering communication between these different traditions, I will sketch some of these connections, including recent work on modal and substructural logics; the interactions between intensionality, equality and identity; the relationship between constructive and classical logic; theories of propositions and speech acts; and the relationship between purely formal and applied logic and type theory. The is a presentation at the TYPES 2025, at the University of Strathclyde, Glasgow. The abstract of the presentation is available here, and the slides are here. Here is a recording of the talk.

Must Do “Must Do Better” Better

Thu, 05 Jun 2025 09:00:00 +0100

What can We Mean? On Practices, Norms and Pluralisms

Sat, 19 Apr 2025 00:00:00 +0000

Last century, Michael Dummett argued that the principles of intuitionistic logic are semantically neutral, and that classical logic involves a distinctive commitment to realism. The ensuing debate over realism and anti-realism and intuitionistic logic has now receded from view. The situation is reversed in mathematics: constructive reasoning has become more popular in the 21st century with the rise of proof assistants based on constructive type theory. In this paper, I revisit Dummett’s concerns in the light of these developments, arguing that both constructive and classical reasoning are recognisable and coherent assertoric and inferential practices.

Modal Logic and Contingent Existence

Thu, 10 Apr 2025 15:00:00 +0200

Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent work on modal semantics makes essential use of possible worlds models. These models have proved useful for analysing the structural properties of modal logics, but it is less clear that they fix the meaning of our modal vocabulary, given that we have no grasp of what counts as a possible world, independent of our grasp of what counts as possible. In this talk, I describe an inferentialist semantics for modal and quantificational vocabulary, not as a rival to possible worlds models, but as an explanation of how the concepts we do employ can be modelled using possible worlds. I then use this inferentialist semantics to clarify the contingentist’s commitments, and offer answers to necessitist objections. This talk is a presentation for the DIP Colloquium and the LIRa Seminar at the ILLC at the University of Amsterdam. The slides for this talk are available here.

Modal Logic and Contingent Existence

Tue, 08 Apr 2025 15:30:00 +0200

Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent work on modal semantics makes essential use of possible worlds models. These models have proved useful for analysing the structural properties of modal logics, but it is less clear that they fix the meaning of our modal vocabulary, given that we have no grasp of what counts as a possible world, independent of our grasp of what counts as possible. In this talk, I describe an inferentialist semantics for modal and quantificational vocabulary, not as a rival to possible worlds models, but as an explanation of how the concepts we do employ can be modelled using possible worlds. I then use this inferentialist semantics to clarify the contingentist’s commitments, and offer answers to necessitist objections. This talk is a presentation for the Knowledge in Crisis research cluster in the Philosophy Department at the University of Vienna. The slides for this talk are available here.

Modal Logic and Contingent Existence

Thu, 03 Apr 2025 14:00:00 +0200

Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent work on modal semantics makes essential use of possible worlds models. These models have proved useful for analysing the structural properties of modal logics, but it is less clear that they fix the meaning of our modal vocabulary, given that we have no grasp of what counts as a possible world, independent of our grasp of what counts as possible. In this talk, I describe an inferentialist semantics for modal and quantificational vocabulary, not as a rival to possible worlds models, but as an explanation of how the concepts we do employ can be modelled using possible worlds. I then use this inferentialist semantics to clarify the contingentist’s commitments, and offer answers to necessitist objections. This talk is a presentation at the Department of Logic in the Institute of Philosophy of the Czech Academy of Sciences, in Prague. The slides for this talk are available here.

Modal Logic and Contingent Existence

Mon, 17 Mar 2025 16:00:00 -0500

Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent work on modal semantics makes essential use of possible worlds models. These models have proved useful for analysing the structural properties of modal logics, but it is less clear that they fix the meaning of our modal vocabulary, given that we have no grasp of what counts as a possible world, independent of our grasp of what counts as possible. In this talk, I describe an inferentialist semantics for modal and quantificational vocabulary, not as a rival to possible worlds models, but as an explanation of how the concepts we do employ can be modelled using possible worlds. I then use this inferentialist semantics to clarify the contingentist’s commitments, and offer answers to necessitist objections. This talk is a presentation at the CUNY Logic and Metaphysics Seminar. The slides for this talk are available here. There is a draft paper, which goes into the results I discuss in this talk in more length.

Modal Logic and Contingent Existence

Fri, 14 Mar 2025 15:00:00 -0600

Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent work on modal semantics makes essential use of possible worlds models. These models have proved useful for analysing the structural properties of modal logics, but it is less clear that they fix the meaning of our modal vocabulary, given that we have no grasp of what counts as a possible world, independent of our grasp of what counts as possible. In this talk, I describe an inferentialist semantics for modal and quantificational vocabulary, not as a rival to possible worlds models, but as an explanation of how the concepts we do employ can be modelled using possible worlds. I then use this inferentialist semantics to clarify the contingentist’s commitments, and offer answers to necessitist objections. This talk is a Presentation at the University of Calgary Philosophy Department. The slides for this talk are available here. There is a draft paper, which goes into the results I discuss in this talk in more length.

Mathematical Practice, Proof Assistants and Meaning

Thu, 13 Mar 2025 15:30:00 -0600

Abstract: Twenty-first century mathematics has seen the rise of the proof assistant, and mathematical practice has changed significantly with the rise of these new tools. One consequence of this change, for the philosopher of mathematics, is the wider adoption in mathematics of intuitionistic logic, since many mathematicians now rush to formalise results in the language of constructive type theory (the framework underlying popular proof assistants such as Agda and Lean). In this talk, I will describe this relatively modern phenomenon, and relate it to considerations central to philosophical discussions in the twentieth century, about how to provide a neutral framework for addressing questions in metaphysics. The contemporary use of proof assistants can provide a useful new angle on these longstanding questions, not only about how we can come to know the truths of mathematics, but also for the very content of whatever we can think and say. This talk is the 2024-2025 Calgary Mathematics & Philosophy Lecture. The slides for this talk are available here. Some of the ideas in this talk are discussed in more length in my paper ``What Can We Mean? On Practices, Norms and Pluralisms.''

Modal Logic and Contingent Existence

Fri, 07 Mar 2025 16:00:00 -0800

Abstract: In this talk, I will defend contingentism, the idea that some things exist contingently. It might be surprising that this needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the view that everything that exists, exists necessarily. Almost all recent work on modal semantics makes essential use of possible worlds models. These models have proved useful for analysing the structural properties of modal logics, but it is less clear that they fix the meaning of our modal vocabulary, given that we have no grasp of what counts as a possible world, independent of our grasp of what counts as possible. In this talk, I describe an inferentialist semantics for modal and quantificational vocabulary, not as a rival to possible worlds models, but as an explanation of how the concepts we do employ can be modelled using possible worlds. I then use this inferentialist semantics to clarify the contingentist’s commitments, and offer answers to necessitist objections. This talk is a presentation at the Orange County Inland Empire Seminar series in History and Philosophy of Mathematics at Chapman University. The slides for this talk are available here.

Upcoming trip to North America

Fri, 28 Feb 2025 22:00:00 +1000

In a few days, I’m heading off to North America, for a brief trip to give a few talks, at Chapman University in California, Calgary University in Alberta, and finally, a brief stop at CUNY in New York. I’m looking forward to meeting new friends and catching up with old ones, as well as the chance to talk with smart people about my research.

Meanwhile, I’ve managed to do a bit of reading during February, as I’ve been preparing for this trip.

Proofs with Star and Perp

Tue, 25 Feb 2025 00:00:00 +0000

In this paper, I show how to incorporate insights from the model-theoretic semantics for negation (insights due the late J. Michael Dunn, among others, in his paper “Star and Perp: Two Treatments of Negation”), into a proof-first understanding of the semantics of negation. I then discuss the how a logical pluralist may understand the underlying accounts of proofs and their significance.

Defining Rules for Quantifiers and Identity

Thu, 20 Feb 2025 02:00:00 +0100

Abstract: Suppose we have a language involving non-denoting singular terms. (The language of everyday mathematics provides one example. Terms like $\frac{n}{m}$ and $\lim_{x\to\infty} f(x)$ do not denote, for appropriate choices of $m$ and of $f$.) It is not too difficult to define inference rules for an appropriately free logic that incorporates non-denoting terms. If $t$ does not denote, then $\phi(t)$ need not entail $\exists x\phi(x)$, and neither is $\phi(t)$ entailed by $\forall x\phi(x)$. We now have a very good understanding of well-behaved cut-free sequent calculi for a variety of different free logics, appropriate for languages with non-denoting singular terms. There is a tradition, in proof-theoretic semantics, of thinking of well-behaved inference rules for some concept as defining that concept. If the rules are well-enough behaved (whatever that notion of good behaviour might be—whether a notion of harmony, or conservative extension and unique definability, or something else—then we are tempted to take the concept that can be introduced by such rules to be defined by them, and hence to be apt for introduction to our vocabulary. In this talk, I will show how, in the natural sequent calculus for negative free logic, we are able to define “outer” quantifiers $\Pi$ and $\Sigma$ where $\phi(t)$ entails $\Sigma x\phi(x)$, and is entailed by $\Pi x\phi(x)$, for any term $t$, whether denoting or not, since the rules for such quantifiers meet the strictures for defining concepts in proof-theoretic semantics mentioned above.

Books Read: January 2025

Fri, 31 Jan 2025 15:52:00 +1000

This month’s reading was mostly nonfiction, featuring Kevin Hart’s Contemplation (The Movements of the Soul), Haruki Murakami’s essay collection Novelist as a Vocation, the harrowing Chasing Shadows: The Life and Death of Peter Roebuck by Tim Lane and Elliot Cartledge, Drew Neil’s Practical Vim (I’m trying to brush up my text editing skills), and finally, Timothy Larsen’s George MacDonald in the Age of Miracles. I rounded off the month enjoying my next two instalments of Dan Moren’s Galactic Cold War series, the short story Showdown and the rollicking novel The Aleph Extraction. That was a fun read on the flight to Australia.

Modal Logic and Contingent Existence (Generality and Existence 2)

Mon, 20 Jan 2025 00:00:00 +0000

In this paper, I defend contingentism, the natural idea that some things exist contingently. Had my parents not met, I would not have existed. It is perhaps surprising that an everyday idea like contingentism needs defence, but natural reasoning principles concerning possibility and necessity on the one hand, and the existential and universal quantifiers on the other, have led some to necessitism, the alternate view, that everything that exists, exists necessarily. Almost all recent work on the semantics of possibility, necessity and the quantifiers—and its metaphysics—makes essential use of possible worlds models. These models have proved useful for analysing the formal and structural properties of modal logics, but it is less clear that these models help fix the meaning of our modal vocabulary, given that we have no grasp of what counts as a possible world independent of our grasp of what counts as possible. In this paper, I develop an alternate inferentialist semantics for the modal and quantificational vocabulary, not as a rival to possible worlds models, but as an explanation of how it is that the concepts we do employ can be modelled using possible worlds. I then use this inferentialist semantics to clarify the contingentist’s commitments, and offer answers to necessitist objections.

Books Read: December 2024

Wed, 01 Jan 2025 20:00:00 +0100

Happy New Year, everyone!

As 2024 draws to a close, I’ve finished another month of reading, so let’s close out my log of books read over 2024 with a short description of December’s reading.

First up, I enjoyed reading the second entry in Dan Moren’s Galactic Cold War series: The Bayern Agenda. As with the previous entry, this was a fun spy thriller, with a cold war science fictional setting. Dan Moren has a deft hand as an author, weaving a plot which balances ratcheting tension and building suspense, while treating the reader—and his characters—with respect. I will enjoy reading the remaining entries in the series in the coming months.

My second novel was very different: I re-read George MacDonald’s Lilith, a fantasy, first published in 1895. MacDonald was a Scottish author (and Congregational minister) who mentored Lewis Carroll (Charles Dodgson), and whose work inspired and influenced other religiously inflected writers of fantasy, such as J. R. R. Tolkien, Madeleine L’Engle and C. S. Lewis. Lilith is an odd book: it’s the story of Mr Vane, a man whose life centres on his library (and, we later discover, his love of horses), and who has few significant relationships with other people. His library seems to be haunted by its former librarian, and soon, Mr Vane is transported to another world, many adventures are had, mistakes are made and lessons are learned. (I’m being very general and schematic here. I’d rather leave it for you to read to learn the details for yourself.) What most interested me most in this re-reading was MacDonald’s view of conversion and personal transformation. MacDonald was a universalist (one who believes that God will save everyone in the end), and in Lilith, MacDonald’s universalism is on display, telling a tale in which the resolution is not one where the antagonists are destroyed, but rather, move towards reconciliation.

Another semester done

Fri, 20 Dec 2024 16:35:00 +0000

I’ve completed the moderation of the exam for Intermediate Logic, and with that, the final administrative responsibilities for this semester are complete. Now it’s time to take a short break over Christmas and the New Year, and then to start a semester of research leave. I’m looking forward to time set aside to think, to write, and to talk to colleagues, near and far. I have some trips lined up, to North America, and to continental Europe, in the months ahead. I’ll post notice of these here, as the details are ironed down. In the mean time, I’m looking forward to having that time to think and to write. An in-place away notice

What Can We Mean?

Fri, 13 Dec 2024 15:15:00 +0100

Abstract: What does the semantically anti-realist revisionary programme of Michael Dummett have to do with contemporary work on proof assistants? What are mathematicians doing when they encode their proofs in these proof assistants, based on constructive type theory? What does all this have to do with the(?) norms of assertion? (And are these norms of assertion relative or absolute, anyway?) Can I explain all of this, coherently, to a general audience of philosophers in five minutes? The answer to the last of these questions is almost certainly no, but I’m going to give it a red hot go. This talk is a five-minute lightning presentation at the Winter All-Arché Research Day. A (one page) handout is here.

Congratulations, John!

Fri, 29 Nov 2024 10:25:00 +0000

My last PhD student at the University of Melbourne has completed his project, and is now Dr John Cleary. Congratulations, John!

It was so much fun to help supervise your project. I’ve learned a lot about Albert Lautman, and his account of the development of mathematics and the dialectic of ideas, problems and mathematical progress.

Graham Priest is in the house

Mon, 25 Nov 2024 15:45:32 +0100

This week, Aaron Cotnoir’s Instruments of Unity project and I are hosting a short visit from our friend (and my PhD supervisor), Professor Graham Priest. It’s always enjoyable to spend time with him, and tomorrow, we’re going to teach a the second-last lecture class for my Intermediate Logic cohort together, on the liar paradox and non-classical logic.

Today, he gave a talk on nothing and its paradoxical properties.

Graham Priest and the Inclosure Schema

Podcast recommendation: Marooned on Mars

Mon, 25 Nov 2024 10:25:00 +0000

I mentioned yesterday that this month I’ve enjoyed rereading Kim Stanley Robinson’s Mars Trilogy. This time around, after completing my re-read, I’ve enjoyed listening to Marooned on Mars, a podcast devoted to Kim Stanley Robinson’s fiction. The initial conceit of the podcast was that the hosts, Matt Hauske and Hilary Strang (two humanities academics, based in Chicago) would take a section from the Mars Trilogy, one episode at a time, and discuss it, drawing out themes, pointing out connections, and generally, enjoying talking about the work.

Books Read: September, October, November 2024

Sun, 24 Nov 2024 17:55:00 +0000

This teaching semester has been keeping me so busy that I have not kept up with my monthly reading logs. I’ve had enough time to read, but I haven’t found the time to keep you, my reader, up with what I’ve been reading. I’ll attempt to remedy this now, by giving a very brisk run-down of my reading over the last three months.

This last three months has been dominated by fiction reading, so let me start with the little pile of non-fiction that I enjoyed. First, for the philosophy, I enjoyed Todd May’s Friendship in an age of Economics, a nice little work in moral/social psychology on the value of friendship and its usefulness as giving us insight into value that does not register in our econometric age. For theology, I enjoyed one longer book, Women and the Gender of God, by Amy Peeler, and two very slim books, Why Did Jesus Have to Die?, by Jane Williams and Passions of the Soul, by Jane’s husband, Rowan Williams. The little Buddhist text on mindul living, The Practice of Not Thinking, by Ryunosuke Koike was a fun read, too. The final non-fiction book from the last three months was very different to all of the others: Daniel P. Friedman and David Thrane Christiansen’s fun little dialogue The Little Typer was a sweet little introduction to dependent type theory, which I’ve been thinking about lately, and I plan to think about this some more in the coming year.

Defining Quantifiers

Sat, 16 Nov 2024 17:00:00 +0100

Abstract: Suppose we have a language involving non-denoting singular terms. (The language of everyday mathematics provides one example. Terms like $\frac{n}{m}$ and $\lim_{x\to\infty} f(x)$ do not denote, for appropriate choices of $m$ and of $f$.) It is not too difficult to define inference rules for an appropriately free logic that incorporates non-denoting terms. If $t$ does not denote, then $\phi(t)$ need not entail $\exists x\phi(x)$, and neither is $\phi(t)$ entailed by $\forall x\phi(x)$. We now have a very good understanding of well-behaved cut-free sequent calculi for a variety of different free logics, appropriate for languages with non-denoting singular terms. There is a tradition, in proof-theoretic semantics, of thinking of well-behaved inference rules for some concept as defining that concept. If the rules are well-enough behaved (whatever that notion of good behaviour might be—whether a notion of harmony, or conservative extension and unique definability, or something else—then we are tempted to take the concept that can be introduced by such rules to be defined by them, and hence to be apt for introduction to our vocabulary. In this talk, I will show how, in a simple negative free logic, we are nonetheless able to—in some sense—define “outer” quantifiers $\Pi$ and $\Sigma$ where $\phi(t)$ entails $\Sigma x\phi(x)$, and is entailed by $\Pi x\phi(x)$, since the rules for such quantifiers meet all the strictures for defining concepts in proof-theoretic semantics mentioned above.

What Can We Mean? — on practices, norms and pluralism

Mon, 28 Oct 2024 18:00:00 +0100

Abstract: Michael Dummett, in a presentation to the Aristotelian Society 65 years ago (in 1959) inaugurated a long-running debate over semantic realism and anti-realism, and the issue of settling an appropriate logic as a necessary prolegomenon to fruitful discussion metaphysics. Dummett argued that the principles of intuitionistic logic are semantically neutral, but that essentially classical principles carry distinctive metaphysical commitment. This debate reached its peak in the 1980s and 1990s, but has receded from view in recent years. The situation is curiously reversed in mathematics. Constructive mathematics (in which the underlying logic is intuitionistic) was very much a minority mathematical tradition in the second half of the 20th Century, but has undergone a significant renaissance in recent years, with the advent of proof assistants, such as Agda and Lean. Now, many mathematicians are busy representing informal mathematical proofs in constructive dependent type theory, in which distinctively classical logical principles, if used, are optional extra commitments, and not part of the underlying logic. In this talk, I will attempt to make sense of this state of affairs, revisiting Dummett’s original concerns in the light of the advent of proof assistants. Along the way, we will encounter issues around logical pluralism, the open texture of concepts, the norms of inference and assertion, and the role of computational devices in our expressive and justificatory practices.

Generics: Inference & Accommodation

Thu, 10 Oct 2024 00:00:00 +0000

Generic claims, such as Birds fly, Men are violent, and Mosquitos carry Ross River Fever, seem pervasive across human thought and talk. We use generic claims to express our understanding of the world around us and our place in it. These generic claims are useful even though they admit exceptions. We can agree that birds fly, even though emus don’t. Mosquitos carry Ross River Fever, but not those in Africa. And you can agree that men are violent while conceding that not all men are, or even that most are. Generics remain important in our thought and talk in the presence of these counter-instances. Generic claims express rules of thumb, ways to see the world around us, and they provide heuristics for navigating that world. Generics also play a significant role in our maintaining the boundaries of social kinds, and in our attempts to shift those boundaries. Arguments about contested generic claims can produce much more heat than light. When the topic of men’s violence against women is raised, it is a common refrain to hear the defensive retort “not all men,” as if that were an objection to the claim of male violence. It seems clear that generics play a significant role, particularly, in the ideologies of our social worlds, of characterising different social kinds and expressing our default orientations toward them, and towards ourselves as members of those kinds.

What do calculators tell us about meaning?

Mon, 07 Oct 2024 19:30:00 +0100

Abstract: When I use a calculator to tell me that 245 × 46 = 11,270, I learn something that I didn’t know before, even though calculators don’t have any beliefs or knowledge. Even small children know how to count things, and it is through our own capacity to enumerate and count things that we learn basic arithmetic. Calculators do not count things in any sense like we do, yet we can use them to learn arithmetic facts. Calculators are one a simple example of the growing phenomenon of meaning at the boundary between humans and machines. In this talk I’ll draw out some lessons from recent work using computers to augment human reasoning to help us clarify what we are doing when we’re making claims about the world and trying to reason about them. This talk was a presentation at the University of St Andrews PhilSoc (the undergraduate Philosophy Student Society).

PY2010: Intermediate Logic

Mon, 16 Sep 2024 00:00:00 +0000

Books Read: August 2024

Wed, 11 Sep 2024 21:25:00 +0100

Here’s August’s book haul: This month I enjoyed three novels. The most experimental of which was Olga Ravn’s The Employees: A workplace novel of the 22nd Century, which has the form of a series of witness statements from the crew of a ship, now far away from earth. The workers, both human and artificial, have been tending a number of exotic objects from the planet New Discovery, and they find their lives changed in subtle and not-so subtle ways. The novel touches on workplace oppression, freedom, and longing for what is absent.

The other two novels were Lords of Uncreation, the final entry in a series by Adrian Tchaikovsky, and The Mercy of Gods, the first entry in a new saga by James S. A. Corey Both Tchaikovsky and Corey are adept at weaving together a compelling tale at interplanetary scale. I particularly enjoyed The Mercy of Gods, which starts off with the petty academic politics of research teams competing for funding, before all hell literally breaks loose with an invasion from an implacable colonising force. We get to see our research collective deal with the PTSD resulting from witnessing the destruction of their entire way of life and being swept away to another world to do the coloniser’s bidding.

Models for Identity in Three-Valued Logics

Thu, 05 Sep 2024 14:50:00 +0100

Abstract: There is a natural way to interpret the propositional connectives and quantifiers in terms of the three semantic values 0, i, and 1, where 0 and 1 are understood as falsity and truth, and i is understood as some intermediate value. These three-valued valuations do not, by themselves, determine a logic, because for that, you need to settle how models are used to provide a counterexample to a sequent. If you take a counterexample to the argument from A to B to be a model that assigns A the value 1 and B some value other than 1 (either 0 or i), the resulting logic is Kleene’s strong three-valued logic, K3. If a counterexample is a model assigning A the value 1 or i and B the value 0, the resulting logic is Priest’s logic of paradox, LP. If a counterexample is a model assigning A the value 1 and B the value 0, then the result is the logic ST of Strict–Tolerant validity. The three logics are different generalisations of two-valued Boolean logic to a tri-valuational setting. The logic ST is distinctive, in that it is, in some sense, a reformulation of classical logic—every classically valid sequent in this language is ST-valid—but since ST allows for strictly non-classical models, there are ST theories which are not classical theories.

Books Read: July 2024

Thu, 22 Aug 2024 21:40:00 +0100

July and August have been really busy, not with teaching (it’s the summer teaching break), but with research and research supervision, a little bit of holiday travel, and various life things taking up my time and attention. I have had time to read, but not so much time to write paragraphs about each book. Instead of skipping the books-of-the-month post entirely, here’s a stripped back version with a link and a sentence for each book.

July was Kierkegaard month. I enjoyed reading through: Fear and Trembling, which I’d known of for many years, but never read through. He’s a striking prose stylist, as well as a thoughtful reader of the biblical Abraham and Isaac story. Alongside, I read Jeffrey Hanson’s Kierkegaard and the Life of Faith, which is not only a commentary and analysis of Fear and Trembling, but an extensive treatment of Kierkegaard’s teleological suspension of the ethical, and a thoughtful and sympathetic treatment of Kierkegaard’s account of the life of faith.

Questions, Justification Requests, Inference, and Definition

Fri, 16 Aug 2024 00:00:00 +0000

In this paper, I examine connections between the speech acts of assertion, denial, polar questions and justification requests, and the common ground. When we pay attention to the structure of norms governing polar questions, we can clarify the distinction between strong and weak denial, together with the parallel distinction between strong and weak assertion, and the distinct way that these speech acts interact with the common ground. In addition, once we pay attention to the distinct norms concerning justification requests, we can give a distinctive answer to Carroll’s puzzle concerning the force of the logical must, and the sense in which certain rules for logical concepts can indeed count as definitions. This paper appears in a Synthese edited collection on non-assertoric speech acts.

Substructural Logics

Thu, 15 Aug 2024 00:00:00 +0000

Substructural logics are non-classical logics weaker than classical logic, notable for the absence of structural rules present in classical logic. These logics are motivated by considerations from philosophy (relevant logics), linguistics (the Lambek calculus) and computing (linear logic). In addition, techniques from substructural logics are useful in the study of traditional logics such as classical and intuitionistic logic. This article provides an overview of the field of substructural logic. (Previous versions: July 2000; February 2018.)

Finitude, Eternity, Love, the Good, and Martin Hägglund’s This Life

Sat, 27 Jul 2024 00:00:00 +0000

Martin Hägglund’s This Life is an important treatise on metaphysics, philosophical anthropology, and political philosophy. It is also a critique of religious orientations to the world. In this paper, I reflect on Hägglund’s account of value and our finitude, and on his criticism of Martin Luther King, Jr’s political theology. According to Hägglund, King’s appeal to God when calling for justice should be replaced by an appeal to our communal norms. To defer to God is at best a colourful way of depicting our own commitments. At worst, it has no determinate content, and it threatens to absolve us of making justice here and now. I will show that while Hägglund’s account is a salutary corrective to a pervasive kind of bad faith, this criticism misses its target. Any identification of God’s justice with our communal norms is to mischaracterise concepts like these. They are ideals that direct our attention outside ourselves, in the same way that love orients us toward an other. Hägglund’s account points us to the crucial concept of dependence, and that once we understand this better, we will see that a religious orientation toward the infinite and secular faith are compatible.

Books Read: June 2024

Wed, 10 Jul 2024 22:58:00 +0100

June was also an enjoyable month for reading. This month’s reading was dominated by Diarmaid MacCulloch’s 864 page doorstopper Reformation: Europe’s House Divided. Having been educated in Australia, the view of history I was taught was oriented around the colonisation of the continent by the British and its aftermath. As far as the religious and social history of Europe was concerned, and the division of the Western Church that predated that colonisation, I knew some of the details, but I had no idea of how the centuries-long convulsion in church and state that was the reformation (and the counter-reformation) spread across Europe, over the 16th and 17th Centuries. MacCulloch’s painstakingly researched and very readable history helped me understand how the historical contingencies and the different twists and turns of these times have led to the distinctively modern world that took shape in the reformation’s wake.

What Do We Mean? Semantics, Practices and Pluralism

Wed, 03 Jul 2024 15:00:00 +0100

Abstract: In this informal talk, I will revisit some longstanding issues in philosophical logic in the light of some contemporary developments. The longstanding issues? (1) Michael Dummett’s challenge in The Logical Basis of Metaphysics to the effect that to get anywhere in fundamental issues of metaphysics we would do well to attend to the fundamental commitments of our theory of meaning—and that those concerns lead to the conclusion that we can find common ground in intuitionistic logic, not classical logic. (2) The issue of pluralism (or monism) about logical consequence. Contemporary work in logic is filled with a range of different (and seemingly opposed) accounts of what follows from what. Many different kinds of logical pluralism have arisen to attempt to make sense of the diversity of logical analyses, and just as many defences of logical monism have been offered. The contemporary developments? The rise of dependent type theory in computer science and the consequent rise of proof assistants in the formalisation of mathematics. Different proof assistants (Agda, Idris, Lean, Isabelle, Rocq/Coq) make different choices in the formal representation of mathematical reasoning, but the predominant choice of these proof assistants is to represent proofs constructively, in what amounts to intuitionistic logic and not classical logic.

What ‘No’ Does

Fri, 28 Jun 2024 11:30:00 +0100

Abstract: In this 5 minute lightning talk, I will introduce the difference between strong and weak denial, and use it to clarify the claims I made in “Multiple Conclusions” about the connection between assertion, denial and logical consequence. The talk is a presentation at the 2024 Arché Day. The handout for the talk is available here.

Books Read: May 2024

Tue, 25 Jun 2024 21:25:32 +0100

May was a great reading month, but I forgot to upload my notes until June was nearly done. This month I read nine books, but one was an unpublished draft of a novel that a friend is writing, so I won’t say more about that, until there is more news to share. The other fiction reads this month could not be more different from each other. Beyond the Light Horizon concludes Ken MacLeod’s most recent politically charged Scotland-based near-future science fiction trilogy. The ending was a bit rushed for my taste, but it was an enjoyable ride along the way.

Alexis Wright’s Praiseworthy could not be more different. A gargantuan (736 page) novel, it reads like a stream of consciousness. It is set in the north of Australia, with a cast of indigenous characters making their way in a surreal world falling apart around them. I was enchanted by the rhythm and the texture and the distinctive voice of the words, sentences and paragraphs, while I struggled to hold the whole thing in view. It’s not an easy read, but it is up there with the most memorable novels I’ve read in the last few years.

Proof Theory

Mon, 10 Jun 2024 00:00:00 +0000

This is an intensive class on logical and philosophical issues in proof theory, taught at nls2024 as part of the Reykjavik Summer of Cool Logic 2024. In this course, I’ll introduce natural deduction and sequent calculus for classical, constructive and substructural logics, motivating and explaining how key results (normalisation for natural deduction proofs and cut elimination for sequent calculus derivations) may be proved, and how these interact with the presence or absence of the structural rules of weakening and contraction. We will also take a look at different proof systems designed to model modal and other intensional logics. Along the way we will see (1) the difference between multiplicative and additive rules for connectives (2) different ways to understand harmony between introduction and elimination rules (or left and right rules in the sequent calculus) (3) the ways in which rules for a connective may be understood as defining the concept introduced; (4) the connection between proof dynamics, dialogue and speech acts; (5) the relationship between proof search and model construction, and (6) connections between structural rules, paradoxes and fixed points. Class Resources If you are attending the class, you can download the (password-protected, for the moment) class notes here. I’ll upload the slides after each class.

A Chinese Translation!

Tue, 21 May 2024 16:05:32 +0100

Earlier this month I got a lovely package in the mail. A Chinese translation of my intro logic text. My colleague and friend Min Xu completed his translation of my text, and it’s now available. The beginning of Chapter 5.

Books Read: April 2024

Wed, 01 May 2024 21:25:32 +0100

Another month, and another pile of books I’ve managed to read. This month’s reading started on the Isle of Iona, where we had a short Easter break. On the last day on the island, and then on the ferry journeys back to the “mainland”, I finished The Life of St Columba, who served as the Abbot of Iona and, as legend has it, brought Christianity to Scotland, from Ireland, dying in 597.

Natural Deduction Proof for Substructural, Constructive and Classical Logics

Wed, 01 May 2024 16:00:00 +0100

Abstract: Since the 1990s, we have seen how to understand a very wide range of logical systems (classical logic, intuitionistic logic, dual intuitionistic logic, relevant logics, linear logic, the Lambek calculus, affine logic, orthologic and more) by way of the distinction between operational and structural rules. We can have one set of rules for a connective (say, conjunction, negation, or the conditional) in a sequent calculus, and get different logical behaviour depending on the shape of the sequents allowed and the structural rules governing those sequents. In this talk, I will consider the relationship between the “big four” traditional substructural logics—intuitionistic, relevant, affine and linear—corresponding to the four options for including or excluding the structural rules of weakening and contraction, in the setting of Gentzen/Prawitz-style natural deduction proofs for implication and the simply typed λ calculus. Such a natural deduction setting—in which proofs have any number of premises and a single conclusion—has a natural bias toward constructive, or intuitionistic logic. I will show how the choice of whether to “go classical”, expanding the structural context to allow for more than one formula in positive position is orthogonal to the choice of the other structural rules, so that even in the context of natural deduction proofs, the familiar pair of traditional implication introduction and elimination rules gives rise to eight different propositional logics, four of which are “classical” and four of which are “constructive”.

Logic (Chinese Translation)

Wed, 01 May 2024 00:00:00 +0000

This is a translation into Chinese of my introductory textbook Logic. It is available from Huazhong University of Science & Technology Press.

λμ: Relating Constructive, Classical and Substructural Logics

Tue, 16 Apr 2024 17:00:00 +0100

Abstract: Since the 1990s, we have seen how to understand a very wide range of logical systems (classical logic, intuitionistic logic, dual intuitionistic logic, relevant logics, linear logic, the Lambek calculus, affine logic, orthologic, ST, TS and more) as consisting of a single broad family of connectives which are set in different structural contexts given a particular choice of the shape of sequents, and the structural rules governing those sequents. The sequent calculus has proved to be a useful lens through which to view a broad range of logical systems, both in terms of their formal properties, and their interpretation. In this talk, I will consider the relationship between the “big four” traditional substructural logics corresponding to the four options for including or excluding the structural rules of weakening and contraction in the setting of Gentzen/Prawitz-style natural deduction proofs for implication, and the simply typed A calculus. Such a natural deduction setting in which proofs have any number of premises and a single conclusion has a natural bias toward constructive, or intuitionistic logic. I will show how the choice of whether to “go classical”, expanding the structural context to allow for more than one formula in positive position is orthogonal to the choice of the other structural rules, and that each of the four constructive substructural logics have their classical counterpart inside them-not only at the level of provability, but also at the level of proofs-by way of a natural generalisation of the double negation translation of classical logic inside intuitionistic logic.

Proofs, Rules, and Meanings, Arché Workshop

Sat, 13 Apr 2024 13:46:32 +1000

I’ve just emerged from two intense days of proof theory. Three of my graduate students, Sophie Nagler, Viviane Fairbank and Francisca Silva, organised a two-day workshop on proof theory and its connections to philosophy and other fields.

Some of the workshop participants at the end of a busy day of proof theory.

λμ: Relating Constructive and Classical Logics

Thu, 11 Apr 2024 09:30:00 +0100

Abstract: Since the 1990s, we have seen how to understand a very wide range of logical systems (classical logic, intuitionistic logic, dual intuitionistic logic, relevant logics, linear logic, the Lambek calculus, affine logic, orthologic, ST, TS and more) as consisting of a single broad family of connectives which are set in different structural contexts given a particular choice of the shape of sequents, and the structural rules governing those sequents. The sequent calculus has proved to be a useful lens through which to view a broad range of logical systems, both in terms of their formal properties, and their interpretation. In this talk, I will consider the relationship between the “big four” traditional substructural logics corresponding to the four options for including or excluding the structural rules of weakening and contraction in the setting of Gentzen/Prawitz-style natural deduction proofs for implication, and the simply typed A calculus. Such a natural deduction setting in which proofs have any number of premises and a single conclusion has a natural bias toward constructive, or intuitionistic logic. I will show how the choice of whether to “go classical”, expanding the structural context to allow for more than one formula in positive position is orthogonal to the choice of the other structural rules, and that each of the four constructive substructural logics have their classical counterpart inside them-not only at the level of provability, but also at the level of proofs-by way of a natural generalisation of the double negation translation of classical logic inside intuitionistic logic.

Books Read: March 2024

Sun, 07 Apr 2024 13:46:32 +1000

Lately, I’ve been keeping track of my book reading, and to help focus my reflection (and as an aid to my own memory), I have taken to writing a few lines for each book I complete. March was a particularly good book-reading month, and since I’ve not posted anything on this website for more than year, I thought I’d share last month’s reading notes.

First, over March I read a few books that are broadly philosophical:

PY4612: Advanced Logic

Mon, 15 Jan 2024 00:00:00 +0000

Collection Frames for Distributive Substructural Logics

Mon, 18 Dec 2023 00:00:00 +0000

We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalization of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for the basic positive substructural logic B+, that collection frames on multisets are sound and complete for RW+ (the relevant logic R+, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic R+. (The completeness of set frames for R+ is, currently, an open question.)

The Philosophical Significance of the Paradoxes

Wed, 13 Dec 2023 00:00:00 +0000

In this essay, I examine the significance of the semantic, set-theoretic and sorites paradoxes for a number of different philosophical issues concerning logic, including the choice of a logical system, the epistemology of logic, and the boundary–if there is one–between logical and non-logical concepts. Along the way, I consider the difference between revisionary logical proposals motivated by expanding the range of models (e.g. by adopting three-valued valuations), and those motivated by restricting the structure of proofs (e.g. by restricting the application of the structural rules of contraction or cut). I also argue that whether to conserve or to revise logical principles in the light of the paradoxes is orthogonal to the question of whether to be an exceptionalist or an anti-exceptionalist about logic. All four combinations these positions have been staked out in recent work: anti-exceptionalist conservative (Tim Williamson) and anti-exceptionalist revisionist (Graham Priest), exceptionalist conservative (Per Martin-Löf), exceptionalist revisionist (Uwe Petersen).

Finitude, Eternity, Love, the Good and Martin Hagglünd’s ‘This Life’

Fri, 03 Nov 2023 15:30:00 +0000

Abstract: Martin Hägglund’s This Life: Secular Faith and Spiritual Freedom (Knopf, 2019) is an important and insightful treatise on metaphysics, philosophical anthropology, and political economy. It is also a trenchant critique of a religious orientation to the world. In this paper I will reflect on Hägglund’s account of value and our finitude, paying special attention to his criticism of one of his targets, the political theology of Martin Luther King, Jr. According to Hägglund, King’s appeal to God when elaborating the need for justice would better be replaced by an appeal to our own communal norms. To defer to God is at best, a colourful way of depicting our own commitments, and at worst, God’s providence acts as an empty signifier that threatens to absolve us of the hard work of making justice in the here and now. I will show that while Hägglund’s account is a salutary corrective to a pervasive kind of bad faith, this criticism goes only so far. Any identification of God’s justice with communal norms, or of truth with our best theory, is to mischaracterise these concepts. They function as ideals that direct our attention outside ourselves and beyond our own conceptions, in the same way that love orients us toward a beloved.

The Semantics and Psychology of Negation: The Australian Plan, Negation as Failure, and Card Selection Tasks

Thu, 19 Oct 2023 15:15:00 +0100

Abstract: In this talk, I will explain how two different kinds of semantics for negation can be used to help us understand some puzzles in the psychology of reasoning. I introduce the “Australian Plan” semantics for negation, which generalises the semantics of negation as found in “worlds” semantics for intuitionistic logic, relevant logics and other substructural logics; and semantics from logic programming and default logic which treat negation as failure. I provide a framework in which two kinds of semantic clauses for negation coexist, and use this to model to give a possible account of why reasoners fare differently in card selection tasks depending on whether the target property is phrased positively or negatively. This talk reports joint work with Francesco Berto. The talk is a presentation at the University of Stirling Philosophy Seminar. The slides of the talk are available here.

The Semantics and Psychology of Negation: The Australian Plan, Negation as Failure, and Card Selection Tasks

Wed, 20 Sep 2023 15:00:00 +0100

Abstract: In this talk, I will explain how two different kinds of semantics for negation can be used to help us understand some puzzles in the psychology of reasoning. I introduce the “Australian Plan” semantics for negation, which generalises the semantics of negation as found in “worlds” semantics for intuitionistic logic, relevant logics and other substructural logics; and semantics from logic programming and default logic which treat negation as failure. I provide a framework in which two kinds of semantic clauses for negation coexist, and use this to model to give a possible account of why reasoners fare differently in card selection tasks depending on whether the target property is phrased positively or negatively. This talk reports joint work with Francesco Berto. The talk was a presentation in the Arché Metaphysics and Logic seminar series in the Philosophy Department at the University of St Andrews. The slides of the talk are available here.

PY2010: Intermediate Logic

Mon, 11 Sep 2023 00:00:00 +0000

py2010: Intermediate Logic is a University of St Andrews undergraduate subject in which we cover important results in logic to philosophy students. It’s taught by Greg Restall, together with a committed crew of postgraduate students. The subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. I’m using the recently published text Logical Methods, written with my colleague Shawn Standefer.

The Semantics and Psychology of Negation: The Australian Plan, Negation as Failure, and Card Selection Tasks

Fri, 08 Sep 2023 11:20:00 +0200

Abstract: In this talk, I will explain how two different kinds of semantics for negation can be used to help us understand some puzzles in the psychology of reasoning. I introduce the “Australian Plan” semantics for negation, which generalises the semantics of negation as found in “worlds” semantics for intuitionistic logic, relevant logics and other substructural logics; and semantics from logic programming and default logic which treat negation as failure. I provide a framework in which two kinds of semantic clauses for negation coexist, and use this to model to give a possible account of why reasoners fare differently in card selection tasks depending on whether the target property is phrased positively or negatively. This talk reports joint work with Francesco Berto. The talk was a presentation 15th Conference of the Italian Society for Analytic Philosophy. The slides of the talk are available here.

Finitude, Eternity, Love, the Good and Martin Hagglünd’s ‘This Life’

Fri, 01 Sep 2023 09:00:00 +0100

Looking at Logic(s)

Mon, 07 Aug 2023 00:00:00 +0000

This is an interview with me, conducted by Daniel Nellor, for the volume What are They Thinking? Conversations with Australian Philosophers. The topics range from the scope of philosophy, the place of logic in philosophy, logical pluralism, and more.

Review of The Contradictory Christ

Mon, 07 Aug 2023 00:00:00 +0000

This is short review of Jc Beall’s The Contradictory Christ, OUP, 2021.

Exploring Three-Valued Models for Identity

Thu, 20 Jul 2023 10:00:00 +0100

Abstract: There is a very natural way to interpret the propositional connectives and quantifiers, relative to the algebra of three semantic values, {0, i, 1} where 0 and 1 are understood as the traditional values of falsity and truth, and the third value is some intermediate value. The evaluation clauses do not, by themselves, determine the logic, because for that, you need to determine how models are used to provide a counterexample to a sequent. If a counterexample is given by a model that assigns every premise the value 1 and assigns every conclusion a value other than 1, the resulting logic is Kleene’s strong three-valued logic, K3. If a counterexample is a model assigning every premise the value 1 or i and every conclusion the value 0, the resulting logic is Priest’s logic of paradox, LP. If a counterexample is a model assigning every premise the value 1 and every conclusion the value 0, you get the logic ST of Strict-Tolerant validity. ST is distinctive, in that it is, in some sense, classical logic—every classically valid sequent in this language is ST-valid—but since it has strictly non-classical models, there are ST theories which are not classical theories. What does this mean for the logic of identity in a three-valued context?

Structural Rules in Natural Deduction with Alternatives

Tue, 18 Jul 2023 00:00:00 +0000

Natural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proof system in which the connective rules are unmodified from the usual Prawitz introduction and elimination rules–the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules fixed, and varying purely structural rules. The key result of this paper is that the two principles that introduce kinds of irrelevance to natural deduction proofs: (a) the rule of explosion (from a contradiction, anything follows); and (b) the structural rule of vacuous discharge; are shown to be two sides of a single coin, in the same way that they correspond to the structural rule of weakening in the sequent calculus. The paper also includes a discussion of assumption classes, and how they can play a role in treating additive connectives in substructural natural deduction. This paper appears in a special issue of the Bulletin of the Section of Logic on Bilateralism and Proof-Theoretic Semantics.

Numbers, the World, and God: Varieties of Semantic Anti-Realism

Thu, 22 Jun 2023 10:15:00 +0100

Abstract: In this talk, I illustrate some of the key ideas of the distinctively Grammatical Thomist kind of apophaticism developed by Simon Hewitt, in his book Negative Theology and Philosophical Analysis: Only the Splendour of Light. In particular, I explore the kind of semantic anti-realism at the heart of the work, contrasting and complementing Hewitt’s use of ideas from Wittgenstein, Dummett and McCabe with some themes from the recent work of the normative pragmatism of Robert Brandom. I use the prosaic examples of properly semantically antirealist ways to understand the semantics of our talk of numbers and of the world, to illustrate the general strategy. Then, using Brandom’s distinction between sense dependence and reference dependence, I show how the proponent of such a non-referential semantics can respond to the criticism that any commitment to semantic anti-realism for some topic (whether numbers, the world or God) is to conceive of the subject matter of that topic as dependent on or subservient to our own interests, concerns or concepts. I end this talk with some questions for Hewitt, concerning the behaviour of kind talk and the concept of identity invovled in his explanation of why God is totally unlike any other thing, and concerning what else may be distinctive about the God concept.

Proofs for Relevant Consequence, with Star and Perp

Thu, 09 Mar 2023 10:00:00 +0100

Abstract: In this talk, I show how to incorporate insights from the model-theoretic semantics for negation (insights due to J. Michael Dunn, in his paper “Star and Perp: Two Treatments of Negation”), into a proof-first understanding of the semantics of negation. I then discuss how a logical pluralist may understand the underlying accounts of proofs and their significance. The result is a new perspective from which to view the connection between relevance and different notions of logical consequence. The talk was a face-to-face presentation at the Logic, Reasoning, and Justification Workshop hosted by the Philosophy Department at the University of Bergen, in Norway. The slides for the talk are available here.

Interview with the Undergraduate Philosophy Journal of Australasia

Tue, 07 Mar 2023 08:42:32 +0000

A few months ago, Anna Day, Eloise Hickey, Mark Rothery, and James Cafferky from the Undergraduate Philosophy Journal of Australasia gave me an opportunity to ramble on about my early days as a mathematics and philosophy student in the 1980s and 1990s, my current research interests, and what I’m thinking about now. They asked thoughtful questions and managed to edit the interview into something coherent. Check it out, if only for the photos of me in my teens and twenties, then stay for the philosophy.

A Brush with Fame

Thu, 26 Jan 2023 21:02:27 +0000

In my PY1012 Reasoning lecture this evening, I used a slide with a photo of Sally Haslanger and a short section from her book Resisting Reality to give an example of an argument to a universal generalisation. (I’ve been teaching reasoning using examples from throughout the philosophical canon.) After class a young student came up to me, all excited, asking: do you know Professor Haslanger!? (I had described her as a colleague.) I explained that yes, we’d met and I’d known her for some time. She reacted as if I had been in the presence of a rock star, and that she was now a little bit closer to true greatness as a result. When I said I loved Sally’s work, and then added that Kate Manne had been a student of mine at the University of Melbourne before she went to MIT to study with Sally, the student was absolutely beside herself. It’s a delight to exploit my own personal contacts with colleagues and friends, to help a student feel a little closer to what she truly treasures in the sometimes humdrum matter of teaching intro reasoning.

PY4601: Paradoxes

Tue, 17 Jan 2023 00:00:00 +0000

py4601: Paradoxes is an honours Philosophy module at the University of St Andrews. It’s coordinated by my colleague, Patrick Greenough, and I’m teaching a small slice at the end on the liar paradox. If you’d like to see what I am covering, you can see some slides and notes here. Here’s what we’re covering in the whole module: A paradox is a plausible argument for an absurd conclusion. Better still: a paradox is an apparently plausible argument, with apparently plausible premises, which leads to an apparently absurd conclusion using apparently valid reasoning. In this module, we are going to look at three groups of paradoxes (and the connections between them): Epistemological Paradoxes (e.g. various forms of Scepticism, Lottery Paradox, Dogmatism Paradox). Paradoxes of Vagueness and Indeterminacy (The Sorites Paradoxes, The Problem of the Many, The Open Future, The Ship of Theseus, The Chicken and Egg Paradox). Semantic Paradoxes (The Liar Paradox, The Truth-Teller, Postcard Paradox, The No-No Paradox, The Prover Paradox, Curry’s Paradox, Yablo’s Paradox). This module is a mixture of meta-philosophy (the philosophy of the nature of philosophical problems and the methods we can or should use to address these problems) plus first-order philosophy (what is the best solution to, e.

PY1012: Reasoning

Mon, 16 Jan 2023 00:00:00 +0000

py1012: Reasoning introduces the essential concepts and techniques of critical reasoning, formal propositional logic, and basic predicate logic. Among the central questions are these: what distinguishes an argument from a mere rhetorical ploy? What makes an argument a good one? How can we formally prove that a conclusion follows from some premises? In addressing these questions, we will also cover topics such as ambiguity, argument forms and analyses, induction compared to deduction, counterexamples, truth-tables, natural deduction, and quantification.

Kicking off Semester 2 in St Salvator's Chapel

Sun, 15 Jan 2023 16:20:58 +0000

I grew up in Australia: my university training and my initial academic positions took place in the explicitly secular institution of the Australian university. So, it’s an uncanny experience to arrive in St Andrews to become a part of a university in a town marked by martyrdom, in which the Chaplaincy plays a central and visible role. University functions, including graduations, are opened with prayers in Latin. There are regular services in Chapel, including graduation services, and many involve an procession of academics, in robes. The separation of “church” and “state” is nowhere near as sharp here in St Andrews as it was in Australia. The university is explicitly pluralist, and the chaplains work very hard to make space for students of all faiths and none. And at the same time, this place owns its Christian heritage. So, it was to my surprise – and my trepidation – that I was invited to preach the sermon at the Sunday chapel service at the start of this new semester. I seem to have survived the experience of doing something in my work context that is far outside the everyday responsibility of my academic role as a Professor of Philosophy.

Come and See! (John 1:29-42)

Sun, 15 Jan 2023 11:00:00 +0100

I normally don’t speak from notes, but I do know that if I get up in front of a group to speak, my natural duration is the lecture, and at 45 to 50 minutes, that just won’t do for a sermon at chapel. To prevent an over-long talk, I took the time to write things down, and edit it to an appropriate length. Now that I have it, I may as well share the text here. * * * Welcome to this new semester! Happy New Year! A new year and a new semester is a time for reorientation. As a recent migrant to Scotland from the southern hemisphere, my life over the past year and half, has been filled with reorientation. I am still getting used to a different calendar where the Christmas break is in winter and it’s the short break between semesters, so I find myself having to reorient – especially when, after celebrating Christmas and the New Year, I look over at the calendar and notice that I’m meant to be back in the classroom on January 16! (My body still expects that I should be sweltering in the heat and watching the cricket for weeks on end.

Erdős Number: 3

Sat, 07 Jan 2023 09:29:30 +0000

According to the AMS’s handy Mathematics Collaboration Distance calculator, my Erdős number is down to three, given the following path: Vedran Čačić, Pavel Pudlák, Greg Restall, Alasdair Urquhart, Albert Visser, “Decorated linear order types and the theory of concatenation,” Logic Colloquium 2007, p. 1–13, ed. F. Delon, U. Kohlenbach, P. Maddy and F. Stephan, Cambridge University Press, 2010. Noga Alon and Pavel Pudlák, “Equilateral Sets in $l^n_p$,” Geometric & Functional Analysis, 13 (2003), 467-482. Noga Alon and Paul Erdős, “Disjoint edges in geometric graphs,” Discrete and Computational Geometry 4 (1989), no. 4, 287–290. This makes my Erdős number equal to my Trotsky number, but that is another story.

Wombat, Conditional, or Inference?

Wed, 04 Jan 2023 16:41:24 +0000

As my colleague and PY1012 Reasoning co-lecturer, Franz Berto knows, it’s never too early to introduce your students to wombats, or to the difference between a conditional and an inference. A slide from my first week’s PY1012 lecture. Yes, next semester’s classes are just about to start, and I’m in the depths of preparation.

Logical Methods Publication Day

Tue, 03 Jan 2023 06:55:00 +0000

Today MIT Press releases our book, Logical Methods into the big wide world. It was an absolute delight to work on this long-term project with my co-author and friend, Shawn Standefer.

A Stack of Copies of Logical Methods

Logical Methods

Tue, 03 Jan 2023 00:00:00 +0000

As the cover blurb says Logical Methods is an accessible introduction to philosophical logic, suitable for undergraduate courses and above. Rigorous yet accessible, Logical Methods introduces logical tools used in philosophy—including proofs, models, modal logics, meta-theory, two-dimensional logics, and quantification—for philosophy students at the undergraduate level and above. The approach developed by Shawn Standefer and I developed is distinct from other texts because it presents proof construction on equal footing with model building and emphasizes connections to other areas of philosophy as the tools are developed. Throughout, the material draws on a broad range of examples to show readers how to develop and master tools of proofs and models for propositional, modal, and predicate logic; to construct and analyze arguments and to find their structure; to build counterexamples; to understand the broad sweep of formal logic’s development in the twentieth and twenty-first centuries; and to grasp key concepts used again and again in philosophy. The text was developed through years of teaching intermediate (second-year) logic at the University of Melbourne. If you’d like to read more of my reflections on that experience, see this news entry from 2019. It is available from MIT Press.

Time for a little grease and oil change

Sun, 01 Jan 2023 20:08:30 +0000

2022 has been another big year, not that you’d know it from looking around the news section of this website. Settling in to St Andrews has taken up a lot of my energy (in a good way), and I’ve been having too much fun writing things and giving talks to spend time updating this website.

With the break between Christmas and New Year, I finally had time to clean up a bit of the mess around here, and update things.

True Contradictions in Theology?

Sat, 31 Dec 2022 00:00:00 +0000

In The Contradictory Christ, Jc Beall argues that paraconsistent logic provides a way to show how the central claims of Christology can all be true, despite their paradoxical appearances. For Beall, claims such as “Christ is peccable” and “Christ is impeccable” are both true, with no change of subject matter or ambiguity of meaning of any term involved in each claim. Since to say that Christ is impeccable is to say that Christ is not peccable, these two claims are contradictory, and so, for Beall the conjunction “Christ is peccable and Christ is not peccable” is a true contradiction. This is a radical and original view of the incarnation, and a revisionary view of what is permissible for theological reasoning. Here, I will examine the term “contradiction” that plays such a central role in Beall’s account. I will argue that in Beall’s own conceptual framework, our everyday concept of contradiction bifurcates into two different senses: negation-contradiction and unsatisfiability. I will show that a theologian who avails herself of Beall’s paraconsistent logic when making theological claims may have cast off the shackles of having to make sure that her commitments avoid negation-contradiction, but the heavy burden of ensuring that her commitments are jointly satisfiable remains.

Natural Deduction with Alternatives: on structural rules, and identifying assumptions

Thu, 15 Dec 2022 16:00:00 +0100

Abstract: In this talk, I will introduce natural deduction with alternatives, explaining how this framework provides a simple, well-behaved, single conclusion natural deduction system for a range of logical systems, including classical logic, (classical) linear logic, relevant logic and affine logic, in addition to the familar intuitionistic restrictions of these systems. Each of these proof systems have identical connective rules. As we expect in substructural logics, different logical systems are given by varying the structural rules in play. The distinctly classical behaviour of these systems is given by the presence of alternatives (formulas in consequent, or positive position, other than the conclusion of the proof) in addition to assumptions (formulas in antecedent, or negative position). Unlike multiple conclusion proof systems, this proof system is single conclusion, but unlike traditional natural deduction à la Gentzen or Prawitz, the context in which that formula is proved consist of formulas ruled in (assumptions) and formulas ruled out (alternatives). The result is a proof system that is mildly bilateralist. I will introduce this framework, and show how the presence of alternatives in natural deduction can give us a new angle from which to view the impact of the structural rules of weakening and contraction, and the difference between multiplicative and additive connectives

Collection Frames: What, How and Why?

Fri, 18 Nov 2022 13:30:00 +0200

Abstract: In this talk, I give a breezy introduction to Collection Frames (joint work with Shawn Standefer), with an emphasis on how they are technically equivalent to, but conceptually simpler than Routley–Meyer ternary relational frames. The talk is an online presentation at the New Directions in Relevant Logic Online Workshop. The slides for the talk are available here.

Natural Deduction with Alternatives: on structural rules, and identifying assumptions

Wed, 09 Nov 2022 14:30:00 +0100

PY3100: Reading Philosophy 1—Texts in Language, Logic, Mind, Epistemology, Metaphysics and Science

Tue, 13 Sep 2022 00:00:00 +0000

py3100: Reading Philosophy 1–Texts in Language, Logic, Mind, Epistemology, Metaphysics and Science is designed to develop the philosophical skills students have acquired over the first two years of their philosophy study, and acquaint them with key works in core areas of philosophy. The module involves close study of philosophical texts – historical and contemporary – that address a variety of topics within metaphysics, epistemology, the philosophies of logic and language, mind and science. Students will be required to carry out close study and discussion of these texts in staff-led weekly workshops, thereby furthering their skills of critical evaluation and analysis. Students will also take turns in presenting papers to the workshop, in pair-groups, which will help them to develop important communication skills and provide an opportunity for teamwork.

PY4638: Philosophy of Religion

Tue, 13 Sep 2022 00:00:00 +0000

py4638: Philosophy of Religion aims to provide a philosophical understanding of the phenomenon of religion and its relation to other central human activities, studying such topics as religious and cultural diversity, religious experience, belief and justification, faith and reason, religious language, religion and metaphysics, or religion and science. In 2022, we will be exploring one important topic in the philosophy of religion from three different perspectives. We will explore the notion that ultimate reality—or God—is in some sense ineffable, beyond our grasp, or transcendent. Understanding God or the ground of being as utterly transcendent is an important aspect of not just one religious tradition, and not just one cultural grouping, and reflective religious thinkers and philosophers through history have had very many interesting things to say about ineffability and transcendence. When we attempt to make sense of religious claims about transcendence, we are brought right up to key philosophical questions about the nature of reality, our capacity to understand reality, and the power and limits of our language. The module has three connected parts: First, Greg will examine the interconnected notions of transcendence, ineffability, and simplicity; whether it is consistent to describe something as truly ineffable; and how coming to terms with religious use of claims like these might lead to reconsidering fundamental assumptions about our language.

Classical Logic and Intuitionistic Logic: looking both ways

Tue, 06 Sep 2022 09:00:00 +0200

Abstract: We know a great many technical results concerning the relationship between classical logic and intuitionistic logic, whether in the propositional, first-order or higher-order languages. We also know quite a lot about the relationship between intuitionistic and classical theories. In this talk, I will explore some of what these results might mean, from the perspective of partisans of one side or other of the divide, and what kinds of pluralism might be tenable, in the light of the existing and long-standing practices of classical and constructive mathematics. The talk is an invited presentation at PhDs in Logic XIII in Turin. The slides for the talk are available here.

The Many Uses of Proofs: logic and philosophy, language and more

Mon, 13 Jun 2022 14:00:00 +0000

Abstract: This talk is a free-wheeling introduction to my research, starting from work in substructural logics and logical pluralism, and ending at the many uses of proofs, including giving an account of how our modal vocabulary has the meaning that it does, and the connections between proof norms and the semantics and pragmatics of dialogue. The talk is a face-to-face presentation at the University of St-Andrews Computer Science Department’s Research in School day. The slides for the talk are available here.

True Contradictions? Why, and Why Not?

Tue, 07 Jun 2022 14:00:00 +0000

Abstract: In this talk, I introduce the difference between paraconsistency (adopting a logic for which a contradiction need not entail everything) and dialetheism (the notion that there are true contradictions), and I explain some reasons why one might take there to be true contradictions. I focus on Jc Beall’s recent work on contradictory Christology as one such motivation, and discuss some attractions of the view, as well as some shortcomings to be further explored. The talk is a face-to-face presentation at the University of St Andrews Philosophy Department’s “Reflectorium”, its regular work-in-progress day. The slides for the talk are available here.

Exploring Three-Valued Models for Identity

Mon, 18 Apr 2022 14:00:00 +0000

A little book on Proofs and Models

Tue, 05 Apr 2022 12:00:00 +0000

Late last month, my little manuscript on Proofs and Models in Philosophical Logic was published by Cambridge University Press. This, like all entries in the new Cambridge Elements series, is a tiny little manscript, with an aim to give students, and researchers in allied fields, a quick, accessible introduction to a research topic and current methods. My mansucript is a breezy 84 pages, and it tries to introduce the role of proofs and models in contemporary philosophical logic, with a focus on work on the paradoxes.

You can download the book for free on the Cambridge University Press website until April 13. After that, you’ll need to either buy a hardcopy or an electronic copy, or access it with some institutional subscription.

Proofs and Models in Philosophical Logic

Fri, 25 Mar 2022 00:00:00 +0000

This is a short book, in the Cambridge Elements series in Philosophical Logic. This is a general introduction to recent work in proof theory and model theory of non-classical logics, with a focus on the application of non-classical logic to the semantic paradoxes and (to a lesser extent), the sorites paradox. After a short introduction motivating general notions of proof and of models, I introduce and motivate a simple natural deduction system, and present the structure of the liar paradoxical argument (concerning truth) and Curry’s paradox (concerning class membership). I introduce and motivate the notion of a structural rule, in both natural deduction and the sequent calculus and I compare and contrast the different approaches to substructural treatments of the paradox, contrasting the roles that contraction, cut and identity play in the derivations of the paradoxes. In the next section, I introduce model theoretic treatments of the paradoxes, introducing supervaluations, and three-valued treatments of vagueness, and of the semantic paradoxes. I explain the fixed-point model construction that shows how to construct three-valued models for theories of truth, which can be used to then give models for different logics: K3 (with truth-value gaps), LP (with truth-value gluts) and ST (which supports all of classical logic, at the cost of invalidating the cut rule).

Natural Deduction with Alternatives: on structural rules, and identifying assumptions

Thu, 17 Mar 2022 16:30:00 +0100

Justification Requests, Inference and Definitions

Fri, 28 Jan 2022 15:00:00 -0700

Abstract: In this talk, I examine some of the interconnections between speech acts, such as assertion and denial, inference, justification requests, and the common ground. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground. In addition, we can give a distinctive answer to Carroll’s puzzle concerning the force of the logical must, and a sense in which certain rules for logical concepts can indeed count as definitions. The talk is an online presentation at the Philosophy Department at the University of Calgary. The slides for the talk are available here.

PY1012: Reasoning

Thu, 13 Jan 2022 00:00:00 +0000

PY4634: Philosophical Logic

Thu, 13 Jan 2022 00:00:00 +0000

py4638: Philosophical Logic focuses on some of the main philosophical questions that have been raised concerning the central notions of logic. We’ll be examining contemporary debates on notions such as logical consequence, the normative status of logic, its epistemology, the meaning of the logical constants, logical pluralism, and higher-order logics.

Proofs with Star and Perp: pluralism and proofs for different logics

Fri, 10 Dec 2021 13:00:00 +0100

Abstract: In this talk, I show how to incorporate insights from the model-theoretic semantics for negation (insights due the late J. Michael Dunn, among others), into a properly proof-theoretic understanding of the semantics of negation. I then discuss the different ways a logical pluralist may understand the underlying accounts of proofs and their significance. The talk is a presentation at the Current Debates in the Philosophy of Logic Seminar, European Network for the Philosophy of Logic. The slides for the talk are available here.

Assertions, Denials, Questions, Answers, and the Common Ground

Wed, 20 Oct 2021 14:00:00 +0100

Abstract: In this talk, I examine some of the interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground. With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent X ⊢ Y as enjoining us to not assert each member of X and deny each member of Y is altogether too weak to explain the inferential force of logical validity. Deriving X ⊢ A should tell us, after all, something about justifying A on the basis of X, rather than merely prohibiting A’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.

Worlds: Possible and Impossible

Mon, 04 Oct 2021 15:00:00 +0100

Abstract: In this talk, I reflect on the role of worlds–possible worlds and impossible worlds—both in the semantics of various kinds of languages and logics, and in broader issues in metaphysics. I will argue that, given very modest assumptions concerning the role of worlds in semantics, that any defender of possible worlds in such a role should be equally comfortable with impossible worlds. However, this argument for impossible worlds does not transfer straightforwardly to logically impossible worlds. So, in the second part of the talk I will consider what we might say, for (or against) properly logically impossible worlds. The talk is a presentation at the Arché Logic and Metaphysics Seminar, University of St Andrews. The slides for the talk are available here.

PY3100: Reading Philosophy 1—Texts in Language, Logic, Mind, Epistemology, Metaphysics and Science

Mon, 13 Sep 2021 00:00:00 +0000

PY4612: Advanced Logic

Mon, 13 Sep 2021 00:00:00 +0000

py4612: Advanced Logic applies the tools of formal logic to make logic itself the object of study. We will explore the power and limits of logical tools and techniques. The main goals of the module will be to come to grips with some standard ‘metatheoretical’ results about logic: (1) the Soundness and Completeness Theorems, which together show that proofs and models can be used analyse the same consequence relation in two very different ways. (2) The Compactness Theorem and the Löwenheim–Skolem Theorems, which explore some of the limits of first-order classical predicate logic for classifying infinite structures. And most importantly (3) we will work through Gödel’s celebrated Incompleteness Theorems and come to grips with what they mean. Along the way, there will be some preparatory discussion of elementary set theory, proof theory, model theory, and recursion theory. Kurt Gödel, seated

Comparing Rules for Identity in sequent systems and natural deduction

Mon, 06 Sep 2021 14:00:00 +0100

Abstract: It is straightforward to treat the identity predicate in models for first order predicate logic. Truth conditions for identity formulas are given by a natural clause: a formula s = t is true (or satisfied by a variable assignment) in a model if and only if the denotations of the terms s and t (perhaps relative to the given variable assignment) are the same. On the other hand, finding appropriate rules for identity in a sequent system or in a natural deduction proof setting leaves a number of questions open. Identity could be treated with introduction and elimination rules in natural deduction, or left and right rules, in a sequent calculus, as is standard for familiar logical concepts. On the other hand, since identity is a predicate and identity formulas are atomic, it is also very natural to treat identity by way of axiomatic sequents, rather than by inference rules. I will describe and discuss this phenomenon, and explore the relationships between different formulations of rules for the identity predicate, and attempt to account for some of the distinctive virtues of each different formulation. The talk is an invited online presentation for the Tableaux 2021. The slides for the talk are available here.

Proofs and Models in Philosophical Logic

Fri, 03 Sep 2021 11:45:00 +0100

Abstract: In this talk, I will draw out three different ways that soundness and completeness—and the relationship between proofs and models—can teach us in something about classical propositional logic, the semantics of modal logic, and the metaphysics of quantified modal logic. The talk is an online presentation for the British Logic Colloqium. The slides for the talk are available here.

Leaving Melbourne

Wed, 30 Jun 2021 09:00:00 +1000

As June 2021 turns to a close, this is my last official day at The University of Melbourne. I’ve taught my last classes, the marking for the semester is all done, I’ve wound up all my committee work, I’ve supervised my last undergraduate theses, and wrapped up all the end-of-semester administration. I’m now packing up my office (which I’ve rarely seen over the last 18 months) and tying up lots of loose ends. If I weren’t starting a new position, things would be falling eerily silent, with no Semester 2 subjects to prepare, no committee work to do, and no students to supervise. And with my new academic year starting in September, I do have a few moments to pause, to breathe, and to reflect on my 19 years at the University of Melbourne, before I head off on the next adventure.

Natural Deduction with Alternatives: on structural rules, and identifying assumptions

Fri, 25 Jun 2021 14:10:00 +1100

Abstract: In this talk, I will introduce natural deduction with alternatives, explaining how this framework provides a simple, well-behaved, single conclusion natural deduction system for a range of logical systems, including classical logic, (classical) linear logic, relevant logic and affine logic, in addition to the familar intuitionistic restrictions of these systems. Each of these proof systems have identical connective rules. As we expect in substructural logics, different logical systems are given by varying the structural rules in play. The distinctly classical behaviour of these systems is given by the presence of alternatives (formulas in consequent, or positive position, other than the conclusion of the proof) in addition to assumptions (formulas in antecedent, or negative position). Unlike multiple conclusion proof systems, the proof system is single conclusion, since one formula in positive position is singled out as the conclusion. The context in which that formula is proved consists, in general, of formulas ruled in (assumptions) and formulas ruled out (alternatives). In sequent systems, and in some natural deduction systems that use labels, the structural rules of contraction and weakening govern an explicitly represented structure, such as a set or multiset or sequence of formulas occurring in each sequent. In this natural deduction framework, the structural rules have their force at the point of discharge, or more generally, at any point at which it is important to determine whether two occurrences of the same formula (in positive position, or in negative position) are the same assumption or the same alternative.

Geometric Models for Relevant Logics

Fri, 14 May 2021 00:00:00 +0000

Alasdair Urquhart’s work on models for relevant logics is distinctive in a number of different ways. One key theme, present in both his undecidability proof for the relevant logic R, and his proof of the failure of interpolation in R, is the use of techniques from geometry. In this paper, inspired by Urquhart’s work, I explore ways to generate natural models of R from geometries, and different constraints that an accessibility relation in such a model might satisfy. I end by showing that a set of natural conditions on an accessibility relation, motivated by geometric considerations, is jointly unsatisfiable.

Platonism, Nominalism, Realism, Anti-Realism, Reprentationalism, Inferentialism and all that

Thu, 06 May 2021 16:15:00 +1100

My usual talk (a close-up view of the Old Quad and Arts West at the University of Melbourne). Abstract: In this talk, I will place contemporary research in philosophical logic in a wider historical and philosophical context, showing how recent work in logic connects to the rivalry between Platonism and Nominalism, or realism and anti-realism in metaphysics, and between representationalism and inferentialism in the the philosophy of language. Along the way, I will touch on the contemporary resurgence of interest in Carnap’s logical positivism, and Robert Brandom’s turn toward Hegel. This talk (a view of the globe, with eastern Australia at dusk). Here is the PhilEvents link to the event.

Comparing Rules for Identity in Sequent Systems and Natural Deduction

Wed, 21 Apr 2021 19:00:00 +1100

Abstract: It is straightforward to treat the identity predicate in models for first order predicate logic. Truth conditions for identity formulas are straightforward. On the other hand, finding appropriate rules for identity in a sequent system or in natural deduction leaves many questions open. Identity could be treated with introduction and elimination rules in natural deduction, or left and right rules, in a sequent calculus, as is standard for familiar logical concepts. On the other hand, since identity is a predicate and identity formulas are atomic, it is possible to treat identity by way of axiomatic sequents, rather than inference rules. In this talk, I will describe this phenomenon, and explore the relationships between different formulations of rules for the identity predicate, and attempt to account for some of the distinctive virtues of each different formulation. The talk was an online presentation for the Proof Theory Virtual Seminar The slides for the talk are available here. Here is the recording of the talk:

PHIL30043: The Power and Limits of Logic

Mon, 01 Mar 2021 00:00:00 +0000

PHIL30043: The Power and Limits of Logic (or, as I like to call it, Kurt Gödel’s Greatest Hits) is a University of Melbourne undergraduate subject. It covers the metatheory of classical first order predicate logic, beginning at the Soundness and Completeness Theorems, Compactness, Cantor’s Theorem, the Downward Löwenheim–Skolem Theorem, Recursive Functions, Register Machines, Representability, the Indefinability of Truth and the Undecidability of Predicate Logic, and ending up at Gödel’s Incompleteness Theorems and Löb’s Theorem. Kurt Gödel, seated The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here. I make use of video lectures I have made freely available on YouTube. Outline The course is divided into four major sections, taught over 12 weeks. Here is a list of all of the videos, in case you’d like to follow along with the content. Soundness and Completeness The Language of Predicate Logic Proofs for Predicate Logic Models for Predicate Logic and Soundness Completeness for Predicate Logic Countability and Compactness Identity and Functions Countability and Diagonalisation Compactness and Countable Models Computability Recursive Functions Register Machines Register Machines compute the Recursive Functions Non-Recursive Functions Undecidability and Incompleteness Theories of Arithmetic Diagonalisation and its Consequences Provability Predicates, and Beyond

UNIB10002: Logic, Language and Information

Mon, 01 Mar 2021 00:00:00 +0000

UNIB10002: Logic, Language and Information is a University of Melbourne undergraduate breadth subject, introducing logic and its applications to students from a wide range of disciplines in the Arts, Sciences and Engineering. I coordinate this subject with my colleague Dr. Jen Davoren, with help from Prof. Lesley Stirling (Linguistics), Dr. Peter Schachte (Computer Science) and Dr. Daniel Murfet (Mathematics). The subject is taught to University of Melbourne undergraduate students. Details for enrolment are here.

An Inferentialist Account of Identity and Modality

Tue, 16 Feb 2021 10:00:00 +0100

Abstract: In this talk I will show how defining rules in a hypersequent setting can give a uniform proof-theoretic semantics of identity and modality, allowing – equally naturally – for (1) modal operators for which identity statements are necessary (if true), and (2) modal operators for which identity statements can be contingently true. The talk is an online presentation for the ERC EXPRESS Project in Amsterdam, and the PHILMATH Seminar in Paris. The slides for the talk are available here. Here is the recording of the talk:

Natural Deduction with Alternatives

Fri, 06 Nov 2020 12:30:00 +1100

Abstract: In this talk, I will introduce natural deduction with alternatives, explaining how this framework can provide a simple well-behaved single conclusion natural deduction system for a range of logical systems, including classical logic, (classical) linear logic, relevant logic and affine logic, by varying the policy for managing discharging of assumptions and retrieval of alternatives. Along the way, the talk will touch on (1) the connection between normalisation of a natural deduction proof and cut elimination in a corresponding sequent calculus; (2) the separation between the operational rules governing the connectives and the “antecedently given context of deducibility”, to borrow a phrase from Nuel Belnap’s essay, “Tonk, Plonk and Plink” (1962); and (3) the sense in which the operational rules for a connective might be understood as providing a definition of that connective. Greg Restall—Natural Deduction with Alternatives from logicmelb on Vimeo. The talk is an online presentation at our Applied Logic Workshop. The slides for the talk are available here.

Speech Acts & the Quest for a Natural Account of Classical Proof

Fri, 18 Sep 2020 16:10:00 -0700

Abstract: It is tempting to take the logical connectives, such as conjunction, disjunction, negation and the material conditional to be defined by the basic inference rules in which they feature. Systems of “natural deduction” provide the basic framework for studying these inference rules. In natural deduction proof systems, well-behaved rules for the connectives give rise to intuitionistic logic, rather than classical logic. Some, like Michael Dummett, take this to show that intuitionistic logic is on a sounder theoretical footing than classical logic. Defenders of classical logic have argued that some other proof-theoretical framework, such as Gentzen’s sequent calculus, or a bilateralist system of signed natural deduction, can provide a proof-theoretic justification of classical logic. Such defences of classical logic have significant shortcomings, in that the systems of proof offered are much less natural than existing systems of natural deduction. Neither sequent derivations nor signed natural deduction proofs are good matches for representing the inferential structure of everyday proofs. In this paper I clarify the shortcomings of existing bilateralist defences of classical proof, and, making use of recent results in the proof theory for classical logic from theoretical computer science, I show that the bilateralist can give an account of natural deduction proof that models our everyday practice of proof as well as intuitionist natural deduction, if not better.

Speech Acts & the Quest for a Natural Account of Classical Proof

Tue, 15 Sep 2020 00:00:00 +0000

It is tempting to take the logical connectives, such as conjunction, disjunction, negation and the material conditional to be defined by the basic inference rules in which they feature. Systems of “natural deduction” provide the basic framework for studying these inference rules. In natural deduction proof systems, well-behaved rules for the connectives give rise to intuitionistic logic, rather than classical logic. Some, like Michael Dummett, take this to show that intuitionistic logic is on a sounder theoretical footing than classical logic. Defenders of classical logic have argued that some other proof-theoretical framework, such as Gentzen’s sequent calculus, or a bilateralist system of signed natural deduction, can provide a proof-theoretic justification of classical logic. Such defences of classical logic have significant shortcomings, in that the systems of proof offered are much less natural than existing systems of natural deduction. Neither sequent derivations nor signed natural deduction proofs are good matches for representing the inferential structure of everyday proofs. In this paper I clarify the shortcomings of existing bilateralist defences of classical proof, and, making use of recent results in the proof theory for classical logic from theoretical computer science, I show that the bilateralist can give an account of natural deduction proof that models our everyday practice of proof as well as intuitionist natural deduction, if not better.

Teaching During a Pandemic

Thu, 06 Aug 2020 22:10:00 +1000

As I write this, the first week of the second semester of 2020 is nearing its end, and I’ve taught my first two seminars in Logical Methods, my main undergraduate teaching responsibility for this semester. Melbourne has just entered Stage 4 of its lockdown, as we attempt to deal with the ongoing community transmission of COVID-19. The streets are quiet, it has been over four months since I’ve been on campus, and all my teaching is done from the chair at my desk in my study, peering into the 15 inch screen of my laptop, with the green cyclops dot in the middle of the top screen bezel showing that yet again, my image is being transmitted across the internet, to students scattered across Melbourne, across Australia, and across the world.

PHIL20030: Logical Methods

Tue, 28 Jul 2020 00:00:00 +0000

PHIL20030: Logical Methods is a University of Melbourne undergraduate subject introducing logic to philosophy students. It’s taught by Greg Restall. The subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. I’m using an introductory text Logical Methods, written with my colleague Shawn Standefer for this course. Here’s the outline of the subject. Preliminaries Introduction Arguments and Trees Sentences and Formulas Propositional Logic Connectives: and & if Conjunction Conditional Biconditional More connectives: not & or Negation and falsum Disjunction Our System of Proofs Facts about proofs & provability Facts about provability Normalisation The Subformula Property Consequences of Normalisation Models & counterexamples Models and truth tables Counterexamples and validity Model-theoretic validity Soundness & completeness Soundness Completeness Proofs first or models first? Heyting algebras Modal Logic Necessity & possibility Possible worlds models Validity Strict conditionals and ambiguities Propositions Another notion of necessity Equivalence relations and epistemic logic Actuality & two-dimensional logic Actuality models and double indexing Validity Fixity and diagonal propositions Real world validity Natural deduction for modal logics Natural deduction for S4 Natural deduction for S5 Features of S5 Predicate Logic Quantifiers Syntax Natural deduction for CQ What is provable? Generality and eliminating detours Models for first-order logic Models and assignments of values Substitution Counterexamples and validity Compactness and what this means One novelty in our approach to the subject is the balance between proof theory and model theory.

Notes from a Pandemic

Wed, 27 May 2020 21:19:47 +1000

I’ve been up to a few things during the pandemic. Quite a few things, it seems. Here are links to some of the traces you can find elsewhere on the internet.

I wouldn’t say that I’ve become good at using Zoom, but I have been doing a heck of a lot of it. My three subjects for this semester moved online, and running seminars, workshops, classes over Zoom has become a part (only a part) of keeping the ship going. I don’t record those classes (for obvious reasons!) but I have recorded a couple of research seminars presentations I’ve made over the last few weeks.

Assertions, Denials, Questions, Answers, and the Common Ground

Tue, 05 May 2020 16:00:00 +1100

Proofs and Models in Naive Property Theory: A Response to Hartry Field's “Properties, Propositions and Conditionals”

Thu, 02 Apr 2020 00:00:00 +0000

In our response Field’s “Properties, Propositions and Conditionals”, we explore the methodology of Field’s program. We begin by contrasting it with a proof-theoretic approach and then commenting on some of the particular choices made in the development of Field’s theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field’s response to Restall’s “What are we to accept, and what are we to reject, when saving truth from paradox?”.

Geometric Models for Relevant Logics

Fri, 20 Mar 2020 11:00:00 +1100

Abstract: Alasdair Urquhart’s work on models for relevant logics is distinctive in a number of different ways. One key theme, present in both his undecidability proof for the relevant logic R, and his proof of the failure of interpolation in R, is the use of techniques from geometry. In this talk, inspired by Urquhart’s work, I explore ways to generate natural models of R from geometries, and different constraints that an accessibility relation in such a model might satisfy. I end by showing that a set of natural conditions on an accessibility relation, motivated by geometric considerations, is jointly unsatisfiable This talk is an online presentation for the Melbourne Logic Group. The slides for the talk are here. There is a (low resolution) recording of the talk on YouTube if you’d like to see what you missed.

UNIB10002: Logic, Language and Information

Wed, 04 Mar 2020 00:00:00 +0000

PHIL40013: Uncertainty, Vagueness and Disagreement

Tue, 03 Mar 2020 00:00:00 +0000

PHIL40013: Uncertainty, Vagueness and Disagreement is a University of Melbourne honours seminar subject for fourth-year students. Our aim in the Honours program is to introduce students to current work in research in philosophy of logic and language. In 2020, we’re covering the connections between speech acts, epistemology and normative theory. Introduction and overview, background Speech acts: what are they? J. L. Austin, How to Do things with Words, Clarendon Press, Oxford, 1962. [Read Lecture 9] H. P. Grice, “Logic and Conversation,” pages 41–58 in Syntax and Semantics: Speech Acts, edited by P. Cole and J. L. Morgan, Academic Press, New York, 1975. Sarah E. Murray and William B. Starr, “Force and Conversational States,” pages 202–236 in New Work on Speech Acts, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018. [Read Sections 9.1 and 9.2] Nuel Belnap “Declaratives are not Enough”, Philosophical Studies 59:1 (1990) 1–30. Mark Lance and Rebecca Kukla “Leave the Gun; Take the Cannoli! The Pragmatic Topography of Second-Person Calls” Ethics 123:3 (2013) 456–478. Mark Lance and Rebecca Kukla Yo! and Lo! The Pragmatic Topography of the Space of Reasons, Harvard University Press, 2009. [Read Chapter 1] Craige Roberts “Speech Acts in Discourse Context”, pages 317–359 in New Work on Speech Acts, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018.

PHIL30043: The Power and Limits of Logic

Mon, 02 Mar 2020 00:00:00 +0000

PHIL30043: The Power and Limits of Logic (or, as I like to call it, Kurt Gödel’s Greatest Hits) is a University of Melbourne undergraduate subject. It covers the metatheory of classical first order predicate logic, beginning at the Soundness and Completeness Theorems, Compactness, Cantor’s Theorem, the Downward Löwenheim–Skolem Theorem, Recursive Functions, Register Machines, Representability, the Indefinability of Truth and the Undecidability of Predicate Logic, and ending up at Gödel’s Incompleteness Theorems and Löb’s Theorem. Kurt Gödel, seated The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here. The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here. I make use of video lectures I have made freely available on YouTube. Outline The course is divided into four major sections, taught over 12 weeks. Here is a list of all of the videos, in case you’d like to follow along with the content. Soundness and Completeness The Language of Predicate Logic Proofs for Predicate Logic Models for Predicate Logic and Soundness Completeness for Predicate Logic Countability and Compactness Identity and Functions Countability and Diagonalisation Compactness and Countable Models Computability Recursive Functions Register Machines Register Machines compute the Recursive Functions Non-Recursive Functions Undecidability and Incompleteness Theories of Arithmetic Diagonalisation and its Consequences Provability Predicates, and Beyond

A Place for Logic in the Humanities?

Fri, 21 Feb 2020 10:30:00 +1100

Abstract: Logic has been an important part of philosophy in the western tradition since the work of Aristotle in the 4th Century BCE. Developments of the 19th and the 20th Century have seen an incredible flowering of mathematical techniques in logic, and the discipline transformed beyond recognition into something that can seem forbiddingly technical and formal. The discipline of logic plays a vital role in mathematics, linguistics, computer science and electrical engineering, and it may seem that it no longer has a place within the humanities. In this short talk, I’ll argue that this perception is misplaced and dangerous, and that in a time of increasing specialisation and differentiation between the cultures of the humanities, the sciences, and of engineering, logic not only has much to give to the humanities, it also has much to learn from them. The talk is an presentation at The Ballarat Philosophy Symposium 2020, at Federation University, Ballarat.

Two Negations are More than One

Thu, 12 Dec 2019 00:00:00 +0000

In models for paraconsistent logics, the semantic values of sentences and their negations are less tightly connected than in classical logic. In “American Plan” logics for negation, truth and falsity are, to some degree, independent. The truth of ${\mathord\sim}p$ is given by the falsity of $p$, and the falsity of ${\mathord\sim}p$ is given by the truth of $p$. Since truth and falsity are only loosely connected, $p$ and ${\mathord\sim}p$ can both hold, or both fail to hold. In “Australian Plan” logics for negation, negation is treated rather like a modal operator, where the truth of ${\mathord\sim}p$ in a situation amounts to $p$ failing in certain other situations. Since those situations can be different from this one, $p$ and ${\mathord\sim}p$ might both hold here, or might both fail here. So much is well known in the semantics for paraconsistent logics, and for first degree entailment and logics like it, it is relatively easy to translate between the American Plan and the Australian Plan. It seems that the choice between them seems to be a matter of taste, or of preference for one kind of semantic treatment or another. This paper explores some of the differences between the American Plan and the Australian Plan by exploring the tools they have for modelling a language in which we have two negations.

Generics: Inference & Accommodation

Fri, 06 Dec 2019 16:00:00 -0500

In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Generics exhibit this behaviour because they make inferences and explanations explicit, and inferences and explanations have exactly the same sort of behaviour as generics. Given the connection between generics and inference, we will be able to see how inference is involved in the process of accommodation, which plays a significant role in how we manage dialogue and conversation. A generic of the form Fs are Gs can enter the common ground when we allow the inference from Fx to Gx to pass without question in conversation. With this connection in hand, I will begin to explore what this means for social kind generics and how we use them. This is a talk for the Constructing Social Hierarchy 2 Workshop at MIT in December 6, 2019.

What's So Special About Logic? Practices, Rules and Definitions

Wed, 04 Dec 2019 19:30:00 -0500

Abstract: Over the last century or so, the discipline of logic has grown and transformed into a powerful set of tools and techniques that find their use in fields as far apart as philosophy, mathematics, computer science, electrical engineering and linguistics. Is there anything distinctive about logic and its results, or is it just another kind of abstract mathematics, or another kind of empirical scientific theory? In this talk I’ll explain why the distinctive subject matter of logical theory means that the tools of logic (proofs and models) can play a special role in our thought and in our talk. This explanation will turn crucially on our practices of assertion and denial, and how it can constrain those practices by using rules and definitions. The talk is the 21st Annual Alice Ambrose Lazerowitz/Thomas Tymocko Logic Lecture at Smith College. The slides for the talk are available here.

Negation on the Australian Plan

Mon, 02 Dec 2019 00:00:00 +0000

We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompat_ibility_ is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan.

Teaching Logical Methods

Thu, 14 Nov 2019 20:01:45 +1100

It’s been a big year. At the start of 2019, Shawn Standefer and I decided to throw all our cards in the air and upend the curriculum for the Level 2 logic unit in the philosophy program here at Melbourne. We wrote 200 pages of a draft textbook (while I really should have been finishing my other book). Shawn designed and implemented a whole raft of multiple choice practice questions, and we worked on a range of class activities to help our class of 60 students grapple with the material. I recorded hours of video lectures covering the content. We stuffed all of this into the LMS. And we spent hours in the classroom teaching 60 students the ins and outs of proof theory and model theory for propositional logic, modal logic (including two-dimensional modal logic), and first-order predicate logic. Like I said, it was a big year putting all of this together. Now we’ve wrapped up our first semester teaching the new unit, so we can sit back, breathe and reflect on how things went.

What's So Special About Logic? Practices, Rules and Definitions

Fri, 01 Nov 2019 16:00:00 +1100

Abstract: Over the last century or so, the discipline of logic has grown and transformed into a powerful set of tools and techniques that find their use in fields as far apart as philosophy, mathematics, computer science, electrical engineering and linguistics. Is there anything distinctive about logic and its results, or is it just another kind of abstract mathematics, or another kind of empirical scientific theory? In this talk I’ll explain why the distinctive subject matter of logical theory means that the tools of logic (proofs and models) can play a special role in our thought and in our talk. This explanation will turn crucially on our practices of assertion and denial, and how it can constrain those practices by using rules and definitions. The talk is an presentation at Melbourne Logic Day 2019. The slides for the talk are available here.

Assertions, Denials, Questions, Answers, and the Common Ground

Mon, 30 Sep 2019 12:15:00 +0000

Collection Frames for Substructural Logics

Thu, 26 Sep 2019 10:30:00 -0100

Abstract: In this talk I present a new frame semantics for positive substructural and relevant propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation is the use of a single accessibility relation to relate collections of points to points. Different logics are modelled by varying the kinds of collections featuring in the relation: for example, they can be sets, multisets, lists or trees. In this talk I will focus on multiset frames, which are sound and complete for the logic RW+ (positive multiplicative and additive linear logic with distribution for the additive connectives, or equivalently, the relevant logic R+ without contraction). This is joint work with Shawn Standefer. The talk is a presentation at the LanCog Workshop on Substructural Logics at the Facultate de Letras at the University of Lisbon. The slides for the talk are available here.

PHIL20030: Meaning, Possibility and Paradox

Mon, 29 Jul 2019 00:00:00 +0000

PHIL20030: Meaning, Possibility and Paradox is a University of Melbourne undergraduate subject introducing logic to philosophy students. It’s taught by Greg Restall and Shawn Standefer. This year, we have completely revised our curriculum. Now the subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. We’re writing an introductory text Logical Methods, which we’re trialling with this class, as well as producing explanatory videos to use along with the text. Here’s the outline of the subject. Preliminaries Introduction Arguments and Trees Sentences and Formulas Propositional Logic Connectives: and & if Conjunction Conditional Biconditional More connectives: not & or Negation and falsum Disjunction Our System of Proofs Facts about proofs & provability Facts about provability Normalisation The Subformula Property Consequences of Normalisation Models & counterexamples Models and truth tables Counterexamples and validity Model-theoretic validity Soundness & completeness Soundness Completeness Proofs first or models first? Heyting algebras Modal Logic Necessity & possibility Possible worlds models Validity Strict conditionals and ambiguities Propositions Another notion of necessity Equivalence relations and epistemic logic Actuality & two-dimensional logic Actuality models and double indexing Validity Fixity and diagonal propositions Real world validity Natural deduction for modal logics Natural deduction for S4 Natural deduction for S5 Features of S5 Predicate Logic Quantifiers Syntax Natural deduction for CQ What is provable?

PHIL40013: Uncertainty, Vagueness and Disagreement

Mon, 29 Jul 2019 00:00:00 +0000

PHIL40013: Uncertainty, Vagueness and Disagreement is a University of Melbourne honours seminar subject for fourth-year students. Our aim in the Honours program is to introduce students to current work in research in philosophy of logic and language. In 2019, we’re covering the connections between speech acts, epistemology and normative theory. Introduction and overview, background Speech acts: what are they? J. L. Austin, How to Do things with Words, Clarendon Press, Oxford, 1962. [Read Lecture 9] H. P. Grice, “Logic and Conversation,” pages 41–58 in Syntax and Semantics: Speech Acts, edited by P. Cole and J. L. Morgan, Academic Press, New York, 1975. Sarah E. Murray and William B. Starr, “Force and Conversational States,” pages 202–236 in New Work on Speech Acts, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018. [Read Sections 9.1 and 9.2] Nuel Belnap “Declaratives are not Enough”, Philosophical Studies 59:1 (1990) 1–30. Mark Lance and Rebecca Kukla “Leave the Gun; Take the Cannoli! The Pragmatic Topography of Second-Person Calls” Ethics 123:3 (2013) 456–478. Mark Lance and Rebecca Kukla Yo! and Lo! The Pragmatic Topography of the Space of Reasons, Harvard University Press, 2009. [Read Chapter 1] Craige Roberts “Speech Acts in Discourse Context”, pages 317–359 in New Work on Speech Acts, edited by Daniel Fogal, Daniel Harris and Matthew Moss, Oxford University Press, 2018.

Assertions, Denials, Questions, Answers, and the Common Ground

Sat, 22 Jun 2019 12:11:45 +0100

Assertions, Denials, Questions, Answers, and the Common Ground

Tue, 18 Jun 2019 12:16:00 +0100

Abstract: In this talk, I examine interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground. With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent X ⊢ Y as enjoining us to not assert each member of X and deny each member of Y is altogether too weak to explain the inferential force of logical validity. Deriving X ⊢ A should tell us, after all, something about justifying A on the basis of X, rather than merely prohibiting A’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims. I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with justification requests — questions aiming at eliciting reasons for assertions or denials.

Assertions, Denials, Questions, Answers, and the Common Ground

Fri, 07 Jun 2019 12:16:00 -0600

Isomorphisms in a Category of Proofs

Fri, 29 Mar 2019 17:00:00 +0100

Abstract: In this talk, I show how a category of classical proofs can give rise to three different hyperintensional notions of sameness of content. One of these notions is very fine-grained, going so far as to distinguish $p$ and $p\land p$, while identifying other distinct pairs of formulas, such as $p\land q$ and $q\land p$; $p$ and $\neg\neg p$; or $\neg(p\land q)$ and $\neg p\lor\neg q$. Another relation is more coarsely grained, and gives the same account of identity of content as equivalence in Angell’s logic of analytic containment. A third notion of sameness of content is defined, which is intermediate between Angell’s and Parry’s logics of analytic containment. Along the way, we show how purely classical proof theory gives resources to define hyperintensional distinctions thought to be the domain of properly non-classical logics. This is a talk at the Third Tübingen Conference on Proof-Theoretic Semantics, 27–30 March 2019. The slides are available here, while a handout is here.

Collection Frames for Substructural Logics

Fri, 15 Mar 2019 11:00:00 -0600

Abstract: In this talk I present a new frame semantics for positive substructural and relevant propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation is the use of a single accessibility relation to relate collections of points to points. Different logics are modelled by varying the kinds of collections featuring in the relation: for example, they can be sets, multisets, lists or trees. In this talk I will focus on multiset frames, which are sound and complete for the logic RW+ (positive multiplicative and additive linear logic with distribution for the additive connectives, or equivalently, the relevant logic R+ without contraction). This is joint work with Shawn Standefer. The talk is a Melbourne Logic Seminar. The slides for the talk are available here.

PHIL30043: The Power and Limits of Logic

Tue, 05 Mar 2019 00:00:00 +0000

PHIL30043: The Power and Limits of Logic is a University of Melbourne undergraduate subject. It covers the metatheory of classical first order predicate logic, beginning at the Soundness and Completeness Theorems (proved not once but twice, first for a tableaux proof system for predicate logic, then a Hilbert proof system), through the Deduction Theorem, Compactness, Cantor’s Theorem, the Downward Löwenheim–Skolem Theorem, Recursive Functions, Register Machines, Representability and ending up at Gödel’s Incompleteness Theorems and Löb’s Theorem. Kurt Gödel, seated The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here. I make use of video lectures I have made freely available on Vimeo. Outline The course is divided into four major sections and a short prelude. Here is a list of all of the videos, in case you’d like to follow along with the content. Prelude Logical Equivalence Disjunctive Normal Form Why DNF Works Prenex Normal Form Models for Predicate Logic Trees for Predicate Logic Completeness Introducing Soundness and Completeness Soundness for Tree Proofs Completeness for Tree Proofs Hilbert Proofs for Propositional Logic Conditional Proof Hilbert Proofs for Predicate Logic Theories Soundness and Completeness for Hilbert Proofs for Predicate Logic Compactness Counting Sets Diagonalisation Compactness Non-Standard Models Inexpressibility of Finitude Downward Löwenheim–Skolem Theorem Computability Functions Register Machines Recursive Functions Register Machine computable functions are Recursive The Uncomputable Undecidability and Incompleteness Deductively Defined Theories The Finite Model Property Completeness Introducing Robinson’s Arithmetic Induction and Peano Arithmetic Representing Functions and Sets Gödel Numbering and Diagonalisation Q (and any consistent extension of Q) is undecidable, and incomplete if it’s deductively defined First Order Predicate Logic is Undecidable True Arithmetic is not Deductively Defined If Con(PA) then PA doesn’t prove Con(PA)

UNIB10002: Logic, Language and Information

Mon, 04 Mar 2019 00:00:00 +0000

Generality and Existence I: Quantification and Free Logic

Fri, 01 Mar 2019 00:00:00 +0000

In this paper, I motivate a cut free sequent calculus for classical logic with first order quantification, allowing for singular terms free of existential import. Along the way, I motivate a criterion for rules designed to answer Prior’s question about what distinguishes rules for logical concepts, like ‘conjunction’ from apparently similar rules for putative concepts like ‘tonk’, and I show that the rules for the quantifiers—and the existence predicate—satisfy that condition.

Generality and Existence 2: Modality and Quantifiers

Thu, 21 Feb 2019 08:15:00 -0600

Abstract: In this talk, I motivate and define a cut free sequent calculus for first order modal predicate logics, allowing for singular terms free of existential import. I show that the cut rule is admissible in the cut-free calculus, and explore the relationship between contingent ‘world-bound’ quantifiers and possibilist ‘world-undbound’ quantifiers in the system. This is a talk for the Association for Symbolic Logic at the Central Division meeting of the American Philosophical Association, at Denver, Colorado. The slides for the talk are available here, and a version formatted for printing, as a handout, is available here.

Summer Reading 2018-2019

Sun, 27 Jan 2019 20:49:05 +1100

This summer break, I set aside some time to turn off my devices, unplug from the internet, and read some honest-to-goodness books. Some I received from friends and family as Christmas or Birthday gifts (thanks, Sharon, Zac, Neil!), and some I had accumulated on my “to-read” pile waiting for just the right time. Here are some short reviews of my summer reading pile, in case you’d like to follow along.

New Work for a (Formal) Theory of Grounds

Fri, 14 Dec 2018 11:00:00 +1100

Abstract: In this talk, I provide two different models for a theory of grounds meeting the following desiderata:$\def\yright{\succ}$ Grammar: There are objects, which we call grounds, which can be grounds for propositions or grounds against propositions. Derivation: A derivation of a sequent $X\yright A,Y$ gives us a systematic way to construct a ground for $A$ out of grounds for each member of $X$ and grounds against each member of $Y$, and a derivation of a sequent $X,A\yright Y$ gives us a systematic way to construct a ground against $A$ out of grounds for each member of $X$ and grounds against each member of $Y$. So, a derivation of $\yright A$ gives us a way to construct a ground for $A$, and a derivation of $A\yright$ gives us a way to construct a ground against $A$. Interpretation: This theory can be interpreted in an epistemic sense, where grounds are our means to access the truth or falsity of a proposition, or a metaphysical sense, where grounds show how a proposition is made true by the world. Grasp: Grounds are the kinds of things we can possess. Hyperintensionality: Not every ground is a ground for every tautology. A ground for $A$ need not also be a ground for each logical consequence of $A$.

Truth and Stereotypes

Tue, 30 Oct 2018 19:00:00 +1100

Abstract: Our thoughts and our conversations are filled with generalisations. From everyday trivialities such as birds fly or trams are crowded to contested claims such as women are oppressed or Muslims are peace-loving, we think and communicate using generalisations and stereotypes. This way of understanding the world is useful and pervasive, but at the same time, it has significant limitations. In this lecture, I will explain some of the surprising features of these generalisations. Then I’ll apply some of the tools developed by philosophers of language over the last decades, in order to understand why generalisations and stereotypes are so pervasive; why they can behave so strangely and can sometimes lead us astray; and finally, to learn how we can use generalisations and stereotypes productively in our thinking and our communication. This is a free public lecture at the University of Melbourne, held at 7pm in the Kathleen Fitzpatrick Lecture Theatre (Arts West). Although it’s free, it’s a good idea to book tickets. The slides for the talk are available here.

Philosophy in Public

Sun, 28 Oct 2018 13:34:27 +1100

Last Wednesday, I went down to the studios at ABC Southbank, to be interviewed by Libbi Gorr for ABC Radio Melbourne’s Sunday program. As I made my way through the building, and settled into the little studio, I thought I heard a familiar voice, faintly in the distance. Libbi explained that this was Kevin Rudd (the former Prime Minister), who was being interviewed in the next room. Unlike the former PM, I wasn’t doing the rounds of media because I had a book to promote. But I was doing promotion in my own small way. The University of Melbourne’s Media Office does a good job at getting the word out about public lectures, and the description for the lecture I’m giving on Tuesday night apparently appealed to Libbi Gorr and her producer, and they thought it would be fun to interview me for the Sunday program, so on Tuesday midday, I get an email from the Media Office asking if I’d be up for an interview in the studio with Libbi, talking about Truth and Stereotypes. So, that’s why I found myself in the studio having a fun 20 minute conversation with Libbi about stereotypes, thoughts, language and communication, and the possibility of objectivity and agreement (or disagreement) when we’re all situated in different places and have different perspectives.

With help from Hugo, GitHub, Netlify, Working Copy and Shortcuts, I might update this website more frequently

Sun, 21 Oct 2018 16:39:00 +1100

If you’ve been following my travels, you’ll get some sense that this has been a busy year. I’ve done lots of writing on my book, and I’ve managed to give lots of talks, both in the US and in Argentina, as well as at home. I haven’t posted here for nearly a year–writing elsewhere has been a higher priority. However, this weekend, I’ve made a few changes to the website which means that I might post here a little more often. The site is produced by Hugo, a really sweet static site generator. Until yesterday, if I wanted to update my site, what I did was Write files on whatever device I was using–most probably my Mac, but maybe my iPad–and push them to my my Git repository, which contains the source for the whole website. Then, on my Mac, sync up the repository. Run hugo to update the generated files. Sync the result up to GitHub. That worked fine, but I needed to do steps 2–4 on my Mac, and I don’t always have my Mac with me. Sometimes I prefer to write on my (smaller, less fiddly an distracting) iPad, and sometimes I only have my phone with me, and it’d be nice to update files on the website without having to run through my Mac to do that.

Accommodation, Inference, Generics and Pejoratives

Fri, 12 Oct 2018 10:00:00 +1100

Abstract: In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Given the connection between generics and inference, I’ll go on to consider how inference relates to the process of accommodation, which plays a significant role in how we manage dialogue and conversation. This, in turn, helps shed some light on some different ways expressions can involve pejorative force, and can inform options for how our vocabulary and our concepts can be revised or reformed. This is a talk for a workshiop on Social Ontology at the University of Melbourne. The slides for the talk are available here, and the handout is here.

Accommodation, Inference, Generics and Pejoratives

Fri, 05 Oct 2018 15:00:00 +1100

Abstract: In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Given the connection between generics and inference, I’ll go on to consider how inference relates to the process of accommodation, which plays a significant role in how we manage dialogue and conversation. This, in turn, helps shed some light on some different ways expressions can involve pejorative force, and can inform options for how our vocabulary and our concepts can be revised or reformed. This is a talk for the University of Queensland Philosophy Seminar Series (3pm-5pm, Fridays). The slides for the talk are available here, and the handout is here.

Defining Rules, Proofs and Counterexamples

Fri, 03 Aug 2018 17:45:00 -0300

Abstract: In this talk, I will present an account of defining rules, with the aim of explaining these rules they play a central role in analytic proofs. Along the way, I’ll explain how Kreisel’s squeezing argument helps us understand the connection between an informal notion of validity and the notions formalised in our accounts of proofs and models, and the relationship between proof-theoretic and model- theoretic analyses of logical consequence. This is a talk for the VII Workshop on Philosophical Logic. The slides for the talk are available here, and the handout is here.

Proof Theory, Rules and Meaning — an introduction

Mon, 30 Jul 2018 16:00:00 -0300

Abstract: I introduce the key themes from my book-in-progress, Proof Theory, Rules and Meaning. This is a talk for the symposium on the manuscript held at the Argentinean Society of Philosophical Analysis (SADAF) in Buenos Aires, in July 2018. The slides for the talk are available here.

PHIL30043: The Power and Limits of Logic

Wed, 25 Jul 2018 00:00:00 +0000

PHIL20030: Meaning, Possibility and Paradox

Tue, 24 Jul 2018 00:00:00 +0000

PHIL20030: Meaning, Possibility and Paradox is a University of Melbourne undergraduate subject. The idea that the meaning of a sentence depends on the meanings of its parts is fundamental to the way we understand logic, language and the mind. In this subject, we look at the different ways that this idea has been applied in logic throughout the 20th Century and into the present day. In the first part of the subject, our focus is on the concepts of necessity and possibility, and the way that ‘possible worlds semantics’ has been used in theories of meaning. We will focus on the logic of necessity and possibility (modal logic), times (temporal logic), conditionality and dependence (counterfactuals), and the notions of analyticity and a priority so important to much of philosophy. In the second part of the subject, we examine closely the assumption that every statement we make is either true or false but not both. We will examine the paradoxes of truth (like the so-called ‘liar paradox’) and vagueness (the ‘sorites paradox’), and we will investigate different ways attempts at resolving these paradoxes by going beyond our traditional views of truth (using ‘many valued logics’) or by defending the traditional perspective.

What Proofs are For

Tue, 12 Jun 2018 10:00:00 +1100

Abstract: In this short talk, I present a new account of the nature of proof, with the aim of explaining how proof could actually play the role in reasoning that it does, and answering some long-standing puzzles about the nature of proof. Along the way, I’ll explain how Kreisel’s Squeezing argument helps us understand the connection between an informal notion of of validity and the notions formalised in our accounts of proofs and models, and the relationship between proof-theoretic and model-theoretic analyses of logical consequence. This is a talk for the Melbourne–Glasgow Formal Philosophy Workshop. The slides for the talk are available here, and the handout is here.

Isomorphisms in a Category of Proofs

Fri, 20 Apr 2018 14:00:00 -0400

Abstract: In this talk, I show how a category of formulas and classical proofs can give rise to three different hyperintensional notions of sameness of content. One of these notions is very fine-grained, going so far as to distinguish $p$ and $p\land p$, while identifying other distinct pairs of formulas, such as $p\land q$ and $q\land p$; $p$ and $\neg\neg p$; or $\neg(p\land q)$ and $\neg p\lor\neg q$. Another relation is more coarsely grained, and gives the same account of identity of content as equivalence in Angell’s logic of analytic containment. A third notion of sameness of content is defined, which is intermediate between Angell’s and Parry’s logics of analytic containment. Along the way we show how purely classical proof theory gives resources to define hyperintensional distinctions thought to be the domain of properly non-classical logics. This is a talk for the MIT SLLERG Group. The slides can be downloaded here, but the handout (4 pages) is best for printing out and reading, so it’s probably better that you download and print that.

Accommodation, Inference, Generics and Pejoratives

Thu, 19 Apr 2018 12:00:00 -0400

Abstract: In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Given the connection between generics and inference, I’ll go on to consider how inference relates to the process of accommodation, which plays a significant role in how we manage dialogue and conversation. This, in turn, helps shed some light on some different ways expressions can involve pejorative force, and can inform options for how our vocabulary and our concepts can be revised or reformed. This is a talk for the MIT Philosophy Work in Progress series (1pm-2pm, Thursdays). The slides for the talk are available here, and the handout is here.

Accommodation, Inference, Generics and Pejoratives

Wed, 18 Apr 2018 12:15:00 -0400

Abstract: In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Given the connection between generics and inference, I’ll go on to consider how inference relates to the process of accommodation, which plays a significant role in how we manage dialogue and conversation. This, in turn, helps shed some light on some different ways expressions can involve pejorative force, and can inform options for how our vocabulary and our concepts can be revised or reformed. This is a talk for the University of Connecticut Philosophy Brown Bag. The slides for the talk are available here, and the handout is here.

Accommodation, Inference, Generics and Pejoratives

Fri, 13 Apr 2018 15:30:00 -0400

Abstract: In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Given the connection between generics and inference, I’ll go on to consider how inference relates to the process of accommodation, which plays a significant role in how we manage dialogue and conversation. This, in turn, helps shed some light on some different ways expressions can involve pejorative force, and can inform options for how our vocabulary and our concepts can be revised or reformed. This is a talk for the University of Pittsburgh Philosophy Colloquium. The slides for the talk are available here, and the handout is here.

Isomorphisms in a Category of Proofs

Thu, 12 Apr 2018 16:30:00 -0400

Abstract: In this talk, I show how a category of formulas and classical proofs can give rise to three different hyperintensional notions of sameness of content. One of these notions is very fine-grained, going so far as to distinguish $p$ and $p\land p$, while identifying other distinct pairs of formulas, such as $p\land q$ and $q\land p$; $p$ and $\neg\neg p$; or $\neg(p\land q)$ and $\neg p\lor\neg q$. Another relation is more coarsely grained, and gives the same account of identity of content as equivalence in Angell’s logic of analytic containment. A third notion of sameness of content is defined, which is intermediate between Angell’s and Parry’s logics of analytic containment. Along the way we show how purely classical proof theory gives resources to define hyperintensional distinctions thought to be the domain of properly non-classical logics. This is a talk for the CMU Pure and Applied Logic Seminar Series. The slides can be downloaded here, but the handout (4 pages) is best for printing out and reading, so it’s probably better that you download and print that.

Isomorphisms in a Category of Proofs

Mon, 09 Apr 2018 16:15:00 -0400

Abstract: In this talk, I show how a category of classical proofs can give rise to three different hyperintensional notions of sameness of content. One of these notions is very fine-grained, going so far as to distinguish $p$ and $p\land p$, while identifying other distinct pairs of formulas, such as $p\land q$ and $q\land p$; $p$ and $\neg\neg p$; or $\neg(p\land q)$ and $\neg p\lor\neg q$. Another relation is more coarsely grained, and gives the same account of identity of content as equivalence in Angell’s logic of analytic containment. A third notion of sameness of content is defined, which is intermediate between Angell’s and Parry’s logics of analytic containment. Along the way we show how purely classical proof theory gives resources to define hyperintensional distinctions thought to be the domain of properly non-classical logics. This is a talk for the CUNY Graduate Center Logic and Metaphysics Seminar. The slides can be downloaded here, but the handout (4 pages) is best for printing out and reading, so it’s probably better that you download and print that.

What Proofs are For

Fri, 06 Apr 2018 16:15:00 -0400

Abstract: In this short talk, I present a new account of the nature of proof, with the aim of explaining how proof could actually play the role in reasoning that it does, and answering some long-standing puzzles about the nature of proof. Along the way, I’ll explain how Kreisel’s Squeezing argument helps us understand the connection between an informal notion of of validity and the notions formalised in our accounts of proofs and models, and the relationship between proof-theoretic and model-theoretic analyses of logical consequence. This is a talk for the NYU Philosophy Department Brown Bag Series. The slides for the talk are available here, and the handout is here.

Accommodation, Inference, Generics and Pejoratives

Wed, 28 Mar 2018 12:00:00 -0400

Abstract: In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Given the connection between generics and inference, I’ll go on to consider how inference relates to the process of accommodation, which plays a significant role in how we manage dialogue and conversation. This, in turn, helps shed some light on some different ways expressions can involve pejorative force, and can inform options for how our vocabulary and our concepts can be revised or reformed. This is a talk for the CUNY Graduate Center Philosophy Colloquium. The slides for the talk are available here, and the handout is here.

Accommodation, Inference, Generics and Pejoratives

Thu, 22 Mar 2018 16:15:00 +1100

Abstract: In this talk, I aim to give an account of norms governing our uses of generic judgements (like “kangaroos have long tails”, “birds lay eggs”, or “logic talks are boring”), norms governing inference, and the relationship between generics and inference. This connection goes some way to explain why generics exhibit some very strange behaviour: Why is it, for example, that “birds lay eggs” seems true, while “birds are female” seems false, despite the fact that only female birds lay eggs? Given the connection between generics and inference, I’ll go on to consider how inference relates to the process of accommodation, which plays a significant role in how we manage dialogue and conversation. This, in turn, helps shed some light on some different ways expressions can involve pejorative force, and can inform options for how our vocabulary and our concepts can be revised or reformed. This is a talk for the University of Melbourne Thursday Philosophy Seminar. The slides for the talk are available here, and the handout is here.

Truth Tellers in Bradwardine's Theory of Truth

Wed, 21 Mar 2018 00:00:00 +0000

Stephen Read’s work on Bradwardine’s theory of truth is some of the most exciting work on truth and insolubilia in recent years. Read brings together modern tools of formal logic and Bradwardine’s theory of signification to show that medieval distinctions can give great insight into the behaviour of semantic concepts such as truth. In a number of papers, I have developed a model theory for Bradwardine’s account of truth. This model theory has distinctive features: it serves up models in which every declarative object (any object signifying anything) signifies its own truth. This leads to a puzzle: there are good arguments to the effect that if anything is a truth-teller, it is false. This is a puzzle. What distinguishes paradoxical truth-tellers from benign truth tellers? It is my task in this paper to explain this distinction, and to clarify the behaviour of truth-tellers, given Bradwardine’s account of signification.

Isomorphisms in a Category of Propositions and Proofs

Fri, 02 Mar 2018 11:00:00 +1100

Abstract: In this talk, I show how a category of propositions and classical proofs can give rise to three different hyperintensional notions of sameness of content. One of these notions is very fine-grained, going so far as to distinguish $p$ and $p\land p$, while identifying other distinct pairs of formulas, such as $p\land q$ and $q\land p$; $p$ and $\neg\neg p$; or $\neg(p\land q)$ and $\neg p\lor\neg q$. Another relation is more coarsely grained, and gives the same account of identity of content as equivalence in Angell’s logic of analytic containment. A third notion of sameness of content is defined, which is intermediate between Angell’s and Parry’s logics of analytic containment. Along the way we show how purely classical proof theory gives resources to define hyperintensional distinctions thought to be the domain of properly non-classical logics. This is a talk for the Melbourne Logic Seminar. The slides of the talk are available here, and the handout is here.

Community (the twelfth of twelve things I love about philosophical logic)

Sat, 30 Dec 2017 17:12:57 +1100

I love the way I’ve met so many different people through working in logic, that I’ve made good friends, good colleagues, good teachers and mentors. I’ve been part of an enterprise that’s larger than any one person. I have been shaped by that community, and have had the opportunity to made some small mark on it myself. Logic, like any other academic discipline, has a history. The activities of doing logic — of studying, researching and teaching — are spread out through time. Those activities are also, equally, spread out in space. Logic is done in many different places, in many different contexts, by many different individuals, and research teams. As I’ve already pointed out, those activities are shaped by different disciplinary connections (into philosophy, mathematics, computer science, linguistics, engineering, etc.), but they are also shaped by the emphases of different research groups and traditions. Research, these days, is dominated not so much by a small number of iconic logicians, but by research groups with distinctive research programmes. Here are some examples: think of the difference between Amsterdam-style modal logic, on the one hand, and exact truthmaking semantics on the other, concerning approaches to intensionality and hyperintensionality, or for examples on the proof theoretical side of the street, the different traditions of Higher Order Type Theory and of Linear Logic and Ludics as frameworks for understanding proof, computation and meaning.

Proof Identity, Aboutness and Meaning

Sat, 11 Nov 2017 09:30:00 +0000

Abstract: This talk is a comparison of how different approaches to hyperintensionality, aboutness and subject matter treat (classically) logically equivalent statements. I compare and contrast two different notions of subject matter that might be thought to be representational or truth first – Aboutness (Princeton University Press, 2014), and truthmakers conceived of as situations, as discussed in my “Truthmakers, Entailment and Necessity.” I contrast this with the kind of inferentialist account of hyperintensionality arising out of the proof invariants I have explored in recent work. This is a talk presented at the Glasgow-Melbourne Formal Philosophy Workshop. The slides are available here.

Negation on the Australian Plan

Fri, 22 Sep 2017 11:00:00 +1100

Abstract: In this talk, I explain the difference between Australian Plan semantics for negation – which treat negation as a kind of negative modality – and semantics based on the American Plan, which conceive of negation in terms of independent truth and falsity conditions. I will update the presentation of the Australian Plan (introduced in the 1970s in early days of the ternary relational semantics for relevant logics), in the light of more recent developments in logic, and defend this updated plan in the face of some recent criticisms due to Michael De and Hitoshi Omori, in their paper “There is More to Negation than Modality.” Along the way, I hope to draw out some insights into what we might want out of a representational semantics for a language with a consequence relation. This talk is based on joint work with Professor Franz Berto, from the University of Amsterdam. This is a talk presented at the Melbourne Logic Seminar. The slides are available here.

Learning and Teaching (the eleventh of twelve things I love about philosophical logic)

Tue, 19 Sep 2017 22:47:48 +1100

Working in philosophical logic, I love the opportunity to learn from so many people through history, and not only to learn, but to pass on a tradition, and to have the opportunity to extend the tradition, and to refine it a little, in passing it on. It’s been a delight to learn from some great figures, the historical figures through their writing, and my contemporaries in person, both as face-to-face teachers (while a student, I learned logic from Sheila Oates-Williams, Neil Williams, Rod Girle, Ian Hinckfuss, and Graham Priest), but the learning doesn’t stop when you finish your degree. I’ve learned much from colleagues (Bob Meyer, Richard Sylvan, John Slaney, Allen Hazen, Graham Priest (again), Zach Weber, Dave Ripley, Shawn Standefer), whose work I admire, and who generously share of their time at whiteboards, in seminars, and in many many conversations. I have also learned a great deal from all of my graduate students, who have sent me in directions I never expected to head. If learning logic is (in part) gaining facility with the concepts you have, as well as acquiring new concepts, then working with others is a very good way to learn. The kinds of knowledge you acquire is not merely knowing that, but it (at least in part) skills to be learned, and they’re learned by practice, and sometimes skilled practice is best acquired when guided by others — explicitly, when the teacher knows she is teaching — or implicitly, when we observe an expert displaying expertise, and we can use their practice to scaffold our own, and learn by imitation.

Possibility (the tenth of twelve things I love about philosophical logic)

Mon, 18 Sep 2017 14:38:20 +1100

In the previous entry I explored the connection between proofs and necessity. Here, I want to spend a little time exploring the other side of the logical street, the connection between models and possibility. As I have already explained, one core insight from 20th Century work in logic is the fundamental duality between proof theory and model theory. You can define logical notions like validity by way of proofs (a valid argument is certified by the existence of some proof) or by way of models (an argument is shown to be invalid by the existence of some model which serves as a counterexample). Exploring proofs gives you can account of the different ways that concepts are tied together. (It gives you an account of what is involved in different kinds of necessary connections. The more different proofs you have, the more connections are possible.) Approaching the validity/invalidity boundary on the other side, by way of models, gives you a very different picture of this boundary. Defining more models means having more counterexamples. Model building is one very fruitful way of articulating what is — and more importantly, what isn’t — a part of a theory. Suppose you are interested in some strange new theory.

Necessity (the ninth of twelve things I love about philosophical logic)

Thu, 14 Sep 2017 23:55:19 +1100

The next two thoughts are motivated by the two complementary aspects of contemporary research in logic, proof theory and model theory. As I try to emphasise to my students, there are two broad ways you can define logical concepts like validity. Following the way of proofs, an argument is valid if there is some proof leading from the premises to the conclusion. Following the way of models, an argument is valid if there is no model in which the premises are true and the conclusion is not. In proof theory, validity is vouchsafed by the existence of something: a proof, which certifies the claim to validity. Invalidity is the absence of such a certificate. In model theory, invalidity is vouchsafed by the existence of something: a model — a counterexample to the claim of validity. Validity is the absence of any such counterexample. It was a great intellectual advance to understand that these are two very different ways to define logical concepts, such as validity, and it was a further advance to be able to rigorously prove that (on certain understandings of logic, such as classical first order predicate logic), these two different kinds of definitions can coincide to determine the same concept.

Attention (the eighth of twelve things I love about philosophical logic)

Wed, 13 Sep 2017 14:01:56 +1100

I’m not totally happy with the word for the next item on the list of twelve things I love about philosophical logic. The word on the list is attention, and it gets at something that I have learned, and which seems to me to be an important distinctive about working in philosophical logic, but I’m not altogether sure that “attention” is the best word for it. Maybe after I’ve explained what I mean, you could suggest a better short label for the phenomenon I’m gesturing towards. Here’s the core idea: when you spend time working with core logical notions such as consequence, consistency, necessity, possibility, model and proof, you notice that you are attending to judgements and thoughts and claims in more than one way. You learn to distinguish between taking a claim to be true, and considering it as possible. You can agree that even though $p$ isn’t true, it is consistent with $q$. You can agree that it’s not true that $p$ while still seriously entertain what it would be like were $p$ to be true. Working with $p$ as an hypothesis is not the same thing as taking it to be true. But even though working under the supposition that $p$ is not the same thing as taking $p$ to be true, it is related intimately to it.

Pragmatics (the seventh of twelve things I love about philosophical logic)

Tue, 12 Sep 2017 12:53:42 +1100

Some of my phrasing in the last two posts about what I love about philosophical logic have emphasised capacities, or abilities. I’ve described the pleasure of the “aha!” moment in terms of the kinds of mastery you acquire in handling the concepts you have, and I described the joys of conceptual expansion in terms of abilities gained. This is to take a pragmatic perspective on logic, to consider the connection to practices and actions. Thinking of things “pragmatically” can be understood in a very crude way, fixing in advance what how you want to measure costs and benefits, and then doing some naïve cost/benefit calculation and then choosing option that somehow maximises benefits and minimises the costs (if that is even possible). This is not what I mean when I consider the connection between logic and pragmatics. I don’t think that the best way to select some logical system or logical theory is on the basis of that kind of cost/benefit analysis. Rather, it’s that there are connections between features of logical systems and the practices of asserting, denying, inferring, questioning, etc. What kind of connections are there? It’s not that the laws of logic are descriptively correct as a theory about how assertion and denial and inference actually work in practice.

Conceptual Expansion (the sixth of twelve things I love about philosophical logic)

Mon, 11 Sep 2017 09:26:50 +1100

There are different delights to be found in working with concepts. It is not all a matter of gaining greater mastery of the concepts you have already acquired. There is also a special delight to be found in acquiring new concepts. I love that feeling of progress when you make a conceptual advance. A common way to do this is to disambiguate, to clarify matters by noticing that what you took to be one thing is really two. This is the clarity gained in uncovering a hidden confusion, the moment when ideas are sharpened and distinguished, when you form a new vocabulary and, as a result, you are able to say things you couldn’t express before. This one way to reap the rewards of conceptual expansion. Our repertoire of concepts is larger than it was. Here’s an example of this phenomenon. (It’s not an uncontroversial example, but it’s one that I find quite compelling.) Consider what it means to suppose something. This is something we do regularly when we’re reasoning, when we’re planning, considering options, or discussing something with people who have different views. Often we “try a claim on for size”, suppose it’s the case, and reason from there. (This kind of dialectical move is something we not only do, it’s also at the heart of different accounts of the structure of proof.

The Moment of Recognition (the fifth of twelve things I love about philosophical logic)

Sat, 09 Sep 2017 14:29:39 +1100

Here is a more personal reflection on what I love in working in philosophical logic. I love the “aha!” moment of recognition. This is the relief of a proof completed, or a counterexample found. It is the delight of gaining clarity into something that you had only dimly understood, or the dawning realisation that an assumption you had made is in fact false and a whole new vista of possibilities opens up to you. The particular kind of “aha” that I mean is the kind where you’re working out of the consequences of something you already know. This can be understood as a kind of mastery that is gained when you become familiar with the conceptual tools you’re using. It is the acquisition of greater skill. This is the “aha” that students in my second year logic class experienced when they figured out for themselves that not all symmetric and transitive relations must be reflexive. In one sense, they already knew the definitions of these concepts (at least, most of them did) and this fact was implicit in what they already knew, but now they had figured this out for themselves — they saw it for themselves. They understood something new about how these concepts fit together, how they relate.

Interdisciplinarity (the fourth of twelve things I love about philosophical logic)

Fri, 08 Sep 2017 14:27:05 +1100

The multiple realisations of a concept in logic often come from different disciplines. One thing I’ve grown to love in philosophical logic is the way different ideas, disciplines and traditions are connected in the space of the wider generality of formal logic. In my own work over the years in substructural logic, logical pluralism and proof theory (among other things), traditions in computer science, linguistics, mathematics and philosophy have all played distinct roles. Each discipline has its own examples, its own traditions, its own heroes, its own villains—and its own concerns. If you are aware of the distinctive features of different traditions, this allows for the strengths of those disciplines to shine, for the insights and examples of one discipline to be brought to bear on the questions and problems of others. If you’re a philosopher, you should, by nature, be interested in more than your own traditions—or at least you should if you understand philosophy in the way Wilfrid Sellars did: The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term. — Wilfrid Sellars, “Philosophy and the Scientific Image of Man”

Multiple Realisability (the third of twelve things I love about philosophical logic)

Thu, 07 Sep 2017 15:34:51 +1100

Closely connected to the notion of abstraction, I love the way that logical concepts are multiply realisable. An abstract structure can be instantiated in different ways, and often in ways completely unforeseen when the original abstraction was made. The twin moves of abstraction (moving from the particular to the general) and concretisation (going back from the general to the particular—perhaps to a new and different particular) in different domains brings different insights, different models, and new connections. These new connections often bring fresh insight. For example, the simple notion of a Boolean algebra can be instantiated as a power set algebra (think of the subsets of a set and the operations of union, intersection and complementation). But this simple idea of a power set Boolean algebra can then be understood in different domains of application: you can think of the underlying set as a domain of objects and the subsets are extensions of different predicates. Or you can think of them as a set of possible worlds, and the subsets are propositions. And so on. Shifting from one representation to another is often conceptually fruitful when thinking about these different domains, or thinking about the ways that the formal techniques are applied.

Abstraction (the second of twelve things I love about philosophical logic)

Wed, 06 Sep 2017 13:00:03 +1100

As I mentioned in the previous entry, philosophical logic uses the tools and techniques from formal logic, and formal logic is nothing if it is not abstract. It gets its power — as well as its weaknesses, to be sure — by abstracting away from specifics and moving to generalities. We explain the virtues of a particular argument (in part) by looking at its form, the structure which is in common to other arguments of the same shape. This goes back, at least, to Aristotle, who taught us that it isn’t a coincidence that both syllogisms All footballers are bipeds. All bipeds have feet. Therefore all footballers have feet. All wombats are cute. All cute things are popular. Therefore all wombats are popular. have similar virtues. At the very least, they’re both valid. They both have the form: All Fs are Gs. All Gs are Hs. Therefore all Fs are Hs. and any syllogisms with that form are valid. Attending to the shape of the reasoning, and “tuning out” concern about whether the premises are true (are all wombats cute? Are all footballers bipeds? — most likely not) and focussing on the form, we see how the premises and conclusions are connected.

The Dialectic (the first of twelve things I love about philosophical logic)

Tue, 05 Sep 2017 13:36:03 +1100

One thing I noticed when making my way from mathematics (my undergraduate degree was a B.Sc. in Mathematics at the University of Queensland) to philosophy was the different approach when doing research in the two disciplines. To put it very coarsely, in mathematics, you prove theorems. In philosophy, you argue about things. The standards for success are very different in philosophy and in mathematics. Witness Norbert Blum’s recent retraction of his paper which purported to prove that P ≠ NP. While philosophers change their minds about things, I don’t recall anyone going so far as to retract a paper that argued for a position they now reject. That’s just not how philosophers work, and nor should they. One of the joys about working in philosophical logic — especially for someone with a relatively short attention span, like me — is that I get to play on both sides of this street. I spend some time as a technical mathematical logician, playing the theorem-proving game, with all of the satisfaction of knowing that I’ve really proved something solid in its own way: a mathematical result. On the other hand, there’s more to life than theorems, and there’s more to understanding than the making of proofs.

Twelve things I love about philosophical logic

Mon, 04 Sep 2017 20:21:54 +1100

Over this last weekend, I spent some time tidying out one of the electronic “junk drawers” of my writing life, a folder full of thousands upon thousands of little scraps of text, ranging from minutes of meetings, recipes I’ve saved, little ideas I came across which I wanted to save, lists of places to visit when travelling, and many other kinds of digital flotsam and jetsam I’ve collected over around 20 years of being online, reading and writing. There was a lot of junk in that big pile of text that I deleted on sight (though there were a few recipes I’m looking forward to trying out in the next little while) but one thing really surprised me. It was a short list, entitled “12 things I love about philosophical logic”. That scrap of writing was about 200 words—the “12 things” are each elaborated with only a sentence or two. I wrote it about five years ago, and I’d totally forgotten about it until coming across it this weekend. Rereading it, the ideas resonated. (My views haven’t shifted that much over five years.) What resonated wasn’t just that I agreed with my earlier self—but that I found the thoughts helpful, and they struck me as the kind of thing that you don’t often hear.

PHIL20030: Meaning, Possibility and Paradox

Tue, 25 Jul 2017 00:00:00 +0000

PHIL40013: Uncertainty, Vagueness and Disagreement

Tue, 25 Jul 2017 00:00:00 +0000

PHIL40013: Uncertainty, Vagueness and Paradox is a University of Melbourne honours seminar subject for fourth-year students. Our aim in the Honours program is to introduce students to current work in research in philosophical logic. Assertions and denials take a stand on something. In 2017, we’re covering the connections between proof theory and philosophy. Here’s the reading list, if you’re interested in following along. Introduction and Overview, Background Introduction to Inferentialism Robert Brandom, Articulating Reasons: an introduction to inferentialism, Harvard University Press, 2000. Introduction and Chapter 1 “Semantic Inferentialism and Logical Expressivism.” The Tonk Debate Arthur Prior, “The Runabout Inference-Ticket”, Analysis 21:2 (1960) 38–39. J. T. Stevenson, “Roundabout the Runabout Inference-Ticket”, Analysis 21:6 (1961) 124–128. Nuel D. Belnap, “Tonk, Plonk and Plink”, Analysis 22:6 (1962) 130–134. Natural Deduction and Normalisation Greg Restall, Proof Theory and Philosophy Draft, Chapter 1. Dag Prawitz, Natural Deduction: A Proof Theoretical Study, Almqvist and Wiksell, 1965. Chapters 1-4. Harmony and Meaning Prawitz “On the Idea of a General Proof Theory”, Synthese 27 (1974) 63–77. Michael Dummett: The Logical Basis of Metaphysics, Harvard University Press, 1991. Chapter 9 “Circularity, Consistency and Harmony” Gillian Russell, “The Justification of the Basic Laws of Logic,” Journal of Philosophical Logic 44:6 (2015) 793–803.

Conditionals in Closed Set Logic

Sat, 22 Jul 2017 14:23:43 +1100

Over the last couple of days on Twitter, I was involved in a thread, kicked off by Dan Piponi, discussing closed set logic—the natural dual of intuitionistic logic in which the law of the excluded middle holds but the law of non-contradiction fails, and which has models in the closed sets of any topological space, as opposed to the open sets, which model intuitionistic logic.

$\def\ydash{\succ}$This logic also has a nice sequent calculus in which sequents have one premise (or zero) and multiple conclusions. In the thread I made the claim that this is a natural and beautiful sequent calculus (it is!) but that the structure of the sequents means that the logic doesn’t have a natural conditional. The dual to the conditional (subtraction) can be defined, for which $A\ydash B\lor C$ if and only if $A-B\ydash C$. But the traditional conditional rules don’t work so well.

I realised, when I thought about it a bit more, that this fact is something I’ve just believed for the last 20 years or so, but I’ve never seen written down, so now is as good as a time, and here is as good as a place as any to explain what I mean.

Typesetting Flow Graphs with tikz

Tue, 11 Jul 2017 11:17:18 +1100

In a few recent papers and talks, I’ve been using flow graphs to display the flow of information in proofs. These are the kinds of things that are easy to draw, but they’re not so straightforward to typeset.

A Concrete Category of Classical Proofs

Wed, 28 Jun 2017 15:00:00 +0200

Abstract: I show that the cut-free proof terms defined in my paper “Proof Terms for Classical Derivations” form a well-behaved category. I show that the category is not Cartesian—and that we’d be wrong to expect it to be. (It has no products or coproducts, nor any initial or final objects. Nonetheless, it is quite well behaved.) I show that the term category is star autonomous (so it fits well within the family of categories for multiplicative linear logic), with internal monoids and comonoids taking care of weakening and contraction. The category is enriched in the category of semilattices, as proofs are closed under the blend rule (also called mix in the literature). This is an invited address, for TACL 2017, in Prague. The slides are available here.

A Category of Classical Proofs

Fri, 19 May 2017 12:00:00 +1100

Abstract: I show that the cut-free proof terms defined in my paper “Proof Terms for Classical Derivations” form a well-behaved category. The talk is intended to be accessible enough for those who don’t know any category theory to follow along. I show that the category is not Cartesian – and that we’d be wrong to expect to be. It has no products or coproducts, nor any initial or final objects. Nonetheless, it is quite well behaved. I show that the term category is star autonomous (so it fits well within the family of categories for multiplicative linear logic), with internal monoids and comonoids taking care of weakening and contraction. The category is enriched in the category of semilattices, as proofs are closed under the blend rule (also called “mix” in the literature). This is a talk presented at the Melbourne Logic Seminar. The slides are available here.

Proof Identity, Invariants and Hyperintensionality

Tue, 07 Mar 2017 14:15:00 +0100

Abstract: This talk is a comparison of how different approaches to hyperintensionality, aboutness and subject matter treat (classically) logically equivalent statements. I compare and contrast two different notions of subject matter that might be thought to be representational or truth first – Aboutness (Princeton University Press, 2014), and truthmakers conceived of as situations, as discussed in my “Truthmakers, Entailment and Necessity.” I contrast this with the kind of inferentialist account of hyperintensionality arising out of the proof invariants I have explored in recent work. This is a talk presented at the Hyperintensionality Afternoon, held by Francesco Berto’s project on the Logic of Conceivability. The slides are available here.

Proof Terms for Classical Derivations

Mon, 06 Mar 2017 16:00:00 +0100

Abstract: I give an account of proof terms for derivations in a sequent calculus for classical propositional logic. The term for a derivation $\delta$ of a sequent $\Sigma \succ\Delta$ encodes how the premises $\Sigma$ and conclusions $\Delta$ are related in $\delta$. This encoding is many–to–one in the sense that different derivations can have the same proof term, since different derivations may be different ways of representing the same underlying connection between premises and conclusions. However, not all proof terms for a sequent $\Sigma\succ\Delta$ are the same. There may be different ways to connect those premises and conclusions. Proof terms can be simplified in a process corresponding to the elimination of cut inferences in sequent derivations. However, unlike cut elimination in the sequent calculus, each proof term has a unique normal form (from which all cuts have been eliminated) and it is straightforward to show that term reduction is strongly normalising—every reduction process terminates in that unique normal form. Further- more, proof terms are invariants for sequent derivations in a strong sense—two derivations $\delta_1$ and $\delta_2$ have the same proof term if and only if some permutation of derivation steps sends $\delta_1$ to $\delta_2$ (given a relatively natural class of permutations of derivations in the sequent calculus).

Logical Pluralism: Meaning, Rules and Counterexamples

Fri, 03 Mar 2017 17:40:00 +0100

Abstract: I attempt to give a pluralist and syntax-independent account of classical and constructive proof, grounded in univocal rules for evaluating assertions and denials for judgments featuring the logical connectives, interpretable as governing warrants for and against claims, and which results in an interpretation of classical and constructive counterexamples to invalid arguments. This is a talk presented at the [Pluralisms Workshop](Logical Pluralism: Meaning, Rules and Counterexamples), hosted at the University of Bonn, March 2-4, 2017. The slides are available here.

Fixed Point Models for Theories of Properties and Classes

Tue, 28 Feb 2017 00:00:00 +0000

There is a vibrant (but minority) community among philosophical logicians seeking to resolve the paradoxes of classes, properties and truth by way of adopting some non-classical logic in which trivialising paradoxical arguments are not valid. There is also a long tradition in theoretical computer science–going back to Dana Scott’s fixed point model construction for the untyped lambda-calculus–of models allowing for fixed points. In this paper, I will bring these traditions closer together, to show how these model constructions can shed light on what we could hope for in a non-trivial model of a theory for classes, properties or truth featuring fixed points.

PHIL30043: The Power and Limits of Logic

Mon, 27 Feb 2017 00:00:00 +0000

UNIB10002: Logic, Language and Information

Mon, 27 Feb 2017 00:00:00 +0000

UNIB10002: Logic, Language and Information is a University of Melbourne undergraduate breadth subject, introducing logic and its applications to students from a broad range of disciplines in the Arts, Sciences and Engineering. I coordinate this subject with my colleagues Dr. Shawn Standefer, with help from Prof. Lesley Stirling (Linguistics), Dr. Peter Schachte (Computer Science) and Dr. Daniel Murfet (Mathematics). The subject is taught to University of Melbourne undergraduate students. Details for enrolment are here. We teach this in a ‘flipped classroom’ model, using resources from our Coursera subjects Logic 1 and Logic 2.

First Degree Entailment, Symmetry and Paradox

Thu, 23 Feb 2017 00:00:00 +0000

Here is a puzzle, which I learned from Terence Parsons in his paper “True Contradictions”. First Degree Entailment (FDE) is a logic which allows for truth value gaps as well as truth value gluts. If you are agnostic between assigning paradoxical sentences gaps and gluts (and there seems to be no very good reason to prefer gaps over gluts or gluts over gaps if you are happy with FDE), then this looks no different, in effect, from assigning them a gap value? After all, on both views you end up with a theory that doesn’t commit you to the paradoxical sentence or its negation. How is the FDE theory any different from the theory with gaps alone? In this paper, I will present a clear answer to this puzzle—an answer that explains how being agnostic between gaps and gluts is a genuinely different position than admitting gaps alone, by using the formal notion of a bi-theory, and showing that while such positions might agree on what is to be accepted, they differ on what is to be rejected.

With Gratitude to Raymond Smullyan

Mon, 13 Feb 2017 22:12:50 +1100

While I was busy writing my most recent paper, “Proof Terms for Classical Derivations”, I heard that Raymond Smullyan had died at the age of 97. I posted a tweet with a photo of a page from the draft of the paper I was writing at the time, expressing loss at hearing of his death and gratitude for his life.

There are many reasons to love Professor Smullyan. I learned combinatory logic from his delightful puzzle book To Mock a Mockingbird, and he was famous for many more puzzle books like that. He was not only bright and sharp, he was also warmly humane. However, the focus of my gratitude was something else. In my tweet, I hinted at one reason why I’m especially grateful for Smullyan’s genius—his deep understanding of proof theory. I am convinced that his analysis of inference rules in the tableaux system for classical logic rewards repeated reflection. (See his First-Order Logic, Chapter 2, Section 1 for details.) I’ll try to explain why it’s important and insightful here.

A New Paper

Mon, 13 Feb 2017 01:32:58 +1100

It’s a new year, and it’s time for a new paper, so here is “Proof Terms for Classical Derivations” I’ve been working on these ideas for about a year, from some rough talks over most of 2016, to many conversations with my colleague Shawn as I attempted to iron out the details, to many more hours in front of whiteboards, I’ve finally got something I’m happy to show in public. The paper still rough, but the ideas are all there, and I think the theorems are all correct. The paper is under 50 pages—but only just! It proposes a new account of proof terms for classical propositional logic. These proof terms give a new account of what it is for one sequent derivation to represent the “same underlying proof” as another. Two derivations represent the same proof if and only if they have the same proof term. In the paper I show that two derivations have the same proof term if and only if one can be permuted into the other, using a natural class of transformations of derivations. Finally, I show that cut elimination for proof terms is confluent and strongly normalising, giving a new account of what it is to evaluate a classical proof, in a way that does not collapse into triviality.

Proof Terms for Classical Derivations

Sun, 12 Feb 2017 00:00:00 +0000

I give an account of proof terms for derivations in a sequent calculus for classical propositional logic. The term for a derivation $\delta$ of a sequent $\Sigma \succ\Delta$ encodes how the premises $\Sigma$ and conclusions $\Delta$ are related in $\delta$. This encoding is many–to–one in the sense that different derivations can have the same proof term, since different derivations may be different ways of representing the same underlying connection between premises and conclusions. However, not all proof terms for a sequent $\Sigma\succ\Delta$ are the same. There may be different ways to connect those premises and conclusions. Proof terms can be simplified in a process corresponding to the elimination of cut inferences in sequent derivations. However, unlike cut elimination in the sequent calculus, each proof term has a unique normal form (from which all cuts have been eliminated) and it is straightforward to show that term reduction is strongly normalising—every reduction process terminates in that unique normal form. Further- more, proof terms are invariants for sequent derivations in a strong sense—two derivations $\delta_1$ and $\delta_2$ have the same proof term if and only if some permutation of derivation steps sends $\delta_1$ to $\delta_2$ (given a relatively natural class of permutations of derivations in the sequent calculus).

Proof Terms as Invariants

Fri, 09 Dec 2016 13:00:00 +1100

This is a talk on proof theory for Melbourne Logic Day. Abstract: In this talk, I will explain how proof terms for derivations in classical propositional logic are invariants for derivations under a natural class of permutations of rules. The result is two independent characterisations of one underlying notion of proof identity. The slides are available.

A Puzzle for Brandom's Account of Singular Terms

Wed, 30 Nov 2016 11:22:42 +1100

I’ve been interested in Robert Brandom’s inferentialism since I picked up a copy of Making it Explicit back in 1996. One interesting component of Brandom’s inferentialism is his account of what it is to be a singular term. There are a number of ways to understand inferentialism, but the important point here is the centrality of material inference to semantics. An inference like “Melbourne is south of Sydney, therefore Sydney is north of Melbourne” is a materially good inference. Material inferences, for Brandom, are not to be understood as grounded in a more primitive notion of logical consequence—we shouldn’t explain the inference in terms of the validity of the form “$a$ is south of $b$, for all $x$ and $y$ if $x$ is south of $y$ then $y$ is north of $x$, therefore, $b$ is north of $a$” and the fact that the extra premise is common knowledge or a part of the norms governing the concepts of north and south. No, according the inferentialist, we are to explain those facts in terms of materially good inferences, and not vice versa. Well, one of the distinctive features of Brandom’s inferentialism is that he takes there to be an inferentialist account of what it is for a term to be a singular term—a name or other device that picks out an object, rather than a predicate that describes something, or some other kind of connective or modifier.

Existence, Definedness and the Semantics of Possibility and Necessity

Fri, 07 Oct 2016 13:30:00 +0800

I’m giving a talk entitled “Existence, Definedness and the Semantics of Possibility and Necessity” at a Workshop on the Philosophy of Timothy Williamson at the Asian Workshop in Philosophical Logic and the Taiwan Philosophical Logic Colloquium at the National Taiwan University. Abstract: In this talk, I will address just some of Professor Williamson’s treatment of necessitism in his Modal Logic as Metaphysics. I will give an account of what space might remain for a principled and logically disciplined contingentism. I agree with Williamson that those interested in the metaphysics of modality would do well to take quantified modal logic—and its semantics—seriously in order to be clear, systematic and precise concerning the commitments we undertake in adopting an account of modality and ontology. Where we differ is in how we present the semantics of that modal logic. I will illustrate how proof theory may play a distinctive role in elaborating a quantified modal logic, and in the development of theories of meaning, and in the metaphysics of modality. The slides are available here. The talk is based on a larger paper, Existence and Definedness.

Existence and Definedness: the semantics of possibility and necessity

Mon, 03 Oct 2016 00:00:00 +0000

In this paper, I will address just some of Professor Williamson’s treatment of necessitism in his Modal Logic as Metaphysics. I will give an account of what space might remain for a principled and logically disciplined contingentism. I agree with Williamson that those interested in the metaphysics of modality would do well to take quantified modal logic—and its semantics—seriously in order to be clear, systematic and precise concerning the commitments we undertake in adopting an account of modality and ontology. Where we differ is in how we present the semantics of that modal logic. I will illustrate how proof theory may play a distinctive role in elaborating a quantified modal logic, and in the development of theories of meaning, and in the metaphysics of modality. The paper was first written for presentation in a workshop on Tim Williamson’s work, at the Asian Workshop in Philosophical Logic and the Taiwan Philosophical Logic Colloquium at the National Taiwan University. The slides from that talk are available here.

Proof Terms are fun

Fri, 02 Sep 2016 17:35:21 +1100

Today, between marking assignments and working through a paper on proof theory for counterfactuals, I’ve been playing around with proof terms. They’re a bucketload of fun. The derivation below generates a proof term for the sequent $\forall xyz(Rxy\land Ryz\supset Rxz),\forall xy(Rxy\supset Ryx),\forall x\exists y Rxy \succ \forall x Rxx$. The playing around is experimenting with different ways to encode the quantifier steps in proof terms. I think I’m getting somewhere with this. (But boy, typesetting these things is not easy.) A whiteboard-to-LaTeX scanner would be really handy right about now. Anybody have one? A photo posted by Greg Restall (@consequently) on Sep 1, 2016 at 4:42pm PDT

Proofs and what they’re good for

Thu, 25 Aug 2016 16:20:00 +1100

I’m giving a talk entitled “Proofs and what they’re good for” at the University of Melbourne Philosophy Seminar on Thursday, August 25, 2016. Abstract: I present a new account of the nature of proof, with the aim of explaining how proof could actually play the role in reasoning that it does, and answering some long-standing puzzles about the nature of proof, including (1) how it is that a proof transmits warrant (2) Lewis Carroll’s dilemma concerning Achilles and the Tortoise and the coherence of questioning basic proof rules like modus ponens, and (3) how we can avoid logical omniscience without committing ourselves to inconsistency. The slides and handout are available.

What Proofs and Truthmakers are About

Fri, 12 Aug 2016 11:00:00 +1100

I was originally scheduled to give a talk entitled “What Proofs are About” at the About Aboutness Workshop at the University of Melbourne on Saturday, July 16, 2016, but my plane back to Melbourne was delayed and I didn’t get to present the paper then. So, I’m presenting it at the Melbourne Logic Seminar instead. Abstract: This talk is a comparison of how three different approaches to subject matter treat some pairs of statements that say different things but are (classically) logically equivalent. The pairs are $p\lor\neg p$ and $\top$ $p\lor(p\land q)$ and $p$ $(p\lor\neg p)\lor(q\lor\neg q)$ and $(p\lor\neg p)\land(q\lor\neg q)$. I compare and contrast the notion of subject matter introduced in Stephen Yablo’s Aboutness (Princeton University Press, 2014), truthmakers conceived of as situations, as discussed in my “Truthmakers, Entailment and Necessity,” and the proof invariants I have explored in recent work. The slides are available here.

First Degree Entailment, Symmetry and Paradox

Wed, 27 Jul 2016 00:36:40 +1100

Talking to Jc Beall during his recent visit to Australia, I got thinking about first degree entailment again.

Here is a puzzle, which I learned from Terence Parsons in his “True Contradictions”. First Degree Entailment (fde) is a logic which allows for truth value gaps as well as truth value gluts. If you are agnostic between assigning paradoxical sentences gaps and gluts (and there seems to be no very good reason to prefer gaps over gluts or gluts over gaps if you’re happy with fde), then this looks no different, in effect, from assigning them a gap value? After all, on both views you end up with a theory that doesn’t commit you to the paradoxical sentence or its negation. How is the fde theory any different from the theory with gaps alone?

I think I have a clear answer to this puzzle—an answer that explains how being agnostic between gaps and gluts is a genuinely different position than admitting gaps alone. But to explain the answer and show how it works, I need to spell things out in more detail. If you want to see how this answer goes, read on.

PHIL20030: Meaning, Possibility and Paradox

Mon, 25 Jul 2016 00:00:00 +0000

PHIL30043: The Power and Limits of Logic

Mon, 25 Jul 2016 00:00:00 +0000

Proof Theory: Logical and Philosophical Aspects

Mon, 11 Jul 2016 00:00:00 +0000

This is an intensive class on logical and philosophical issues in proof theory, taught by Shawn Standefer and me at nasslli 2016. We cover cut elimination, some substructural logics, and hypersequents, with a bit of inferentialism and bilateralism mixed in. Here are the slides for each day of the course. Day 1: Foundations. An introduction to the sequent calculus for intuitionistic and classical logic, cut elimination and some of its consequences. Day 2: Substructural Logics. Structural rules in sequent systems; the case of distribution; different substructural logics and their applications; revisiting cut elimination in substructural logics. Day 3: Beyond Sequents. Sequent sytstems for basic modal logics; three ways to move beyond traditional sequent systems—display logic, labelled sequents and tree hypersequents. Day 4: Hypersequents for S5, Actuality and 2D Modal Logics. From tree hypersequents to simple hypersequents for S5; extending simple hypersequents to model actuality and two-dimensional modal logic. Day 5: Semantics. Normative pragmatics; the scope of rules and definitions; between proofs and models; moving beyond propositional logics. Readings and References Foundations Gerhard Gentzen, “Untersuchungen über das logische Schließen—I”, Mathematische Zeitschrift, 39(1):176–210, 1935. Gerhard Gentzen, The Collected Papers of Gerhard Gentzen, Translated and Edited by M. E. Szabo, North Holland, 1969. Albert Grigorevich Dragalin, Mathematical Intuitionism: Introduction to Proof Theory, American Mathematical Society, Translations of Mathematical Monographs, 1987.

Proofs and what they’re good for

Sun, 03 Jul 2016 13:00:00 +1100

I’m giving a talk entitled “Proofs and what they’re good for” at the 2016 Australasian Association for Philosophy Conference on Monday, July 3, 2016. Abstract: I present a new account of the nature of proof, with the aim of explaining how proof could actually play the role in reasoning that it does, and answering some long-standing puzzles about the nature of proof, including (1) how it is that a proof transmits warrant (2) Lewis Carroll’s dilemma concerning Achilles and the Tortoise and the coherence of questioning basic proof rules like modus ponens, and (3) how we can avoid logical omniscience without committing ourselves to inconsistency. The slides and handout are available.

Terms for Classical Sequents: Proof Invariants and Strong Normalisation

Fri, 01 Jul 2016 16:45:00 +1100

I’m giving a talk entitled “Terms for Classical Sequents: Proof Invariants and Strong Normalisation” at the 2016 Australasian Association for Logic Conference. Abstract: A proof for a sequent $\Sigma\vdash\Delta$ shows you how to get from the premises $\Sigma$ to the conclusion $\Delta$. It seems very plausible that some valid sequents have different proofs. It also seems plausible that some different derivations for the one sequent don’t represent different proofs, but are merely different ways to present the same proof. These two plausible ideas are hard to make precise, especially in the case of classical logic. In this paper, I give a new account of a kind of invariant for derivations in the classical sequent calculus, and show how it can formalise a notion of proof identity with pleasing behaviour. In particular, it has a confluent, strongly normalising cut elimination procedure. The slides for the talk are available here.

I have a campaign poster on my picket fence

Wed, 15 Jun 2016 21:23:20 +1100

For the first time in my home owning career, the picket fence outside my home sports a how-to-vote sign.

This is a change for me. I’m a member of no political party, and I’ve never encouraged my neighbours to vote in any particular way. For almost all of my life, I’ve lived in safe Labor seats, from growing up in working class Brisbane to living in the inner north of Melbourne, my members of Federal Parliament have all been members of the ALP in safe seats. (Only short sojourns in Toowong and Marsfield found me in conservative territory.) My vote in my electorate hasn’t made a difference over the years.

More importantly, I think that there is much more to politics than voting. It’s one thing to distribute a few votes once every few years in to record your voice about how we are governed. There are plenty of other, more effective ways to take part in our common life, both locally, and globally. Broader political action can take many different forms, beyond party politics. Different forms of political action include investigating and exposing injustice or corruption, campaigning for change, making proposals for different ways to do things, protesting, raising awareness, building alternative communities and political structures and resisting injustice—through to covert and overt struggle and revolution. Politics is a complicated business with many different strands.

So, given all of that, why do I have a campaign poster on my fence?

3am Interview on Philosophical Logic

Mon, 06 Jun 2016 17:37:47 +1100

A few weeks ago, Richard Marshall interviewed me for 3am Magazine’s series of interviews with philosophers. If you’re interested in my work on logical pluralism, proof theory and things like that, this interview might be a good place to start. I hope you like it. If you’ve got any questions, please let me know.

Proofs and what they’re good for

Fri, 27 May 2016 14:30:00 +1100

I’m giving a talk entitled “Proofs and what they’re good for” at the Australian Catholic University Philosophy Seminar on May 27, 2016. Abstract: I present a new account of the nature of proof, with the aim of explaining how proof could actually play the role in reasoning that it does, and answering some long-standing puzzles about the nature of proof, including (1) how it is that a proof transmits warrant (2) Lewis Carroll’s dilemma concerning Achilles and the Tortoise and the coherence of questioning basic proof rules like modus ponens, and (3) how we can avoid logical omniscience without committing ourselves to inconsistency. The slides and handout are available.

Proofs and what they’re good for

Wed, 18 May 2016 13:00:00 +1100

I’m giving a talk entitled “Proofs and what they’re good for” at the University of Sydney Philosophy Seminar on May 18, 2016. Abstract: I present a new account of the nature of proof, with the aim of explaining how proof could actually play the role in reasoning that it does, and answering some long-standing puzzles about the nature of proof, including (1) how it is that a proof transmits warrant (2) Lewis Carroll’s dilemma concerning Achilles and the Tortoise and the coherence of questioning basic proof rules like modus ponens, and (3) how we can avoid logical omniscience without committing ourselves to inconsistency. The slides and handout are available.

Review of Thomas Piecha and Peter Schroeder-Heister (editors), Advances in Proof-Theoretic Semantics

Mon, 16 May 2016 00:00:00 +0000

What could you mean by the term “proof-theoretic semantics” (PTS)? At first glance, it could mean either the semantics of proof theory, or perhaps it’s more likely to mean semantics conducted using the tools of proof theory. And that’s the enterprise that the fifteen authors intend to advance in the sixteen papers in this edited collection…

Advice from Undergraduate Logic Students

Thu, 12 May 2016 22:49:30 +1100

Prompted by a colleague and friend (thanks Ruth!), I’ve been asking students who took my upper level logic subjects last year what they learned about how to learn logic. This is helpful for me—I get a better understanding of how students are learning. It’s hopefully helpful for them, too, to reflect more explicitly on how they learn. But most of all, I hope that their reflections will help students who come after them. After all, advice from peers is fresh and direct in ways that advice from someone whose first encounter with the material was nearly 30 years isn’t. I’ve been so impressed with the answers students sent me. They’re thoughtful and insightful and I’m sure the advice will useful for future students. I thought I’d share them, in case it helps you to learn logic, or to teach it to others. Here is advice from two students from my subject phil20030: Meaning, Possibility and Paradox. This is a second year introduction to modal and non-classical logic, with a little bit of philosophy of language along the way. It’s taught using video lectures, a weekly two hour seminar, and students work in teams on weekly to prepare for each session, and as a support and study group outside classes.

Terms for Classical Sequents: Proof Invariants and Strong Normalisation

Tue, 10 May 2016 19:00:00 +1100

I’m giving a talk entitled “Terms for Classical Sequents: Proof Invariants and Strong Normalisation” at the University of Gothenburg Logic Seminar, via Skype. Abstract: A proof for a sequent $\Sigma\vdash\Delta$ shows you how to get from the premises $\Sigma$ to the conclusion $\Delta$. It seems very plausible that some valid sequents have different proofs. It also seems plausible that some different derivations for the one sequent don’t represent different proofs, but are merely different ways to present the same proof. These two plausible ideas are hard to make precise, especially in the case of classical logic. In this paper, I give a new account of a kind of invariant for derivations in the classical sequent calculus, and show how it can formalise a notion of proof identity with pleasing behaviour. In particular, it has a confluent, strongly normalising cut elimination procedure. The slides are available here (for screen), and here is a version for printing.

Terms for Classical Sequents: Proof Invariants and Strong Normalisation

Fri, 06 May 2016 11:00:00 +1100

I’m giving a talk entitled “Terms for Classical Sequents: Proof Invariants and Strong Normalisation” at the Melbourne Logic Seminar. Abstract: A proof for a sequent $\Sigma\vdash\Delta$ shows you how to get from the premises $\Sigma$ to the conclusion $\Delta$. It seems very plausible that some valid sequents have different proofs. It also seems plausible that some different derivations for the one sequent don’t represent different proofs, but are merely different ways to present the same proof. These two plausible ideas are hard to make precise, especially in the case of classical logic. In this paper, I give a new account of a kind of invariant for derivations in the classical sequent calculus, and show how it can formalise a notion of proof identity with pleasing behaviour. In particular, it has a confluent, strongly normalising cut elimination procedure. If you’d like to attend, details are on the PhilEvents entry for the talk. The slides are available here.

On Priest on Nonmonotonic and Inductive Logic

Tue, 08 Mar 2016 00:00:00 +0000

Graham Priest defends the use of a nonmonotonic logic, LPm, in his analysis of reasoning in the face of true contradictions, such as those arising from the paradoxes of self-reference. In the course of defending the choice of this logic in the face of the criticism that LPm is not truth preserving, Priest argued that requirement is too much to ask: since LPm is a nonmonotonic logic, it necessarily fails to preserve truth. In this paper, I show that this assumption is incorrect, and I explain why nonmonotonic logics can nonetheless be truth preserving. Finally, I diagnose Priest’s error, to explain when nonmonotonic logics do indeed fail to preserve truth. Graham Priest has written a very short reply to this article, which also appears in Thought.

UNIB10002: Logic, Language and Information

Mon, 29 Feb 2016 00:00:00 +1100

UNIB10002: Logic, Language and Information is a University of Melbourne undergraduate breadth subject, introducing logic and its applications to students from a broad range of disciplines in the Arts, Sciences and Engineering. I coordinate this subject with my colleagues Dr. Shawn Standefer and Dr. Jen Davoren, with help from Prof. Lesley Stirling (Linguistics), Dr. Peter Schachte (Computer Science) and Dr. Daniel Murfet (Mathematics). The subject is taught to University of Melbourne undergraduate students. Details for enrolment are here. We teach this in a ‘flipped classroom’ model, using resources from our Coursera subjects Logic 1 and Logic 2.

Models for Compatibility

Thu, 11 Feb 2016 16:57:01 +1100

This morning I received an email from Rachael Briggs, asking me some questions about the notion of compatibility as it appears in my paper “Negation in Relevant Logics.” These questions got me to thinking that there were some ideas in that paper and a much-less-read paper of mine “Modelling Truthmaking”, which might be worth reflecting on some more. So I’ll try to do that here. Here’s the background you need to get up to speed. Relevant logics attempt to make sense of the idea that if an argument from premises $\Sigma$ to conclusion $A$ is valid then the premises must somehow be relevant to the conclusion. If the $A$ is a tautology by itself (say, $p\lor\neg p$) then that fact alone is not enough to ensure that it follows from the unrelated premise $q$. $p\lor\neg p$ follows well enough from $p$ (a disjunction follows from a disjunct), and from $\neg p$ (the other disjunct), but if $q$ has nothing to do with it, we can’t necessarily get to $p\lor\neg p$ from there. The same goes for arguments from contradictory premises. $p\land\neg p$ is inconsistent. Classical logic tells us that the argument from $p\land\neg p$ to $q$ is valid because there’s no interpretation to make $p\land\neg p$ true that makes $q$ false—but that’s not because of any connection between $p\land\neg p$ and $q$.

Congratulations to Bruce French, AO

Fri, 29 Jan 2016 00:30:47 +1100

I’ve been away from home in Auckland at a conference over the last couple of days, so I’m a bit behind on the news, but tonight I’ve finally found a few moments to write something in honour of my friend and mentor, Bruce French, who was—three days ago—named an Officer of the Order of Australia in the 2016 Australia Day Honour’s list. Bruce was awarded this honour because of his many decades of work, learning about tropical food plants throughout the Pacific, Asia and Africa, cataloguing that knowledge and making it freely available to anyone who can use it. Bruce French with a young friend, in 2012 I’ve known Bruce since my second year at university, many years ago. During this time, I was involved in a student Christian group, and Bruce worked there with students, encouraging them to explore both their university studies and their faith in equal measure. Bruce had trained as an agricultural scientist, and as a Baptist pastor, and his enthusiasm for tropical plants, and his passion for loving God and loving his neighbour was infectious. Before coming to work at the University, Bruce had spent time in Papua New Guinea, and he’d seen both the incredible knowledge that indigenous people had for the opportunities for nutrition in the plants around them, and the devastating damage done by well-meaning but ignorant attempts at aid in times of famine, where inappropriate Western foodstuffs would be introduced, when local options were much more appropriate to the climate and the nutritional needs of community.

Fixed Point Models for Theories of Properties and Classes

Tue, 26 Jan 2016 13:38:19 +1100

I’m giving a talk entitled “Fixed Point Models for Theories of Properties and Classes” at the Frontiers of Non-Classicality conference at the University of Auckland. Abstract: There is a vibrant (but minority) community among philosophical logicians seeking to resolve the paradoxes of classes, properties and truth by way of adopting some non-classical logic in which trivialising paradoxical arguments are not valid. There is also a long tradition in theoretical computer science—going back to Dana Scott’s fixed point model of the lambda calculus—of constructions allowing for various fixed points. In this talk, I will bring these traditions closer together, to show how these model constructions can shed light on what we could hope for in a non-trivial model of a theory for classes, properties or truth featuring fixed points. The slides are available here.

On Priest on nonmonotonic and inductive logic

Fri, 22 Jan 2016 00:55:52 +1100

Thanks to a recent visit from Jc Beall, I was reminded of a critical discussion between Jc and our colleague and friend Graham Priest in the pages of Analysis. Jc was puzzled by a claim that Graham made in his reply to Jc’s paper, concerning nonmonotonic consequence relations and failures of truth preservation. Here, I’ll explain the disagreement between Jc and Graham, and why Graham’s claim (that all nonmonotonic logics fail to preserve truth) is wrong.

Update on March 9, 2016: I submitted a version of this note to the journal Thought, and it has been accepted for publication. The prepublication version of the paper is archived here.

Generality and Existence 3: Substitution and Identity

Fri, 11 Dec 2015 14:00:00 +1100

I am giving a talk, entitled “Generality and Existence 3: Substitution and Identity” in the Melbourne Logic Workshop 2015 at the Melbourne Logic Group. Abstract: In this talk, extend a sequent proof system for free logic to include an identity predicate. Or rather, two different candidate identity predicates allowing for the substitution of a lesser or greater class of predications. I show that the resulting system is well-behaved proof theoretically (the Cut rule is admissible) and the identity predicates can be well motivated in terms of their defining rules. Workshop Program at the Melbourne Logic Group. Slides.

Generality and Existence 4: Identity and Modality

Thu, 03 Dec 2015 10:45:00 +0000

I’m giving a talk, entitled “Generality and Existence 4: Identity and Modality” in the HPLM Seminar at the University of St Andrews. Abstract: In this talk, extend the proof system for quantified modal logic, with both subjunctive (metaphyisical) and indicative (epistemic) modalities, to include an identity predicate. Or rather, a range of candidate identity predicates allowing for the substitution of a lesser or greater class of predications. I show that the resulting system is well-behaved proof theoretically (the Cut rule is eliminable) and the identity predicates can be well motivated in terms of their defining rules. I end with a discussion of the implications for the ontology of possibilia. Arché at the University of St Andrews. Slides.

Generality and Existence 3: Substitution and Identity

Wed, 02 Dec 2015 16:00:00 +0000

I gave a talk, entitled “Generality and Existence 3: Substitution and Identity” in the Arché Super Special Seminar at the University of St Andrews. Abstract: In this talk, extend the proof system for free logic to include an identity predicate. Or rather, two different candidate identity predicates allowing for the substitution of a lesser or greater class of predications. I show that the resulting system is well-behaved proof theoretically (the Cut rule is eliminable) and the identity predicates can be well motivated in terms of their defining rules. Arché at the University of St Andrews. Slides.

Generality and Existence 2: Modality and Quantifiers

Wed, 02 Dec 2015 10:00:00 +0000

I’m giving a talk, entitled “Generality and Existence 2: Modality and Quantifiers” in the Arché Logic Group Seminar at the University of St Andrews. Abstract: In this talk, I motivate and define a cut free sequent calculus for first order modal predicate logics, allowing for singular terms free of existential import. I show that the cut rule is admissible in the cut-free calculus, and explore the relationship between contingent ‘world-bound’ quantifiers and possibilist ‘world-undbound’ quantifiers in the system. Arché at the University of St Andrews. Slides.

Generality and Existence 1: Quantification and Free Logic

Wed, 25 Nov 2015 09:15:00 +0000

I’m giving a talk, entitled “Generality and Existence 1: Quantification and Free Logic” at a Workshop on Inferentialism, hosted by Arché at the University of St Andrews. Abstract: In this presentation, I motivate a cut free sequent calculus for classical logic with first order quantification, allowing for singular terms free of existential import. Along the way, I motivate a criterion for rules designed to answer Prior’s question about what distinguishes rules for logical concepts, like conjunction from apparently similar rules for putative concepts like tonk, and I show that the rules for the quantifiers—and the existence predicate—satisfy that condition. Details of the workshop program. Draft paper. Slides.

Language, Logic and Existential Commitment

Wed, 18 Nov 2015 11:49:21 +1100

In between wrapping up teaching for the end of Semester 2, and getting ready for a short trip to Scotland, I’m spending some time thinking about free logic and quantified modal logic and identity. This is difficult but exciting terrain to cover. There is no obvious way to tie together the logic of quantifiers, the modalities and identity in a way that commands broad appeal—there is no default quantified modal logic that has the same ‘market reach’ as classical first order logic.

This is a shame, because many important arguments involve quantification, identity and possibility and necessity—and claims about existence and nonexistence. Understanding the logic of these arguments better would help us gain some kind of systematic understanding of the positions in play. In the absence of a clear picture of the logic implicit in our concepts, we’re playing the dialectical game in ignorance of the rules.

Not that there’s anything wrong with that.

Logic at Melbourne finally has a presence on the web

Thu, 29 Oct 2015 17:02:04 +1100

I’ve been at the University of Melbourne since 2002, and the Logic Group has been meeting regularly since my arrival there. While it’s changed significantly in its membership over the last 13 years, one thing has remained constant—we’ve been an informal, friendly bunch of people, so hard at work in teaching and research that we’ve not had time to make a website for the group. Well, thanks to the hardworking Shawn Standefer, that’s changed. Point your browsers to http://blogs.unimelb.edu.au/logic/ to keep up with Logic at Melbourne.

Three Cultures—or: what place for logic in the humanities?

Thu, 29 Oct 2015 00:00:00 +0000

Logic has been an important part of philosophy since the work of Aristotle in the 4th Century BCE. Developments of the 19th and the 20th Century saw an incredible flowering of mathematical techniques in logic, and the discipline transformed beyond recognition into something that can seem forbiddingly technical and formal. The discipline of logic plays a vital role in mathematics, linguistics, computer science and electrical engineering, and it may seem that it no longer has a place within the humanities. In this essay, I show why this perception is misplaced and dangerous, and that in a time of increasing specialisation and differentiation between the cultures of the humanities, the sciences, and of engineering, logic not only has much to give to the humanities, it also has much to learn.

Cian Dorr on Defining Quantifiers (Verbal Disputes Workshop Report #2)

Sat, 24 Oct 2015 15:27:36 +1100

There’s something very attractive in the idea that meanings of logical concepts—such as conjunction, negation, the quantifiers—can be specified by the role they play in proofs. If I learn the rules for conjunction: $ \def\sem#1{[\![#1]\!]} $ \[ \frac{A\land B}{A} \quad \frac{A\land B}{B} \qquad \frac{A\quad B}{A\land B} \] then I’ve “pinned down” what there is to know about the conjunction connective “$\land$”. The quantifiers are an interesting case. The rules for quantifiers, like $(\exists x)$ prove to be more complex. \[ \frac{A(t)}{(\exists x)A(x)} \qquad \frac{ {{{}\atop{\large }}\atop{\large (\exists x)A(x)}} \quad {{{\LARGE A(n)}\atop{\large \vdots}}\atop{\large C}} } {C} \] (where in the second rule, the name $n$ is not present in $C$ or in the other premises of the subproof from $A(n)$ to $C$). In these rules, we unwrap the quantified expression $(\exists x)A(x)$ into a simpler expression which is an instance of the schema $A(x)$. In the introduction rule, $A(t)$ is an instance formed by substituting any singular term $t$ for $x$, while in the elimination rule, it must be a name (or an eigenvariable). The details can be worked out in many different ways, but one feature remains the same: the semantics of quantifiers is tied closely to the semantics of singular terms. Can we avoid this connection? Is there a way to define the semantics of quantifiers in by way of their rules in proof which do not tie them so closely to the semantics of singular terms.

This was one of the goals of Cian Dorr’s talk “Pinning Down the Meanings of Quantifiers” at the Verbal Disputes conference, and is a central theme in his paper “Quantifier Variance and the Collapse Theorems”. There’s a lot in the paper (it’s 67 pages long!), and I won’t engage with most of it, but I’d like to discuss this central theme of the paper here.

Cian Dorr

Fixed Point Models for Theories of Properties and Classes

Wed, 05 Aug 2015 11:00:00 +0100

I’m giving a talk entitled “Fixed Point Models for Theories of Properties and Classes” at the first 2015 meeting of the 15th Congress on Logic, Methodology, and Philosophy of Science, in Helsinki. Abstract: There is a vibrant (but minority) community among philosophical logicians seeking to resolve the paradoxes of classes, properties and truth by way of adopting some non-classical logic in which trivialising paradoxical arguments are not valid. There is also a long tradition in theoretical computer science—going back to Dana Scott’s fixed point model of the lambda calculus—of constructions allowing for various fixed points. In this talk, I will bring these traditions closer together, to show how these model constructions can shed light on what we could hope for in a non-trivial model of a theory for classes, properties or truth featuring fixed points. Details of the session are available at the event page on sched.org. The slides are available here.

PHIL20030: Meaning, Possibility and Paradox

Mon, 27 Jul 2015 00:00:00 +0000

PHIL40013: Uncertainty, Vagueness and Disagreement

Mon, 27 Jul 2015 00:00:00 +1100

PHIL40013: Uncertainty, Vagueness and Paradox is a University of Melbourne honours seminar subject for fourth-year students. Our aim in the Honours program is to introduce students to current work in research in Philosophy. In 2015, I will cover some topic introducing logic and its applications to issues in metaphysics, epistemology or philosophy of language. For further information, contact me. To participate, check the handbook.

Merely Verbal Disputes and Coordinating on Logical Constants

Mon, 06 Jul 2015 16:00:00 +1100

This is a talk at the 2015 Australasian Association for Philosophy Conference at Macquarie University in Sydney. I will attempt to be precise about the notion of a merely verbal disagreement, and explain how we can fix on the meaning of our logical constants can play a coordinating role in dialogue, and how (in a certain sense), it might be impossible to a have a merely verbal dispute concerning logical constants such as conjunction, negation, the quantifiers, and even—if you’re careful—modal operators. Slides and Handout for the conference presentation.

Contingent Existence and Modal Definedness

Thu, 02 Jul 2015 16:00:00 +1100

A presentation for the Australasian Association for Logic Conference at the University of Sydney. I will discuss Tim Williamson’s arguments for the non-contingency of existence (see his Modal Logic as Metaphysics, 2013) and explain how and why they might be reasonably resisted. Along the way, I’ll try to explain the impact of proof theory for theories of meaning and for metaphysics. Handout for the seminar presentation. Slides for the seminar presentation.

Another Day at the Office

Thu, 04 Jun 2015 12:44:46 +1100

The view from my desk. A photo posted by Greg Restall (@consequently) on Jun 3, 2015 at 7:39pm PDT

Adventures in Bi-Intuitionistic Logic

Sat, 30 May 2015 04:04:25 +1100

In yesterday’s Logic Seminar, Tomasz Kowalski introduced some lovely results about analytic cut and interpolation in sequent systems for bi-intuitionistic logic. After the seminar, Lloyd Humberstone, Dave Ripley, Tomasz and I got talking about various features of bi-intuitionistic logic, some of which should be more well known than they are … so I may as well post them here. Read on, if you’re into non-classical logic, algebras and frames.

Verbal Disputes in Oxford (Workshop Report #1)

Wed, 27 May 2015 15:56:45 +1100

I’m back from last week’s quick trip to Oxford for a workshop on Verbal Disputes, just in time to wrap up the last week of teaching in my two undergraduate units as the semester winds up here in Melbourne.

This was an excellent two-day workshop, capably organised by Trevor Teitel, and funded by the Ertegun Graduate Scholarship Programme for the Humanities. There was a lot to think about in the papers presented at the conference, and the informal discussions between talks. In another entry, I’ll comment on Cian Dorr’s talk “Pinning Down the Meanings of Quantifiers.” In this entry I’ll make some general remarks about the workshop and, in particular, about Amie Thomasson’s talk “Metaphysical Disputes and Metalinguistic Negotiation”.

A highlight of a workshop like this is meeting people, maintaining contact with those you’ve met before, and making new connections with new friends and colleagues. While there were lots of different views expressed at the conference, there was enough common ground—and more than enough good will and good humour—to have fruitful and productive conversations. This was a great workshop.

Workshop on Verbal Disputes and their Philosophical Significance

Thu, 21 May 2015 09:20:00 +0000

I’m giving a presentation, entitled “Merely Verbal Disputes and Coordinating on Logical Constants,” on proof theory, modality, existence and verbal disputes at a workshop held at the University of Oxford in May, 2015. I will attempt to be precise about the notion of a merely verbal disagreement, and explain how we can fix on the meaning of our logical constants, how they can play a coordinating role in dialogue, and how (in a certain sense), it might be impossible to a have a merely verbal dispute concerning logical constants such as conjunction, negation, the quantifiers, and even—if you’re really careful—modal operators. The website for the workshop verbaldisputesoxford.wordpress.com contains more details. Slides: Merely Verbal Disputes and Coordinating on Logical Constants—Slides (1.8MB PDF) Handout: Merely Verbal Disputes and Coordinating on Logical Constants—Handout (600KB PDF)

One Quick Trip to Oxford

Sun, 17 May 2015 22:19:26 +1100

I'm here for May 20 and 21

Tomorrow I head off on a flying visit to Oxford for a Workshop on Verbal Disputes and their Philosophical Significance. I’m looking forward to catching up with old friends and making new ones. The lineup for the workshop is fantastic, so it looks like it’ll be a great time. If you’re in the area, please do come along and say “hi”. If you can’t come, I’ve posted the slides and handout for my talk here.

Merely Verbal Disputes and Coordinating on Logical Constants

Fri, 08 May 2015 11:00:00 +1100

This is a dry run for my talk at the Oxford Workshop on Verbal Disputes. I will attempt to be precise about the notion of a merely verbal disagreement, and explain how we can fix on the meaning of our logical constants can play a coordinating role in dialogue, and how (in a certain sense), it might be impossible to a have a merely verbal dispute concerning logical constants such as conjunction, negation, the quantifiers, and even—if you’re careful—modal operators. If you’d like to attend, keep track on the PhilEvents entry for the talk. Slides for the seminar presentation.

Assertion, Denial, Accepting, Rejecting, Symmetry and Paradox

Tue, 05 May 2015 00:00:00 +0000

Proponents of a dialethic or “truth-value glut” response to the paradoxes of self-reference argue that “truth-value gap” analyses of the paradoxes fall foul of the extended liar paradox: “this sentence is not true.” If we pay attention to the role of assertion and denial and the behaviour of negation in both “gap” and “glut” analyses, we see that the situation with these approaches has a pleasing symmetry: gap approaches take some denials to not be expressible by negation, and glut approaches take some negations to not express denials. But in the light of this symmetry, considerations against a gap view point to parallel considerations against a glut view. Those who find some reason to prefer one view over another (and this is almost everyone) must find some reason to break this symmetry. This short paper was written for presentation at AAP2004. After very many revisions, it’s finally going to be published in an edited collection from members and friends of the Arché project on Foundations of Logical Consequence.

Incarnation and Detachment: on surprising connections between logic, semantics, and prayer

Fri, 01 May 2015 11:00:00 +1100

I’m giving a research seminar on May 1 at the Catholic Theological College. The details are listed here in PhilEvents. Abstract: In this talk, I will explore some of the connections between logic and philosophy of language on the one hand, and theology—or religious practice—on the other. Instead of focussing on the role of logic and semantics as a prolegomena to theology, or as a guide or corrective or standard for it—the standard picture of the relationship between logic and theology—I will draw other connections between the two disciplines. First, I will draw on some contemporary insights in the philosophy of language in order to shed light on the structure and semantics of prayer. Second, I will show how there are some significant parallels between the phenomenology or the pragmatics of logical judgements and the kinds of cognitive shifts which are modelled in certain kinds of articulate prayer. The talk will involve more suggestions than arguments, and the aim is to provide opportunities for new and different interactions between philosophy and theology.

Logic: Language and Information 2

Tue, 14 Apr 2015 18:00:00 +1100

Jen Davoren and I are running Logic: Language and Information 2 through Coursera again in 2015. You can enrol here. This course follows on from Logic: Language and Information 1

Sophistry and Argumentation: The Role of Reason in the Examined Life

Mon, 13 Apr 2015 19:00:00 +1100

A presentation to the Philosophy Circle at the Lyceum Club of Melbourne on Sophistry and Argumentation, and on the role of reason in the examined life.

Another View of the Old Quad

Wed, 01 Apr 2015 00:23:10 +1100

Here’s another view of my building at work, in the autumn light.

Old Quad, back entrance

A photo posted by Greg Restall (@consequently) on Mar 31, 2015 at 6:20am PDT

Incarnation and Detachment: Philosophy, Prayer, and Taking Positions

Tue, 24 Mar 2015 19:00:00 +1100

I’m giving a talk entitled “Incarnation and Detachment: Philosophy, Prayer, and Taking Positions” at the first 2015 meeting of the Christian Scholars Network. (We’ll meet at the Dan O’Connell Pub at 7pm on Tuesday March 24. Come early to share dinner before the talk and discussion.) In the talk I’ll discuss some of the parallels between the range of cognitive attitudes to be found in reasoning and philosophical reflection on the one hand, and prayer, on the other. The Christian Scholars Network is a ministry of St. James Old Cathedral. Anyone who is interested in the intersection of faith and intellect is entirely welcome to participate. It’s coordinated by Rev. Dr. Jeffrey Hanson. For more information, see the Network’s website.

Advocacy on Sydney Road

Thu, 19 Mar 2015 12:49:30 +1100

On the afternoon of Sunday March 1, Sydney Road in Brunswick was transformed by its annual Street Party. Over four blocks of this busy inner north thoroughfare was closed to trams and cars, lined by stalls, punctuated with live bands, and filled with crowds. Everyone was sampling wares, eating, drinking and soaking in the atmosphere. It’s a distinctive slice of multicultural Brunswick life, and an opportunity for everyone to get out and sample that Brunswick experience in all its variety.

Live music at the Sydney Road Street Party

Our Thinking Behind Logic Teaching on Coursera

Mon, 16 Mar 2015 14:46:22 +1100

Just today, I stumbled across a video of an interview with me from 2013, explaining our thinking behind teaching introductory logic on Coursera, and the connections between this and classroom teaching. Come for the ideas, stay for the crazy hand gestures.

Come and work with me on proof theory and philosophy

Fri, 06 Mar 2015 12:28:44 +1100

I’m delighted to finally announce that I have a job to offer on my five year ARC funded research project in philosophical logic and its applications, “Meaning in Action: new techniques for language, logic and information.” It’s a two year postdoctoral research position, with a little bit of teaching on the side. If you’re interested in philosophical logic, and would like to join the logic group at Melbourne, this is the job for you.

PHIL30043: The Power and Limits of Logic

Mon, 02 Mar 2015 00:00:00 +1100

PHIL30043: The Power and Limits of Logic is a University of Melbourne undergraduate subject. It covers the metatheory of classical first order predicate logic, beginning at the Soundness and Completeness Theorems (proved not once but twice, first for a tableaux proof system for predicate logic, then a Hilbert proof system), through the Deduction Theorem, Compactness, Cantor’s Theorem, the Downward Löwenheim–Skolem Theorem, Recursive Functions, Register Machines, Representability and ending up at Gödel’s Incompleteness Theorems and Löb’s Theorem. Kurt Gödel, seated The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here. I make use of video lectures I have made freely available on Vimeo. Download the course guide (303KB PDF). Outline The course is divided into four major sections and a short prelude. Here is a list of all of the videos, in case you’d like to follow along with the content. Prelude Logical Equivalence Disjunctive Normal Form Why DNF Works Prenex Normal Form Models for Predicate Logic Trees for Predicate Logic Completeness Introducing Soundness and Completeness Soundness for Tree Proofs Completeness for Tree Proofs Hilbert Proofs for Propositional Logic Conditional Proof Hilbert Proofs for Predicate Logic Theories Soundness and Completeness for Hilbert Proofs for Predicate Logic Compactness Counting Sets Diagonalisation Compactness Non-Standard Models Inexpressibility of Finitude Downward Löwenheim–Skolem Theorem Computability Functions Register Machines Recursive Functions Register Machine computable functions are Recursive The Uncomputable Undecidability and Incompleteness Deductively Defined Theories The Finite Model Property Completeness Introducing Robinson’s Arithmetic Induction and Peano Arithmetic Representing Functions and Sets Gödel Numbering and Diagonalisation Q (and any consistent extension of Q) is undecidable, and incomplete if it’s deductively defined First Order Predicate Logic is Undecidable True Arithmetic is not Deductively Defined If Con(PA) then PA doesn’t prove Con(PA)

UNIB10002: Logic, Language and Information

Mon, 02 Mar 2015 00:00:00 +1100

UNIB10002: Logic, Language and Information is a University of Melbourne undergraduate breadth subject, introducing logic and its applications to students from a broad range of disciplines in the Arts, Sciences and Engineering. I coordinate this subject with my colleague Dr. Jen Davoren, with help from Dr. Lesley Stirling and Dr. Peter Schachte. The subject is taught to University of Melbourne undergraduate students. Details for enrolment are here. We teach this in a ‘flipped classroom’ model, using resources from our Coursera subjects Logic 1 and Logic 2.

Academic Genealogy

Sat, 28 Feb 2015 00:35:19 +1100

I have known about the Mathematical Genealogy Project for quite some time, but, prompted by Richard Zach I notice two new and wonderful things.

First, David Alber has produced in Geneagrapher a neat tool to download genealogy data and save a .dot file, which can be read by Graphviz and displayed as a directed graph.

Here’s a detail from my genealogy. (This image is a link to a pdf file of the entire thing.)

Some of my genealogy

Contingent Existence and Modal Definedness

Thu, 26 Feb 2015 16:00:00 +1100

A seminar for the ANU School of Philosophy. I will discuss Tim Williamson’s arguments for the non-contingency of existence (see his Modal Logic as Metaphysics, 2013) and explain how and why they might be reasonably resisted. Along the way, I’ll try to explain the impact of proof theory for theories of meaning and for metaphysics. Handout for the pre-seminar talk for graduate students, on Meaning Rules and Defining Concepts. Handout for the seminar presentation. Slides for the seminar presentation.

Classes in Semester One

Wed, 25 Feb 2015 23:55:17 +1100

The Power and Limits of Logic

Teaching started for first semester, with the launch of Logic: Language and Information 1 on Coursera. That’s been a lot of fun already, with over 17,000 enrolled. There’s something very exciting about being involved with a large number of students all around the world, choosing to work hard to learn logic for the first time.

Logic: Language and Information 1

Tue, 24 Feb 2015 18:00:00 +1100

Jen Davoren and I are running Logic Language and Information 1 through Coursera again in 2015. You can enrol here.

Ash Wednesday

Wed, 18 Feb 2015 11:59:50 +1100

Rembember that you are dust, and to dust you shall return. Remember that you are dust (@consequently) on Feb 18, 2015 at 5:57am PST

Do LNC and LEM suffice to define negation?

Tue, 17 Feb 2015 23:20:25 +1100

What is negation?

One answer you find in the literature is that negation is the operator that makes each instance of the Law of the Excluded Middle (LEM) and the Law of Non-Contradiction (LNC) turn out to be true. That is, every sentence of the form \[ p\lor \neg p \qquad \neg(p\land\neg p) \] is true, no matter what sentence we use in the place of $p$ (where $\neg$ stands for negation, $\lor$ for disjunction, and $\land$ for conjunction).

This is the wrong way to try to define negation. If you read on, I’ll explain why.

Adding Site Search

Fri, 13 Feb 2015 11:19:07 +1100

I’ve added a search feature to the site in the footer of each page. There seems to be enough on the site that search is genuinely useful. I’ve used DuckDuckGo, which has a comprehensive index of this place, and which can be styled to match the visual treatment of the rest of the site. Please let me know if there are any problems with the implementation.

Another Blast From the Past

Sun, 08 Feb 2015 15:02:19 +1100

A blast from the past

While recovering the archives for this site, I came across another blast from the past: the weblog I ran with Jc during the early days of our research on logical pluralism.

Recovering the Past

Sat, 07 Feb 2015 07:20:10 +1100

If you glance at the news archive, you’ll see that once again it stretches back to the dim past of weblog days, the year 2000. Thanks to the wayback machine, I was able to recover entries from my old blog (powered by Blogger, from well before it was acquired by Google), and fulfil a commitment I made in March 2001 to move the old archives over to consequently.org.

Back in the office

Wed, 04 Feb 2015 12:45:04 +1100

It’s good to be back in the office. The Parkville Campus of The University of Melbourne is a delightful place to work.

It’s February, and I’m back in the office.

A photo posted by Greg Restall (@consequently) on Feb 1, 2015 at 4:36pm PST

Why no comment box?

Tue, 03 Feb 2015 09:52:39 +1100

Previous incarnations of this website provided a facility for comments in the usual way for a weblog in the first decade of the 2000s—a comment form on each entry. This one doesn’t. Why not?

What’s the point of a personal website these days?

Sun, 01 Feb 2015 23:05:00 +1100

My last post to consequently.org was in late in 2010, well over four years ago. You’d be justified in thinking that I’d abandoned the place—but you’d also be wrong. I’ve been tinkering with it behind the scenes. The software I’d been using to maintain the website has long since fallen into disrepair, so updating it meant porting everything to another content management system. At least, it would have to mean that, given that I wasn’t really keen to burn the whole thing to the ground and start again from scratch. This required time.

It didn’t require four years, but it did require an uninterrupted stretch of time to tinker with software, experiment with the design, and finally, piece everything back together.

Before getting back into all of this website business, however, I wanted to answer one question. Is the investment of time worth it? What’s the point of a personal website these days, anyway?

Ternary Relations and Models for Relevant Arithmetics

Mon, 15 Dec 2014 09:00:00 +1100

A seminar at the workshop Algebra and Substructural Logic 5 (La Trobe, December 2015). I explain how the `trace model’ for relevant arithmetic (introduced here) can be extended quite naturally to model a logic including a de Morgan negation. The result is not a model for the logic R, but for its weaker cousin, TW.

Normal Proofs, Cut Free Derivations and Structural Rules

Mon, 01 Dec 2014 00:00:00 +0000

Different natural deduction proof systems for intuitionistic and classical logic—and related logical systems—differ in fundamental properties while sharing significant family resemblances. These differences become quite stark when it comes to the structural rules of contraction and weakening. In this paper, I show how Gentzen and Jaśkowski’s natural deduction systems differ in fine structure. I also motivate directed proof nets as another natural deduction system which shares some of the design features of Genzen and Jaśkowski’s systems, but which differs again in its treatment of the structural rules, and has a range of virtues absent from traditional natural deduction systems.

Living as a Christian Academic

Wed, 17 Sep 2014 09:00:00 +1100

I gave a presentation to postgraduate students in the Monash University Christian Union about life as a Christian academic, and my journey from being an undergraduate in Mathematics at the University of Queensland in the late 1980s, to my current role as Professor of Philosophy at the University of Melbourne.

Towards Fixed Point Models for Theories of Properties and Classes

Fri, 05 Sep 2014 11:00:00 +1100

A seminar at the University of Melbourne Logic Group. I sketch some of the features of models of naïve theories of properties or classes defined by finding a space C that is isomorphic to the class of continuous functions [C ∪ D → Ω].

Ludwig Wittgenstein’s Tractatus Logico-Philosophicus

Tue, 26 Aug 2014 17:00:00 +1100

I gave a masterclass on Ludwig Wittegenstein’s Tractatus Logico-Philosophicus in the University of Melbourne’s 10 Great Books Masterclass in 2014.

Pluralism and Proofs

Sun, 03 Aug 2014 00:00:00 +0000

Beall and Restall’s Logical Pluralism (2006) characterises pluralism about logical consequence in terms of the different ways cases can be selected in the analysis of logical consequence as preservation of truth over a class of cases. This is not the only way to understand or to motivate pluralism about logical consequence. Here, I will examine pluralism about logical consequence in terms of different standards of proof. We will focus on sequent derivations for classical logic, imposing two different restrictions on classical derivations to produce derivations focusr intuitionistic logic and for dual intuitionistic logic. The result is another way to understand the manner in which we can have different consequence relations in the same language. Furthermore, the proof-theoretic perspective gives us a different explanation of how the one concept of negation can have three different truth conditions, those in classical, intuitionistic and dual-intuitionistic models.

PHIL10003: Philosophy, the Great Thinkers

Mon, 28 Jul 2014 00:00:00 +0000

PHIL10003: Philosophy, the Great Thinkers is a University of Melbourne undergraduate subject, introducing philosophy to students through close readings of core texts. I taught with Dr. Chris Cordner and Dr. Ruth Boeker. In my classes, we looked at David Hume’ Dialogues Concerning Natural Religion and Natural History of Religion, with some side trips into hume’s Enquiry, and some Marx, Wittgenstein for good measure. The subject was taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here.

PHIL30043: The Power and Limits of Logic

Mon, 28 Jul 2014 00:00:00 +0000

From Defining Rules to Cut Elimination (and its consequences)

Fri, 04 Jul 2014 11:00:00 +1100

A presentation at the Metaphysical Basis of Logic Workshop at the Northern Institute of Philosophy, University of Aberdeen.

Proof Theory and Philosophy

Wed, 25 Jun 2014 00:00:00 +0000

I taught an intensive class on proof theory and its applications to Philosophy at the NIP Summer School on the Foundations of Logic and Mathematics, June July 2014.

Modal Definedness

Thu, 19 Jun 2014 16:15:00 +0000

A presentation in a workshop in the Philosophy of Logic at Arché. The distinction between defined terms and undefined terms provides a metaphysically “light” way to give a semantics for free logic. Singular terms may be undefined, and if they are undefined, they are not appropriate substitution instances for inference rules for the quantifiers. Sol Feferman, in “Definedness” (Erkenntnis, 1995), provides an elegant system for an extensional negative free logic with undefined terms: it is a very natural model for mathematical reasoning in which we allow undefined terms (like 1/0) and we keep track of the behaviour of such terms by talking of when they are defined and when they are not. Free logics also see use when it comes to quantified modal logic. It is very tempting to conceive of the domain of quantification as varying from world to world, that what exists in one world might fail to exist in another. This seems to be a very different motivation for free logics—terms which denote in this world and which do not denote in another are not undefined at that world. They are defined all too well—defined to denote something that fails to exist at that world. In this talk, I will explain the motivations for these two different approaches to free logic, and show that a sequent calculus for Feferman’s own system (with a metaphysically ’light’ interpretation, that eschews all talk of an outer domain of quantification of non-existent objects) can, with one small change, be naturally extended into a modal hypersequent calculus for a quantified modal logic with a non-constant domain.

Ternary Relations and Models for Relevant Arithmetics

Fri, 30 May 2014 13:30:00 +1100

A seminar at Melbourne Logic Day, May 30, 2014. I explain how the ’trace model’ for relevant arithmetic (introduced here) can be extended quite naturally to model a logic including a de Morgan negation. The result is not a model for the logic R, but for its weaker cousin, TW.

Logic: Language and Information 2

Tue, 22 Apr 2014 00:00:00 +1100

Jen Davoren and I taught Logic Language and Information 2 (an introduction to predicate logic and its applications) through Coursera for the first time in 2014. An archive of the material is available here.

PHIL20030: Meaning, Possibility and Paradox

Mon, 03 Mar 2014 00:00:00 +0000

PHIL20030: Meaning, Possibility and Paradox is a University of Melbourne undergraduate subject. The idea that the meaning of a sentence depends on the meanings of its parts is fundamental to the way we understand logic, language and the mind. In this subject, we looked at the different ways that this idea has been applied in logic throughout the 20th Century and into the present day. In the first part of the subject, our focus is on the concepts of necessity and possibility, and the way that ‘possible worlds semantics’ has been used in theories of meaning. We will focus on the logic of necessity and possibility (modal logic), times (temporal logic), conditionality and dependence (counterfactuals), and the notions of analyticity and a priority so important to much of philosophy. In the second part of the subject, we examined closely the assumption that every statement we make is either true or false but not both. We will examine the paradoxes of truth (like the so-called ‘liar paradox’) and vagueness (the ‘sorites paradox’), and we will investigate different ways attempts at resolving these paradoxes by going beyond our traditional views of truth (using ‘many valued logics’) or by defending the traditional perspective.

PHIL40007: Philosophy of Language and Mind

Mon, 03 Mar 2014 00:00:00 +0000

PHIL40007: Philosophy of Language and Mind is a University of Melbourne honours seminar subject for fourth-year students. Our aim in the Honours program is to introduce students to current work in research in Philosophy. In 2014, I introduced the students to the normative pragmatism and inferentialism of Robert Brandom, through a close reading of his book Articulating Reasons and related literature.

UNIB10002: Logic, Language and Information

Mon, 03 Mar 2014 00:00:00 +0000

UNIB10002: Logic, Language and Information is a University of Melbourne undergraduate breadth subject, introducing logic and its applications to students from a broad range of disciplines in the Arts, Sciences and Engineering. I coordinate this subject with my colleague Dr. Jen Davoren, with help from Dr. Lesley Stirling and Dr. Peter Schachte. The subject was taught to University of Melbourne undergraduate students. Details for enrolment are here. We teach this in a ‘flipped classroom’ model, using resources from our Coursera subjects Logic 1 and Logic 2.

MULT10016: Reason

Mon, 03 Mar 2014 00:00:00 +1100

In 2014, I participated in the Arts Foundation subject MULT10016: Reason. I taught six classes, on Plato, Aristotle and Hume.

Logic: Language and Information 1

Sat, 01 Mar 2014 00:00:00 +1100

Jen Davoren and I taught Logic Language and Information 1 (an introduction to propositional logic and its applications) through Coursera for the first time in 2014. An archive of the material is available here.

Assertion, Denial and Non-Classical Theories

Wed, 31 Jul 2013 00:00:00 +0000

In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naïve theories of classes and truth.

Special Issue of the Logic Journal of the IGPL: Non-Classical Mathematics

Fri, 15 Feb 2013 00:00:00 +0000

A special issue of the Logic Journal of the IGPL, on Non-Classical Mathematics. Editorial Preface, Libor Bĕhounek, Greg Restall, and Giovanni Sambin. "Mathematical pluralism," Graham Priest "A first constructive look at the comparison of projections," Douglas S. Bridges and Luminiţa S. Vîţa "Lipschitz functions in constructive reverse mathematics," Iris Loeb "Constructive version of Boolean algebra," Francesco Ciraulo, Maria Emilia Maietti, and Paola Toto "A generalized cut characterization of the fullness axiom in CZF," Laura Crosilla, Erik Palmgren, and Peter Schuster "Interpreting lattice-valued set theory in fuzzy set theory," Petr Hájek and Zuzana Haniková "On equality and natural numbers in Cantor-Łukasiewicz set theory," Petr Hájek "Identity taken seriously: a non-classical approach," Chris Mortensen "Strong, universal and provably non-trivial set theory by means of adaptive logic," Peter Verdee The entire issue can be found here.

New Waves in Philosophical Logic

Mon, 31 Dec 2012 00:00:00 +0000

Philosophical logic has been, and continues to be, a driving force behind much progress and development in philosophy more broadly. This collection of 12 original papers by 15 up-and-coming philosophical logicians deals with a broad range of topics, including proof-theory, probability, context-sensitivity, dialetheism and dynamic semantics. If the discipline’s past influence on the core areas of philosophy is anything to go by, then the scholars and the ideas in these pages are the ones to watch. Here are the essays in the volume. Introduction, Greg Restall and Gillian Russell How Things Are Elsewhere, Wolfgang Schwarz Information Change and First-Order Dynamic Logic, Barteld Kooi Interpreting and Applying Proof Theories for Modal Logic, Francesca Poggiolesi and Greg Restall The Logic(s) of Modal Knowledge, Daniel Cohnitz From Type-Free Truth to Type-Free Probability, Hannes Leitgeb Dogmatism, Probability and Logical Uncertainty, David Jehle and Brian Weatherson Skepticism about Reasoning, Sherrilyn Roush, Kelty Allen and Ian Herbert Lessons in Philosophy of Logic from Medieval Obligationes, Catarina Dutilh Novaes How to Rule Out Things with Words:Strong Paraconsistency and the Algebra of Exclusion, Francesco Berto Lessons from the Logic of Demonstratives, Gillian Russell The Multitude View on Logic, Matti Eklund

A Cut-Free Sequent System for Two-Dimensional Modal Logic, and why it matters

Thu, 22 Nov 2012 00:00:00 +0000

The two-dimensional modal logic of Davies and Humberstone is an important aid to our understanding the relationship between actuality, necessity and a priori knowability. I show how a cut-free hypersequent calculus for 2d modal logic not only captures the logic precisely, but may be used to address issues in the epistemology and metaphysics of our modal concepts. I will explain how use of our concepts motivates the inference rules of the sequent calculus, and then show that the completeness of the calculus for Davies–Humberstone models explains why those concepts have the structure described by those models. The result is yet another application of the completeness theorem. (This paper was awarded a Silver Medal for the Kurt Gödel Prize in 2011.)

Bradwardine Hypersequents

Fri, 03 Aug 2012 00:00:00 +0000

According to Stephen Read, Thomas Bradwardine’s theory of truth provides an independently motivated solution to the paradoxes of truth, such as the liar. In a series of papers, I have discussed modal models for Read’s reconstruction of Bradwardine’s theory. In this paper, provide a hypersequent calculus for this theory, and I show that the cut rule is admissible in the hypersequent calculus. (This paper is dedicated to Professor Stephen Read, whose work has been a profound influence on my own. His work on relevant logic, on proof theoretical harmony, on the logic of identity and on Thomas Brad- wardine’s theory of truth have been a rich source of insight, of stimulation and of provocation. In an attempt to both honour Stephen, and hopefully to give him some pleasure, I am going to attempt to cook up something original using some of the many and varied ingredients he has provided us. In this paper, I will mix and match ideas and techniques from Stephen’s papers on proof theory, on Bradwardine’s theory of truth, and on identity to offer a harmonious sequent system for a theory of truth inspired by Stephen Read’s recovery of the work of Thomas Bradwardine.)

History of Logical Consequence

Sat, 03 Mar 2012 00:00:00 +0000

Consequence is a, if not the, core subject matter of logic. Aristotle’s study of the syllogism instigated the task of categorising arguments into the logically good and the logically bad; the task remains an essential element of the study of logic. In this essay, we give a quick history of the way logical consequence has been studied from Aristotle to the present day.

Interpreting and Applying Proof Theories for Modal Logics

Sat, 03 Mar 2012 00:00:00 +0000

Proof theory for modal logic has blossomed over recent years. Many ex- tensions of the classical sequent calculus have been proposed in order to give natural and appealing accounts of proof in modal logics like K, T, S4, S5, provability logics, and other modal systems. The common feature of each of these different proof systems consists in the general structure of the rules for modal operators. They provide introduction and elimination rules for statements of the form □􏰅A (and ◇􏰆A), which show how those statements can feature as conclusions or as premises in deduction. These rules give an account of the deductive power of a modal formula 􏰅□􏰅A (and 􏰆◇􏰆A) in terms of the constituent formula A. The distinctive feature for the modal operators in contemporary proof systems for modal logics is that the step introducing or eliminating 􏰅􏰅□􏰅A is at the cost of introducing or eliminating some kind of extra structure in the proof. In this way, the proof rules for modal concepts such as 􏰅􏰅□􏰅 run in parallel with the truth conditions for these concepts in a Kripke model, in which the truth of 􏰅􏰅□􏰅A stands or falls with the truth of A, but at the cost of checking that truth elsewhere, at points in the model accessible from the point at which 􏰅􏰅□􏰅A is evaluated.

On the ternary relation and conditionality

Thu, 02 Feb 2012 00:00:00 +0000

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley–Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing a general conception of conditionality that may unify the three given conceptions.

Molinism and the Thin Red Line

Sun, 01 Jan 2012 00:00:00 +0000

Molinism is an attempt to do equal justice to divine foreknowledge and human freedom. For Molinists, human freedom fits in this universe for the future is open or unsettled. However, God’s middle knowledge – God’s contingent knowledge of what agents would freely do in this or that circumstance – underwrites God’s omniscience in the midst of this openness. In this paper I rehearse Nuel Belnap and Mitchell Green’s argument in ‘Indeterminism and the Thin Red Line’ against the reality of a distinguished single future in the context of branching time, and show that it applies applies equally against Molinism + branching time. In the process, we show how contemporary work in the logic of temporal notions in the context of branching time (specifically, Prior–Thomason semantics) can illuminate discussions in the metaphysics of freedom and divine knowledge.

Anti-Realist Classical Logic and Realist Mathematics

Thu, 13 Oct 2011 00:00:00 +0000

I sketch an application of a semantically anti-realist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically anti-realist defence of mathematical realism. In the paper, I develop the view and compare it to orthodox positions in the philosophy of mathematics.

Logic in Australasia

Sat, 01 Jan 2011 00:00:00 +0000

This is an idiosyncratic history of philosophical logic in Australia and New Zealand, highlighting two significant points of research in Australasian philosophical logic: modal logic and relevant/paraconsistent logic.

On the ternary relation and conditionality

Sat, 25 Dec 2010 00:00:00 +0000

Majer and Peliš have proposed a relevant logic for epistemic agents, providing a novel extension of the relevant logic R with a distinctive epistemic modality K, which is at the one and the same time factive (Kφ → φ is a theorem) and an existential normal modal operator (K(φ ∨ ψ) → (Kφ ∨ Kψ) is also a theorem). The intended interpretation is that Kφ holds (relative to a situation s) if there is a resource available at s, confirming φ. In this article we expand the class of models to the broader class of ‘general epistemic frames’. With this generalisation we provide a sound and complete axiomatisation for the logic of general relevant epistemic frames. We also show, that each of the modal axioms characterises some natural subclasses of general frames.

Always More

Tue, 16 Nov 2010 00:00:00 +0000

A possible world is a point in logical space. It plays a dual role with respect to propositions. (1) A possible world determines the truth value of every proposition. For each world w and proposition p, either at w, p is true, or at w, p is not true. (2) Each set of possible worlds determines a proposition. If S, a subset of W is a set of worlds, there is a proposition p true at exactly the worlds in S. In this paper, I construct a logic, extending classical logic with a single unary operator, which has no complete Boolean algebras as models. If the family of propositions we are talking about in (1) and (2) has the kind of structure described in that logic, then (1) and (2) cannot jointly hold. I then explain what this might mean for theories of propositions and possible worlds.

Testing, testing, 1, 2, 3

Tue, 16 Nov 2010 00:00:00 +0000

Just testing to see if this thing is on. Yes, it seems to be working. How about that!? It’s pretty rusty, but apparently this site still works. That’s good to see.

Decorated Linear Order Types and the Theory of Concatenation

Sun, 03 Oct 2010 00:00:00 +0000

We study the interpretation of Grzegorczyk’s Theory of Concatenation TC in structures of decorated linear order types satisfying Grzegorczyk’s axioms. We show that TC is incomplete for this interpretation. What is more, the first order theory validated by this interpretation interprets arithmetical truth. We also show that every extension of TC has a model that is not isomorphic to a structure of decorated order types. We provide a positive result, to wit, a construction that builds structures of decorated order types from models of a suitable concatenation theory. This construction has the property that if there is a representation of a certain kind, then the construction provides a representation of that kind.

Proof Theory and Meaning: the context of deducibility

Sun, 03 Oct 2010 00:00:00 +0000

I examine Belnap’s two criteria of existence and uniqueness for evaluating putative definitions of logical concepts in inference rules, by determining how they apply in four different examples: conjunction, the universal quantifier, the indefinite choice operator and the necessity in the modal logic S5. This illustrates the ways that definitions may be evaluated relative to a background theory of consequence, and the ways that different accounts of consequence provide us with different resources for making definitions.

Barriers to Consequence

Thu, 15 Jul 2010 00:00:00 +0000

In this paper we show how the formal counterexamples to Hume’s Law (to the effect that you cannot derive a properly moral statement from properly descriptive statements) are of a piece with formal counterexample to other, plausible “inferential barrier theses”. We use this fact to motivate a uniform treatment of barrier theses which is immune from formal counterexample. We provide a uniform semantic representation of barrier theses which has applications in the case of what we call “Russell’s Law” (you can’t derive a universal from particulars) and “Hume’s Second Law” (you can’t derive a statement about the future from statements about the past). We then finally apply these results to formal treatments of deontic logic to show how to avoid formal counterexamples to Hume’s Law in a plausible and motivated manner.

Relevant Agents

Wed, 03 Mar 2010 00:00:00 +0000

On t and u, and what they can do

Tue, 16 Feb 2010 00:00:00 +0000

This paper shows that once we have propositional constants t (the conjunction of all truths) and u (the disjunction of all untruths), paradox ensues, provided you have a conditional in the language strong enough to give you modus ponens. This is an issue for views like those of Jc Beall, Ross Brady, Hartry Field and Graham Priest.

What are we to accept, and what are we to reject, when saving truth from paradox?

Mon, 18 Jan 2010 00:00:00 +0000

In this article, I praise Hartry Field’s fine book Saving Truth From Paradox (Oxford University Press, 2008). I also show that his account of properties is threatened by a paradox, and I explain how we can only avoid this paradox by coming to a clearer understanding the connections between accepting and rejecting (or assertion and denial) and the identity conditions for properties.

Models for Substructural Arithmetics

Thu, 31 Dec 2009 00:00:00 +0000

This paper explores models for arithmetics in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R. This paper is dedicated to Professor Robert K. Meyer.

On Permutation in Simplified Semantics

Tue, 03 Nov 2009 00:00:00 +0000

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Journal of Philosophical Logic 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) and permutation (A → (B → C)) → (B → (A → C)). We show that the modelling conditions for assertion and permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. In this note, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose a correction.

Time Flies

Tue, 20 Oct 2009 00:00:00 +0000

Time flies like an arrow.
Fruit flies like a banana.

If you’ve been following my twitter feed, you’d realise I’m still alive. You wouldn’t think that from the activity – or lack thereof – here. (Though a few papers have appeared – or changed their publication status – on my writing page.)

Here’s where we are: It’s been a busy, eventful semester, and the teaching period is almost done. I’ve had fun teaching proof theory to fourth-year students, tutoring intro philosophy to first years, and supervising graduate students (at last count, I have eight current research students in various stages of the degrees). One of the sadder things to befall us here at Melbourne is the departure of Allen Hazen, who as left our shores for the chillier climes of Edmonton. The Melbourne logic community’s loss is Canada’s gain here.

A Priori Truths

Fri, 17 Jul 2009 00:00:00 +0000

Philosophers love a priori knowledge: we delight in truths that can be known from the comfort of our armchairs, without the need to venture out in the world for cofirmation. This is due not to laziness, but to two different considerations. First, it seems that many philosophical issues aren’t settled by our experience of the world – the nature of morality; the way concepts pick out objects; the structure of our experience of the world in which we find ourselves – these issues seem to be decided not on the basis of our experience, but in some manner by things prior to (or independently of) that experience. Second, even when we are deeply interested in how our experience lends credence to our claims about the world, the matter remains of the remainder: we learn more about how experience contributes to knowledge when we see what knowledge is available independent of that experience. In this essay we will look at the topic of what can be known a priori.

Live from Hejnice

Fri, 19 Jun 2009 00:00:00 +0000

Posting has been light, since I’ve been powering through work at the end of the semester, and getting ready for a quick trip west to Europe, for Non-Classical Mathematics 2009 and Logica 2009, preceded by a quick visit to Dresden to see Heinrich Wansing, and to break up the train trip from Frankfurt to Prague.

So, posting here has paused for a bit, but now that I’m settled in Hejnice and that there’s a wireless connection here, I can deal with some of my backlog of things I’ve promised to post. So, here’s a salad of links for you.

Rumfitt on Multiple Conclusions, Part 2

Tue, 02 Jun 2009 00:00:00 +0000

This is Part 2 of a series of comments on Ian Rumfitt’s paper “Knowledge by Deduction” (Grazer Philosophische Studien, vol. 77 (2008) pp. 61–84). In Part 1, I focussed on Rumfitt’s direct criticism of my approach in “Multiple Conclusions,” and I tried to show that his criticism missed the mark, and that it missed the mark in an important way. The norms of logical consequence and logical coherence apply not only to occurrent beliefs but to all manner of states of accepting and rejecting (or acts of assertion and denial), whether they express our deep standing beliefs or hypotheses we simply entertain lightly.

In this part, I want to consider the comments on the possibility of genuine proofs with multiple conclusions.

Rumfitt on Multiple Conclusions, Part 1

Mon, 01 Jun 2009 00:00:00 +0000

Thanks to Ole Hjortland, I’ve been alerted to Ian Rumfitt’s paper “Knowledge by Deduction” (Grazer Philosophische Studien, vol. 77 (2008) pp. 61–84.). In it, he makes a number of critical comments on multiple conclusion accounts of logical consequence, and in particular, he makes some critical remarks on my paper “Multiple Conclusions.” Now, the criticism of mutiple conclusion consequence isn’t the main point of the paper—the main topic is how one can acquire knowlege by deduction, as the title indicates. On that topic, it’s a really interesting paper, and I hope to comment on those parts at some time.

However, since the paper ends with the sentence

But we have found reason to leave multiple-conclusion logics to the boy racers, and focus on the single-conclusion rules, by following which we can splice together the deliverances of various sources of knowledge to come to know things that we could not know otherwise. (page 83)

I’ve got to respond.

Problems for Naïve Property Theories

Thu, 21 May 2009 00:00:00 +0000

I’ve been thinking about generalisations of Russell’s paradox, cleaning things up so you can’t get around the problem by changing the logic of connectives. I don’t think that mucking around with negation or implication gets to the heart of the issue. (This view is shared by some very insightful people. I haven’t come to it alone.)

Getting around negation and conditionals is surprisingly easy, once you get the proof theory sorted out. I’ve been noodling about with this issue for a year or so now. I presented on this in a talk at the World Congress of Paraconsistency last year, and a bit of it has appeared in my draft discussion of some themes from Hartry Field’s Saving Truth From Paradox.

There, the paradoxical derivations are done in sequent calculi, and they’re not the most perspicuous presentation. I managed to sharpen it up a bit tonight, and the resulting proof is here. It’s not explained in the text of that note: that gives just the definitions and the proof. I hope to get to that soon. But let me use this site to get the ideas out in a rough and ready form.

Bob Meyer

Thu, 07 May 2009 00:00:00 +0000

Earlier today I received the sad news that Bob Meyer, my former colleague at the ANU, and friend, two-time collaborator, died last night, after a long struggle with cancer.

Bob will be sorely missed by many of us. His warmth and humour, his brilliance, and his willingness to talk logic (and much more) to anyone and everyone, will all be impossible to replace. If I manage to show a small fraction of both his logical insight, and his ability to communicate difficult concepts with good humour and wit, I’ll be a happy philosophical logician.

Truth Values and Proof Theory

Fri, 01 May 2009 00:00:00 +0000

In this paper I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, feature as an idealisation of more fundamental logical features arising out of the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical consequence.

More on Words

Fri, 24 Apr 2009 00:00:00 +0000

Allen Hazen pointed me to this nice interview Julian Bagnini conducted with Ernie Lepore, on words. Lepore comes to the same sort of conclusion as Kaplan – that identity conditions for words are tricky. He’s right.

Using Peer Instruction to Teach Philosophy, Logic and Critical Thinking

Mon, 20 Apr 2009 00:00:00 +0000

We explain how Eric Mazur’s technique of Peer Instruction may be used to teach philosophy, logic and critical thinking — to good effect.

Types, Tokens and Names

Fri, 27 Mar 2009 00:00:00 +0000

The quote from yesterday’s quiz was from the inimitable David Kaplan, in the article “Words” (Proceedings of the Aristotelian Society, Supplementary Volume, LXIV 1990). As Robbie mentioned in the comments “Words” is such a cool paper. I want to give an example of how cool it is.

It’s tempting to think of words, repeatable things that the are, as the types of token inscriptions, jottings, arrangements of pixels, utterances and all of the other kinds of ways we find to express words. Kaplan argues in “Words” that this isn’t right. I won’t rehearse the argument here, but I’ll tell you about one example he gives along the way.

Quiz for today

Wed, 25 Mar 2009 00:00:00 +0000

Here is today’s quiz question. Which master of exposition said this, and where? After arguing for years, unconvincingly, that semantic value (properly understood) is not affected by substitution, I hit upon a brilliant, new, and completely successful, strategy: argue, instead, that semantic value is affected by substitution. Here’s a hint: the quote occurs after in the context of a discussion of proper names. (Upon reflection, that’s not much of a hint, is it?) Post your guess as to who I’m quoting, in the comments form on this post. (You can post an informed answer too, if you like, but I suspect a guess would be even more fun.) The most interesting answer will receive a prize in the honest-to-goodness snail mail.

Spandrels of Truth

Wed, 25 Mar 2009 00:00:00 +0000

Whoever thought that teaching in three subjects, chairing one committee, participating in another one with a honking big project to keep on the rails for the next few months, supervising n research students (for a seriously large value of ‘n’) and trying to keep your research ticking over would keep you busy with work? I did think that. But now I know it by acquaintance, and not merely by dispassionate theoretical reflection.

Anyway, I’m not here to moan. I’m here to make you an offer.

Back in the saddle

Tue, 24 Feb 2009 00:00:00 +0000

I’m back in Melbourne. Z and I have been back for over a week – and I’ve been back at work since the middle of last week, slowly setting up while Z has been settling into school. C returns tomorrow, and the place will feel like home at last. The philosophy blogosphere is all a-flutter with the release of the 2009 Leiter Report. We have known for some time that the troubles here at Melbourne would impact our rating and the result is there for all to see. We’re out of the top ranking in Australia, sinking below Monash in the report’s league tables. Still, it’s nice to recognise that in the discipline rankings we’re notable in Applied Ethics, Feminist Philosophy, Mathematical Logic, Philosophy of Mathematics and Philosophical Logic. For a very small core of a department, we do rather well. We have a very nice group from which to rebuild a decent department – the trick will be managing the rebuilding phase. Please wish us luck (or if you can, lend us your support). The trip was wonderful. I needed the break, and I enjoyed visting Pitt, CMU and UConn (thanks Shawn, Anil, Bob, Nuel, Kevin, Steve, Kohei, Horacio, Jc, Katrina, Marcus, Aaron, Colin, Reed, Michael, Scott, and everyone else, for your hospitality), and we had much fun at each stop on the way.

Holiday, in progress

Wed, 14 Jan 2009 00:00:00 +0000

We’re having a wonderful time on our holiday, cruising through the southwest of the USA. The sights have been spectacular, varied and unexpected. The highlight so far was Bryce Canyon. The Grand Canyon is grand, and sublime in its own way. Bryce Canyon was totally unexpected: I had no idea erosion could do just that. Z ran around like a madman enjoying the snow, and the awesome view. But the highlight for Z has been throwing snowballs, something doesn’t feature in the average Melbourne kid’s average day. Bigger photos, for those who want to keep up, are accumulating in the gallery. After Zion National Park, we’re off to Las Vegas. There, we’re looking forward to meeting Greg Frost-Arnold in our couple of days there. (As well as the sensory overload of the Strip, of course.)

Holiday!

Wed, 07 Jan 2009 22:30:40 +1100

We’re going on a holiday! Tomorrow at noon, we’ll be on a Qantas 747 to LA, and at the other end, the three of us will be enjoying four weeks of break. It’s mostly time off for all of us, with a few detours for work talks for me (at Pitt, and at UConn) and for Christine (at Harvard, San Diego, Berkeley and maybe Stanford). It’s our ‘consolation tour’ in place of what was our original plan: a year in the USA, based in Pittsburgh. That fell through—it’s a complicated story, all to do with Melbourne, nothing to do with Pitt, where people have been generous, understanding and gracious. Anyway, I’m not visiting Pitt for a year, but we’re touring the US, having a break, seeing the sights (like the Grand Canyon, Death Valley, and Las Vegas) and visiting friends. At Pittsburgh, I’ll finally get to meet Shawn whose blog I’ve read for ages, and Anil, whose work I’ve read and learned from for much longer. At UConn, I’ll catch up with Jc, which is always a blast, and Marcus, who’s moved there. I’ve also got the honour of giving the inaugural annual lecture of the UConn Logic Group. I won’t be posting here unless I get web access, and I think that this isn’t a priority on the holiday.

Little Snippets of News

Wed, 07 Jan 2009 00:00:00 +0000

Here are some random, miscellaneous items of news, concerning what’s been happening around here, around now. It’s exhausting, but lots of fun, to host visitors. Since just before Christmas, we’ve had visits from two good friends (separately) and four family members (one batch of three and one solo), as well as meals with other friends over the Christmas/Summer period. Much fun was had by all. It’s really easy to share good experiences in Melbourne with family and friends. Highlights have included Day 2 of the Boxing Day test, Shane Warne: the musical, and an evening watching (or re-watching) the Sound of Music and singing along. I’ve finally redesigned the site in a way I’m happy to see and to use. Again. There are two causes, one proximate, and one more long-term. The proximate cause was the superabundance of spam to moderate, on my own server. Logging in and cleaning out spam, arriving at the rate of at least one a minute, was depressing. So no more hosted weblog software for me. Instead, this site is now hand crafted on my computer (using Webby—it’s been fun to learn Ruby) and uploaded to the server as a static site. When I want comments on an item, I can implement them using intense debate or some other commenting service.

Appendix to 'Yo!' and 'Lo!': the pragmatic topography of the space of reasons

Thu, 01 Jan 2009 00:00:00 +0000

The appendix to Kukla and Lance’s book is the preliminary development of a score-keeping semantics that begins with the broader field of vision opened up when we eschew the declarative fallacy and make use of abstract versions of the pragmatic distinctions developed in the body of the book.

Merry Christmas, all

Wed, 24 Dec 2008 00:00:00 +0000

It’s Christmas break here, and I’m taking a break, with visits from family and friends, and then a trip to the USA for a holiday (and for a couple of little side trips to see friends and give a few talks). More about that in a little while. To share the Christmas spirit, I’ll share a favourite carol: The tree of life my soul hath seen, Laden with fruit and always green: The trees of nature fruitless be Compared with Christ the apple tree. His beauty doth all things excel: By faith I know, but ne’er can tell The glory which I now can see In Jesus Christ the apple tree. For happiness I long have sought, And pleasure dearly I have bought: I missed of all; but now I see ’Tis found in Christ the apple tree. I’m weary with my former toil, Here I will sit and rest awhile: Under the shadow I will be, Of Jesus Christ the apple tree. This fruit doth make my soul to thrive, It keeps my dying faith alive; Which makes my soul in haste to be With Jesus Christ the apple tree. The plan is to reboot the blog (and to fix the comment system, which is down for the count just now) for 2009.

Always More...

Tue, 09 Dec 2008 00:00:00 +0000

(What follows is a bundle of ideas I’ve been trying to write up for some time. Instead of making it a fully fledged paper, I’ve decided to rough it out first as a blog post. If anything comes of it, I’ll polish it up. Let me know what you think.) Are there any possible worlds? The idea of a point in logical space – at which every proposition is either True or False – seems at the one and the same time compelling and repellant. The notion plays a vital role in the semantics of logics of modality and conditionality, and so, is compelling. It is hard to take modal logic seriously without points in models that play the role of deciding every statement one way other the other. But to take possible worlds seriously as more than a useful fiction has seemed too great a price for many to pay. This squeamishness seems, to many, to have a distinctly ‘ontological’ flavour. Places at which there are blue swans or in which kangaroos have no tails seems to crowd the halls of being with blue swans and tailless kangaroos. We would do better without such things if we can. If we can explain possible worlds away – as propositions, or stories or abstracta or something else relatively tame – then we should.

New Paper: Assertion, Denial and Non-Classical Theories

Thu, 04 Dec 2008 00:00:00 +0000

I’m on a roll with the writing: I’ve managed to get another paper converted from slides-from-a-talk to real, live, words on a page and actual proofs. The paper I’ve worked on over the last couple of days is from my presentation from WCP4 on non-classical theories like non-classical theories of numbers, classes, or truth. I’m very happy with this one, as it’s an application of my recent work – on proof theory – to a topic I’ve been writing about for some time, non-classical theories like naïve theories of classes or of truth. Here’s the abstract. In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic.

Models for Liars in Bradwardine's Theory of Truth

Wed, 03 Dec 2008 00:00:00 +0000

Stephen Read’s work on Bradwardine’s theory of truth is some of the most exciting work on truth and insolubilia in recent years. In this paper, I give models for Read’s formulation of Bradwardine’s theory of truth, and I examine the behaviour of liar sentences in those models. I conclude by examining Bradwardine’s argument to the effect that if something signifies itself to be untrue then it signifies itself to be true as well. We will see that there are models in which this conclusion fails. This should help us elucidate the hidden assumptions required to underpin Bradwardine’s argument, and to make explicit the content of Bradwardine’s theory of truth.

New Paper: Truth Values and Proof Theory

Tue, 02 Dec 2008 00:00:00 +0000

It’s good to get back into writing. Here’s a paper “Truth Values and Proof Theory” that I’ve been thinking about for a long time. I presented a research seminar on this material last year – it’s taken me this long to write it up, due to other commitments. Here’s the abstract: In this paper I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, feature as an idealisation of more fundamental logical features arising out of the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical consequence. I like these results, as they’re a mix of motivating philosophy, and formal proofs. I think I’m getting a better understanding of the relationship between the [Cut] rule in a sequent calculus and the maximality conditions involved in the behaviour of things like two-valued evaluations, possible worlds and points in model structures for other sorts of logics. It’s not a coincidence that sequents have two sides, and that there are two truth values.

Back! Then off, then back again!

Thu, 27 Nov 2008 15:34:50 +1100

My trip to Guangzhou was super. Thanks, Guoxin, Min and Xuefeng for looking after me and showing me the sights of Guangzhou, and to Professors Ju and Zhu for the invitation. It was a wonderful experience for me, and I look forward to many more visits to China, and to the Institute of Logic and Cognition. Since then, I’ve been off to Canberra for a little workshop on hyperintensionality, organised by David Chalmers. Much fun was had there, too, but I must admit I’m even more confused than I was when I started. It must have been all that talking to Max Cresswell, Lloyd Humberstone and John Bigelow about how semantics works.

Off to Guangzhou

Fri, 14 Nov 2008 21:13:50 +1100

The primary purpose of this site seems to be travel announcements, so rather than break the trend, I’ll post another travel announcement. On Sunday, I’m off to the Institute of Logic and Cognition at Sun Yat-Sen University, to give three lectures on my research. This is my first trip to China, so I’m more excited (and more nervous) than usual for an overseas trip.

Truthmakers, Entailment and Necessity 2008

Fri, 03 Oct 2008 00:00:00 +0000

I update “Truthmakers, Entailment and Necessity” with a response to Stephen Read's wonderful Mind 2000 argument to the effect that I got the disjunction thesis wrong. I think I didn't get it wrong, but figuring out why it's OK to hold that a truthmaker makes a disjunction true iff it makes a disjunct true is not as straightforward as I first thought. Pluralism about truthmaking makes an entrance.

Modal Models for Bradwardine's Theory of Truth

Mon, 15 Sep 2008 00:00:00 +0000

Stephen Read has recently discussed Thomas Bradwardine’s theory of truth, and defended it as an appropriate way to treat paradoxes such as the liar. In this paper, I discuss Read’s formalisation of Bradwardine’s theory of truth, and provide a class of models for this theory. The models facilitate comparison of Bradwardine’s theory with contemporary theories of truth.

Tartu Pluralism Days #3 and #4

Mon, 01 Sep 2008 07:12:17 +1100

Phew! What a long rest-of-the conference! I’m now back in Tallinn, after the rest of the conference: 5 talks on Day 3, and one wrapup talk on Day 4. My comments concerning Days 3 and 4 will not be as extensive as those for Days 1 and 2. I liked these as much as the earlier talks – it’s just that my stamina has flagged. Day 3 The day kicked off with Per Martin-Löf presenting ‘Is Logic about Consequence?’ I really enjoyed finally getting the opportunity to hear Per speak, as his view of logical consequence – a particular constructive type theory in which proof objects play a central role – has always fascinated me, but I’ve not looked into it in any depth. Per argued that logic is not primarily about consequence, but must be also about the act of inference, and that distinguishing content and force in the judgement helps us understand the difference. I think that this is clearly right, and in my work since the Logical Pluralism book I’ve started to attend to these issues. In “Multiple Conclusions” I attended to the connection between valid sequents and the acts of assertion and denial. While the account I give there is not one that Per would endorse, the constellation of issues connecting content and force, and the correctness of the steps we make in inferring conclusions from premises, are certainly in the domain of logic.

Tartu Pluralism Day #2

Fri, 29 Aug 2008 17:46:48 +1100

OK, it’s a short day here today – thankfully, since I’m knackered after Day #1 – with two talks. Last night we dined at an Italian Restaurant, and I learned more about Roger Swyneshed from Stephen Read, than I ever expected to learn. Talks today are: Peter Pagin, on Universalist and Actualist Consequence. Peter connected his work with Kathrin Glüer on relational modality and the semantics of names with considerations concerning the difference between universal validity (truth in all possible worlds whatsoever) and actualist validity (truth in all candidate actual worlds – down the diagonal in a 2D semantics, for example). He indicated that his looks like our distinction, both should count as validity notions which ought to be endorsed. Peter wondered whether this was a problem, and whether there’s a conflict between endorsing universalist validity and actualist validity. This is a good question: I think that this is a good case where being a pluralist makes sense: if we think of validity as classifying deduction steps into those that are valid and those that aren’t, then actualist and universalist validity notions have their place. Take the step from p to actually p. This is OK in one obvious sense (we’d make a mistake to assert p and deny actually p), and bad in another (that step doesn’t work if we are under a counterfactual assumption – we grant that not p, and then consider what would have happened were p the case – we can’t infer from that to actually p).

Tartu Pluralism Day #1

Thu, 28 Aug 2008 23:57:40 +1100

We’re coming semi-live from Tartu, as long as my computer’s battery lasts, anyway. For live off-the-cuff comments, try my experimental twitter stream, which is profoundly silly, but fun to fiddle with on my phone when my fingers are itchy. The lineup today. Me, giving “Pluralism and Proofs.” I think I went well. Look at Ole’s sneak-preview of what I said, which is pretty-well what I said here. The fact that Dag Prawitz and Per Martin-Löf didn’t eat me alive, but were actually interested, was profoundly reassuring. I’ve only done the slides, not a written version of the paper. Graham Priest on “Logical Pluralism as another application of Chunk-and-Permeate.” This is an interesting project, conceiving of logic’s application to a domain of reasoning as variegated – a domain of discourse can be ‘chunked’ into different bits, where different logics are operative, and then ‘permeation’ relations tell us how facts in one chunk transfer over to facts in another. Graham was using this general setup (which he introduced in earlier work with Bryson Brown, on the infinitesimal calculus – one chunk is where the infinitesimal is zero, another chunk is where it’s nonzero, and you only permeate facts about non-infinitesimals, or something like that).

Tartu

Thu, 28 Aug 2008 13:19:08 +1100

If this is Thursday, it must be Tartu. The conference starts today, and I’m up first. The nerves have started to jangle just a bit, since the audience is about as high-powered as any in my experience at international conferences. It’s just as well as those who intimidate me the most this time around (due to their all-round awesomeness combined with the fact that this is the first time I’ve met them) – Per Martin-Löf, Dag Prawitz and Johan van Benthem – are uniformly warm and personable. (I can’t help but wonder if that will make it better or worse when the knife gets sunk in to my position.) Depending on time, attention and lots of other factors, I’ll try blogging a bit from the conference, though I won’t promise to keep up through the entire thing. So, if you’re here and I don’t mention your presentation, please don’t hold that against me. It’s hard enough to keep your attention up over an entire conference, let alone write something sensible, meaningful and worth-posting-on-the-internet about each and every talk at a conference. My talk starts in roughly three hours from now. Wish me luck!

Amsterdam!

Sun, 24 Aug 2008 23:37:56 +1100

I’ve arrived in Amsterdam. Getting here was touch-and-go. My decent wait between my Melbourne → Singapore flight and my Singapore → Amsterdam leg whittled down to around 45 minutes. That involved a trek through the terminal, a tense wait at Transfer Desk C (where people in the queue in front of me were arguing with the staff about visas, and people behind me were concerned that they wouldn’t get on the flight – and were expressing this concern, rather vocally). I got there in the end, but it was a rush. Highlight of the trip? Reading 100 pages of Anil Gupta’s Empricism and Experience. This is very good. It’s (1) logically sharp, (2) applied to philosophical issues of deep concern and most interestingly (3) original and creative, and enlightening. He cuts at deep issues with classical empiricism, at a level of abstraction that – I’m convinced – gets to the heart of the matter in a new and illuminating way. Characterising classical empiricism as the result of a number of theses concerning what is given in experience (that it’s propositional, veridical, and multiply factorisable), and that this combination – rather than the argument from Illusion and the debate surrounding this – that dooms classical empiricism to never explain the epistemic success we have.

Bag packed, let's go!

Sat, 23 Aug 2008 10:43:42 +1100

Bag packed, ticket & passport at the ready, computer charged, phone charged and synced, books to read located, papers to read/referee/assess all printed or downloaded. I must be going on a flight. The trip takes me from Melbourne to Singapore to Amsterdam to Tallinn to Tartu, and then I retrace my steps to get back here in ten days. That’s too much plane and airport time for my money, but I fully expect that this will be worth the discomfort. Oh, the day-and-a-bit stopover in Amsterdam should be worth it too.

Pain, stress, redundancies, another day at the office

Wed, 20 Aug 2008 10:35:03 +1100

I had thought that we (the Faculty of Arts at the University of Melbourne) were trying to keep our difficulties an internal affair, but apparently we’re not. Given that this is all public news, I suppose I could comment. The take-home-message from this memo from the Dean: (1) The budget in the Arts Faculty is not yet in the black, though it’s getting better, and (2) The plan is for more voluntary ‘redundancies’ (this is now Round 3), this time with the shadow of involuntary redundancies is we don’t get enough voluntary ones to right the ship. The discipline of Philosophy took an almighty hit at the end of Round 1. At the end of 2007 we had 5 departures out of a full-time teaching staff of 10.5. So we’ve done rather more than our part of the deal in righting the budgetary situation – except for the very real problem of ‘overcorrecting,’ making it rather more difficult to teach a coherent Philosophy major in the BA (with keen, bright students ready to learn), supervise our wonderful graduate students, and get research done. There are so few philosophers left, this round of redundancies is not directed at us, but we feel for our colleagues down the corridors, and across the campus.

Random interesting fact (one in an intermittent series)

Tue, 19 Aug 2008 08:05:55 +1100

Out of the five Australians who were chosen as nominating editors of this year’s Philosophers’ Annual, four are from The University of Melbourne. The lucky 4 are Hazen, Schroeter (François), Schroeter (Laura), and Restall. (The other Australian is David Chalmers.) 4 out of 28 is a good strike rate (that’s 1 in 7, if you’re too lazy to do the mental arithmetic). This is a teensy bit higher than the representation of University of Melbourne Philosophers in the Whole Wide World.

Breaking Silence

Sat, 16 Aug 2008 20:11:34 +1100

I’ve been pretty quiet here, lately. I’ve been busy with things, getting stuff done. I hope to be posting more soon, when various up-in-the-air things are more settled. Still, here are some recent highlights which are worth sharing. WCP4 was a blast. It was very busy, but lots of fun was had: it was good to catch up with many folks I hadn’t seen for ages including (Diderik Batens, Bryson Brown, Francesco Paoli, Ole Hjortland, Jerry Seligman, Gill Russell and especially JC, who I hadn’t seen in an all-too-long period of time) and to make the acquaintance of new folks (Francesco Berto, David Ripley, Andreas Pietz, Ben Burgis, Patrick Girard and many others). It’s been lots of fun to have Andi Pietz, Dave Ripley and Ole Hjortland hang out with us in Melbourne this semester. I think they’re having fun, and I’m having fun talking logic with them. Reasoning & Uncertainty is my new level 2/3 undergraduate subject. I’ve got a class of 60, doing philosophy of probability, decision theory, etc., and we’re having a lot of fun. Having students turn up to class, enthusiastic, despite 9am lectures, is something to behold. Conrad Asmus is my tutor, and working with him makes things very easy.

Logical Pluralism, in Tartu

Wed, 16 Apr 2008 06:42:39 +1100

I’m off to Tartu in late August, for what looks like a really fun conference on logical pluralism. The list of speakers is extraordinarily high-powered. Having so many smart people talk about logical pluralism is exciting, and not a little bit scary. If you’re interested in coming along, let the organisers know by July 1. If you’re interested in the state of the art on logical pluralism, this conference is the place to be. Thanks, Daniel, Marcus and Peter for organising this. It’s going to be my highlight of the second half of the year.

Informal Logic: now open access

Wed, 09 Apr 2008 11:09:44 +1100

The journal Informal Logic, on argumentation theory and related issues, is now open access and online, after a 27 year history as a print (closed access) journal. The editors have sent out a request for interested parties to subscribe. Subscription is free: just fill in this form on the site. The editors are aiming get enough subscribers to secure funding from the SSHRC to keep the journal going now that they’re not taking paid subscriptions. This is a worthy goal: the more open access journals, the better! So, if you’re at all interested in argumentation theory and informal logic, please sign up.

We're in the news...

Sun, 30 Mar 2008 20:03:58 +1100

Alas, we are in the news for not-very-good reasons. Given the recent retirements in Philosophy at Melbourne, it would be worse than surprising if any of the new redundancies hit Philosophy. However, the atmosphere around here not good – what concerns us most is the scope for new hiring when the Faculty as a whole is in the red so much. It is very hard to effectively plan ahead when so much is in flux and you’re not sure whether your efforts will benefit the discipline. I hope to find out more when we hear what the Dean and the Vice Chancellor has to say at a meeting of all Arts Faculty staff this Tuesday. Let’s hope that they can articulate something that goes beyond slashing positions to rectify a budget hole. I’m sorry to start posting on something so grim, but “trouble at t’mill” has been on our minds since mid-2007. I hope to post more about more enjoyable things soon. Meanwhile, take a look at my writing page. There are quite a few new items there.

A Participatory Theory of the Atonement

Mon, 03 Mar 2008 00:00:00 +0000

We argue that the participatory language, used in the New Testament to describe the the efficacy of Jesus’ death on the cross, is essential for any understanding of the atonement. Purely personal or legal metaphors are incomplete and perhaps misleading on their own. They make much more sense when combined with and undergirded by, participatory metaphors.

Assertion and Denial, Commitment and Entitlement, and Incompatibility (and some consequence)

Mon, 03 Mar 2008 00:00:00 +0000

In this short paper, I compare and contrast the kind of symmetricalist treatment of negation favoured in different ways by Huw Price (in “Why ‘Not’?”) and by me (in “Multiple Conclusions”) with Robert Brandom’s analysis of scorekeeping in terms of commitment, entitlement and incompatibility. Both kinds of account provide a way to distinguish the inferential significance of “A” and “A is warranted” in terms of a subtler analysis of our practices: on the one hand, we assert as well as deny; on the other, by distingushing downstream commitments from upstream entitlements and the incompatibility definable in terms of these. In this note I will examine the connections between these different approaches.

Curry's Revenge: the costs of non-classical solutions to the paradoxes of self-reference

Mon, 03 Mar 2008 00:00:00 +0000

I point out that non-classical “solutions” the paradoxes of self-reference are non-particularly easy to give. Curry’s paradox is very very hard to avoid, if you wish to give a semantically cohrerent picture.

Envelopes and Indifference

Mon, 03 Mar 2008 00:00:00 +0000

We diagnose the two envelope “paradox”, showing how the indifference principle plays a role in prompting the conflicting assignments of the expected outcomes for switching or keeping.

Sorry...

Wed, 13 Feb 2008 22:40:20 +1100

This has been a good day. (Here is the video — in three parts.) As PJK said, when you change the government, you change the country…

Review of Ross Brady, Universal Logic

Sat, 01 Dec 2007 00:00:00 +0000

This solid volume with the ominous black cover and eerie glowing disc lettered with inscrutable strings of characters such as “DNdQ,” “LSDJd,” and “L2LDJQ±” is the fruit of over 30 years of Ross Brady’s logical labours. And it is worth the wait…

Invention is the Mother of Necessity: modal logic, modal semantics and modal metaphysics

Wed, 03 Oct 2007 00:00:00 +0000

Modal logic is a well-established field, and the possible worlds semantics of modal logics has proved invaluable to our understanding of the logical features of the modal concepts such as possibility and necessity. However, the significance of possible worlds models for a genuine theory of meaning–let alone for metaphysics–is less clear. In this paper I shall explain how and why the use of the concepts of necessity and possibility could arise (and why they have the logical behaviour charted out by standard modal logics) without either taking the notions of necessity or possibility as primitive, and without starting with possible worlds. Once we give an account of modal logic we can then go on to give an account of possible worlds, and explain why possible worlds semantics is a natural fit for modal logic without being the source of modal concepts. This paper is an early draft, and comments are most welcome.

Melbourne Philosophy Undergraduate Workshop

Mon, 06 Aug 2007 11:32:02 +1100

Are you an undergraduate student in philosophy, or do you know any undergraduate philosophy students? If so, you might be interested in the Melbourne Philosophy Undergraduate Workshop to be held from September 21 to 23. This will be a chance for Australian and New Zealand undergraduate students who are interested in philosophy, to get together to talk philosophy with the faculty here at the University of Melbourne. Given that we have a broad range of interests, from Continental and Asian Philosophy, through Ethics and Social Philosophy, Philosophy of Science, to Metaphysics and Logic, this will be a fun weekend. The deadline for applying for a grant to come to the workshop is coming up very soon: it’s August 15. For more details of how to be involved, read on… During the weekend of September Friday 21 to Sunday 23, Melbourne University’s School of Philosophy will host the Melbourne Philosophy Undergraduate Workshop, aimed at third and fourth year students who intend to continue in any branch of academic philosophy. Friday evening will begin with a Welcome Meeting followed by dinner on Lygon Street, Carlton. On Saturday, Melbourne faculty members will give presentations and/or lead seminars on a range of topics – from Continental and Asian Philosophy, through Philosophy of Science, to Metaphysics and Logic.

Talk on the Philosopher's Zone

Sun, 05 Aug 2007 17:26:54 +1100

I gave a talk on Logic in Australia at Monash University’s Arts in Action festival in early June. (This was a part of a long-running project on a History of Australasian Philosophy. The talk is now appearing, in two parts, on the (wonderful) ABC Radio National program, The Philosopher’s Zone. The first part was broadcast this weekend, but the audio can be downloaded from their website for the next four weeks. That was the first half of the talk, on possible worlds. The second half, on paraconsistency, will be broadcast next week. In the talk, I made heavy use of digital projection. I think you can follow the talk without it, but if you want to see the pretty pictures and diagrams, you can download the slides of the talk here: Logic in Australia.mov 22.5MB Quicktime file. (Navigate it by clicking or by using the arrow keys.) Logic in Australia.pdf 3.3MB PDF file. (This does not have all of the fancy transitions in the quicktime file.) Logic in Australia.key.zip 3.8MB compressed keynote file. (This has all the fancy transitions, and is the original document, but it requires Apple’s Keynote presentation program to view.)

... and we're back

Sun, 05 Aug 2007 16:36:53 +1100

I’ve been back in Melbourne for a while. The trip was very enjoyable, but I’ve returned to Melbourne with a chest bug, which has meant that I’m not quite up to full speed yet. All non-essential activities (and alas, some essential ones, I fear) are progressing much more slowly than usual. Here are some highlights of the trip. Logica 2007 with so many great people, and then exploring Prague, catching up with Richard, and getting to know more Czech logicians. Overdosing on Medieval logic in Bonn, and especially talking over Bradwardine with Stephen Read. Exploring København with Christine and Zachary. It was my second visit there, and Christine’s umpteenth, so it was a lot easier to get around given than we were more familiar with the place. Navigating our way on the trains and buses from Århus to Legoland in Billund with Zachary, and then enjoying all there was to see and do there. Meeting everyone at Logic Colloquium in Wrocław – and especially thinking through – with Pavel Pudlák, Albert Visser, Alasdair Urquhart and Vedran Čačić – a counterexample to a conjecture of Grzegorczyk’s. The result of our thinking is here. Most of the thinking got done in a hike in the hills near Wrocław.

... arrived!

Mon, 18 Jun 2007 21:19:55 +1100

I’ve arrived in a sunny and summery Prague. The locals tell me they it’s cooler than the unseasonably warm that it’s been in the last few years, but it seems nicely summery for one who has come from a winter. I’m now chatting in the office with Timothy, Jarda and Vladimir, and monopolising Jarda’s computer while the others do work preparing for the conference.

Heading off...

Sun, 17 Jun 2007 14:45:28 +1100

I’m waiting in Gate 9 at the international terminal Melbourne Airport, getting ready to board my first flight on my trip to Europe. It’s not a short hop – Singapore/Frankfurt/Prague, but at least I’ll be in a very nice place for what to all appearances will be a very enjoyable conference. From then, it’s Bonn, and then Denmark (København, and Århus) to meet up with Christine and Zachary, to have a little holiday while Christine does some work. On the way out, it’s Wrocław, where I’ll play a little part in a big conference before heading home with Zachary. Please, let me know if you’re going to be around anywhere in the vicinity during this European trip. I’ll be catching up with some mates, but it’s always fun to meet more friends. (If I owe you an email, that’s because I’ve been ridiculously busy preparing for this trip, teaching courses, looking after the boy, etc… But I’m ploughing through my backlog, and I might just get there in the next few weeks. One can hope!)

Not 'gargoyle' but 'finial'

Thu, 03 May 2007 12:11:54 +1100

You learn something new every day. What we mistakenly called gargoyles are actually finials. (I’ve corrected the caption on the photo – in case you don’t understand what Christine meant.) Christine took that photo on a finial-photographing expedition with Zachary and a friend from his class in school, on Zachary’s birthday a few weeks ago.

Logic Job at Auckland

Thu, 03 May 2007 09:41:45 +1100

As I’ve said before, logic jobs in Philosophy Departments are pretty rare. Well, as Richard mentioned earlier, there’s a logic job in Philosophy at Auckland. A little bird has told me that even though the deadline is close (May 18), they’re keen to get the word out to get as many applicants as possible. So, if you qualify (PhD or equivalent in philosophy, teaching experience, some research publications) then hop to it and apply. The details are here.

Logica 2007 is coming up

Wed, 04 Apr 2007 06:39:32 +1100

The program for Logica 2007 has been released, and it looks great. It looks like it will be seriously good, and it will be my first ever conference held in a monestery.

Book Launch! Inside Lawyers' Ethics

Fri, 30 Mar 2007 06:51:19 +1100

This Tuesday, I’m going to a party: It’s a launch for Christine’s book Inside Lawyers’ Ethics, which she’s coauthored with Adrian Evans from Monash Law School. The book will be launched by Louise Sylvan, from the ACCC. This party will be held at the Monash University Law Chambers in the city. Congratulations, Christine and Adrian! It seems to be the season for book launches. Simon Holt’s God Next Door is launching on April 19 and Matt Carter’s Minds and Computers is launching on April 23. (Activity here on the site will be quiet as I attempt to plough through a big teaching load and rather too many work commitments…)

Proof Theory and Meaning: on second order logic

Sat, 03 Mar 2007 00:00:00 +0000

Second order quantification is puzzling. The second order quantifiers have natural and compelling inference rules, and they also have natural models. These do not match: the inference rules are sound for the models, but not complete, so either the proof rules are too weak or the models are too strong. Some, such as Quine, take this to be no real problem, since they take “second order logic” to be a misnomer. It is not logic but set theory in sheep’s clothing, so one would not expect to have a sound and complete axiomatisation of the theory. I think that this judgement is incorrect, and in this paper I attempt to explain why. I show how on Nuel Belnap’s criterion for logicality, second order quantification can count as properly logic so-called, since the quantifiers are properly defined by their inference rules, and the addition of second order quantification to a basic language is conservative. With this notion of logicality in hand I then diagnose the incompleteness of the proof theory of second order logic in what seems to be a novel way.

Proofnets for S5: sequents and circuits for modal logic

Sat, 03 Mar 2007 00:00:00 +0000

In this paper I introduce a sequent system for the propositional modal logic S5. Derivations of valid sequents in the system are shown to correspond to proofs in a novel natural deduction system of circuit proofs (reminiscient of proofnets in linear logic, or multiple-conclusion calculi for classical logic). The sequent derivations and proofnets are both simple extensions of sequents and proofnets for classical propositional logic, in which the new machinery—to take account of the modal vocabulary—is directly motivated in terms of the simple, universal Kripke semantics for S5. The sequent system is cut-free and the circuit proofs are normalising.

Symbolic Logic

Sat, 03 Mar 2007 00:00:00 +0000

1007 words on symbolic logic – concentrating on the history of 20th Century logic, aimed at an audience of social scientists

In Banff: Monday, Tuesday, Wednesday, Thursday...

Fri, 23 Feb 2007 04:50:12 +1100

Liveblogging talks was a bit beyond me, but I’ve been keeping up with a few notes on the talks here in Banff. They’re not as extensive as my notes on Branden and Delia’s talks, but they’ll give you an idea of the fun we’ve been having here. (After lunch today, there’s a batch of five talks, including mine. I’m not sure when I’ll be able to write up comments on those, since before dinner I have to catch a shuttle to Calgary to catch a plane early tomorrow morning.) So, after the jump, comments on the talks: Monday afternoon JC Beall & Michael Glanzberg Truth and Paradox Apparently, this was a very nice talk, but I wasn’t there! I misread the program, and thought it was at 3pm and not 2pm. So, I missed out on it. At least it gave JC a chance to make some jokes at my expense… I’m looking at the slides to catch up, since I am talking about truth and paradoxes too. Kai Wehmeier: Identity is not a relation Kai’s elegant talk was on Wittgensteinian predicate logic: logic in which distinct variables must denote distinct objects, and in which we do not have an identity predicate.

My talk in Banff

Fri, 23 Feb 2007 03:23:51 +1100

I’ve finished up the slides for my talk in Banff. If you’re interested, the slides are available here Modal Models for Bradwardine’s Truth [1.3MB pdf] It’s a talk giving a modal logic interpretation for a medieval theory of truth due to Thomas Bradwardine, as it’s reconstructed by Stephen Read. This will hopefully help out Steve’s project, by providing tools for exploring the strength of this kind of theory of truth. I’ll write up an extended paper version of this after getting comments from the audience. Since the audience contains people who know a heck of a lot more than me about truth and about modal logic, I’m a little more on the anxious side than usual for such things. Wish me luck!

In Banff: Delia Graff Fara

Tue, 20 Feb 2007 05:37:33 +1100

The next Banff talk is Delia Graff Fara, on “Relative identity and de re modality.” She’s defending the thesis that material objects are identical to the matter of which they’re constituted. For example, the statue Goliath (G) is identical to the lump of clay Lumpel (L). She wants to allow that L = G, while having L and G differing in their temporal and modal properties. So, for example, in the future, a some more clay might become a part of the statue G, while not becoming a part of the lump of clay L. The original lump of clay is still present (and is joined by more clay), the original statue is still present, and is now larger. You can see why this is desirable: after the addition of more clay, there’s still only one statue there, and the old statue did not disappear only to be replaced by a new one. Delia wants to preserve the truth of claims that statues don’t have all their parts essentially, while portions of matter do have their parts essentially, while holding that statues are identical to portions of matter. How can we do this? I find this kind of work interesting, but deeply difficult.

In Banff: Branden Fitelson on Formal Epistemology

Tue, 20 Feb 2007 03:22:30 +1100

The first talk in Banff is by Branden Fitelson, who is giving a ‘“Survey” of Formal Epistemology: some propaganda, and an example’. It’s a part of the general movement towards carving out the discipline of ‘formal epistemology’. It’s a partly political talk, recounting the motivation for Formal Epistemology. He recounted Bob Meyer’s manifesto of the “Logicians Liberation League.” It’s worth recounting the section Branden quoted. Do not be deceived, Establishment pigs (this means you too, Establishment dogs). The subservience of past generations of logicians does not mean that we shall bear forever our treatment as animals (you barnyard fowl). We are human beings (you swine). You are living in a day when logicians will not any longer endure your taunts, your slurs, your insults (you filthy vermin). In the name of A. N. Whitehead and B. Russell we gather; in the spirit of R. Carnap and A. Tarski, we march; by the word of W. V. O. Quine, we shall prevail. Beware you snakes of the Philosophical Power Structure, which you have created and which you maintain to put down the logician; you have caged the eagle of reason, the dove of wisdom, and the lark of a definite, precisely formulated formal system, with exact formation rules, a recursive set of axioms, and clear and cogent rules of inference, and you have made them your pigeons.

In Banff

Tue, 20 Feb 2007 02:50:47 +1100

I’m in sunny and wintry Banff at the Mathematical Methods in Philosophy workshop. The program is very full, and I won’t be able to report on everything going on here. (Heck, I haven’t even reported back from our time in India, despite popular demand.) So, please let me give my apologies in advance if youre talk doesn’t get reported here. It’s probably because I was so interested in it I had to think and not blog.

Off to India...

Tue, 19 Dec 2006 22:17:22 +1100

I’ve been having too much fun and doing too much work lately to post anything here. On the work side, our proposal to teach first-year logic in an interdisciplinary way as a “breadth” subject to all students in the “new generation” degrees from 2008, has passed the first hurdle. We have a bucket of money to develop the course over the next year. Lots of work, but lots of fun. On the fun side, we’ve been getting ready to go to India for our family holiday. Landing in Mumbai, we go straight to Pune for Christmas with friends, and then we wend our way down the south-east coast on our way to Trivandrum. We’re off tomorrow, and we’ll be back in Melbourne on January 15. See you in 2007.

Scenes from an afternoon

Wed, 22 Nov 2006 20:31:49 +1100

My son, Z, asks me Dad, what is Kantian freedom? some time later Dad, what is a conceptual repertoire? No, he’s not a philosophical prodigy. He was curious about what I was reading, and I explained that I had an essay written by a student, and I had to figure out how good it was. It happened to be an essay about John McDowell’s Mind and World, primarily on animal perception and cognition. We ended up having a good conversation about whether or not our cat, Erasmus can think (Z thought ‘yes’, and I confessed being to being thoroughly confused on the matter after having read too many McDowell essays). I’m not a McDowell expert, but I’m the second examiner, brushing up on my knowledge by reading a batch of honours essays. On the whole, they’ve been very good. (Before this week I’d never seen a philosophy essay in the form of an extended interview with Andrew Denton. “So, John, tell us how we can get off the seesaw between the Myth of the Given and the frictionless void…”) After this, it’s my logic honours essays, and three honours theses. Could be another long evening.

Horn tooting

Wed, 15 Nov 2006 11:36:20 +1100

Two bits of horn tooting for today: First, from the famous and notorious Philosophical Gourmet Report. Brian Leiter pointed out that the report hit the newspaper here in Australia. The point of the little note in the Higher Education section of The Australian was that the ANU topped the rankings of Australian philosophy departments (as it has in the past), over Sydney and Melbourne. Curiously the article picks out discipline specific areas where Melbourne scored higher than ANU (Applied Ethics) and where ANU scored higher (Metaphysics). You wouldn’t be surprised that Melbourne did well in Applied Ethics, since we have a centre devoted to it, and there are lots of good things going on in the field. We ranked very well internationally in that area (we end up in the cohort of the top 7-18 departments in the survey). What The Australian’s report didn’t mention was that Melbourne also scored respectably well in Philosophical Logic. (Here, we’re in the cohort of the top 5-12.) I suppose one salient difference is that in philosophical logic, we don’t have a named centre: we just a few people who do it. Here’s the second bit of shameless self-promotion concerning logic at Melbourne: You might have heard about the process of institutional change taking place at this University.

A Philosophical Poll: on a priori knowledge of possibilities

Wed, 01 Nov 2006 16:56:32 +1100

I’d like to guage some philosophical opinions. (I don’t have many of my own. Most of them have been worn down by years of logical abuse.) First, some pre-requisites: Let’s understand ‘possibly’ as a metaphysical sort of possibility. If it helps, think of it as truth in some possible world, no matter how outlandish. Let’s understand a priori knowledge in the way that all of the people who talk about a priori knowledge understand the term. Plausibly, we can’t know that Hesperus is Phosphorus a priori, but, plausibly, we can know the Chinese Remainder Theorem a priori. Here’s the question. Can we have a priori knowledge of any proposition of the form possibly ~p where p is true? (We don’t need to know a priori that p is true.) Speaking rather briskly, do we have a priori knowledge that fatalism is false? (Where ‘fatalism’ is the thesis that everything that is true is metaphysically necessary.) This is fudging a scope distinction, commuting a propositional quantifier under the knowledge operator, but that doesn’t seem like too much of a worry. If you think that there’s a proposition ‘@’ true in the actual world alone (of any of the possible worlds accessible from the actual world) then, the question reduces to this:

Party on Tuesday

Thu, 12 Oct 2006 17:36:51 +1100

You’re all invited to a party on Tuesday, October 17 (5:30pm for 6pm) at the Melbourne University Bookshop to celebrate the publication of five books in Logic and Philosophy here in the Philosophy Department at Melbourne. Here are the books: Models, Truth and Realism, Barry Taylor. Doubt Truth to be a Liar, Graham Priest. In Contradiction, Graham Priest. Logic, Greg Restall Logical Pluralism, JC Beall and Greg Restall. Details of the party are found on this flyer. If you’re planning to come, you need to email the bookshop (the address is on the flyer) to RSVP for catering purposes. I’m really pleased to be a part of such a productive research community, and it’s nice to throw a little party to recognise that. Do come if you can make it: I hope to see you there!

Masses of Formal Philosophy: Question 2

Sat, 16 Sep 2006 13:30:54 +1100

Here’s my (much delayed) answer to the second of Vincent Hendricks and John Symons’ five questions about Formal Philosophy. What example from your work illustrates the role formal methods can play in philosophy? I’ll focus on one example from some of my recent work. In the last few years I have been working on topics in proof theory and connections between the way we can conceive of the structure of proofs and concerns in the theory of meaning. The idea that the meaning of a word or a concept might be usefully explicated by giving an account of its inferential role is a common one – the work of Ned Block, Bob Brandom and Michael Dummett are three very different examples of ways to take this idea seriously. It is a truism that meaning has some sort of connection with use, and use in reasoning and inference is a very important part of any account of use. It has seemed to me that if we are going to take take inferential role as playing its part in a theory of meaning, then we had better use the best available tools for giving an account of proof. The theory of proofs should have something to teach philosophers who have interests in semantics.

An idea...

Mon, 04 Sep 2006 15:43:30 +1100

I just had an Idea today. From here, it seems like a Really Neat Idea. (Having the idea made me remember what it feels like to prove something you’ve been struggling with for a long time: a mix of excitement, wonder, awe, relief, and much else besides. It’s welcome to be reminded of why I like working in logic.) This idea isn’t a new theorem, but what seems to me to be a simpler proof of an already proved theorem. I’ll try writing it up in the next couple of days and I’ll report back. It could dissolve into dust and vapour, but if it survives the writeup process, I’ll be one happy logician. If it doesn’t survive, at least I’ll have learned something new.

Visits

Wed, 02 Aug 2006 09:39:37 +1100

In the Philosophy Department we’re pleased to be hosting some short visits from Gillian Russell, and Jon Cohen. It’s neat to be in a place where smart people come to visit, think and talk to you. Last week we had a couple of logic seminars from neat west-coast of the USA types: Mike Titelbaum from Berkeley and Stanley Peters from Stanford. At drinks after the logic seminar I met Mark Steedman from Edinburgh, and it was nice to have a symmetric reaction “oh, you’re X, I’m Y” for X and Y in {‘MS’,‘GR’}. He was able to fill me in on the very interesting work done on substructrual logics, categorial grammar, and statistical parsing.

Ten Questions about Books

Sat, 29 Jul 2006 22:02:48 +1100

Since Jo asked so nicely, I’ll add my answers to the one book meme that’s going around the place. One book that changed your life Robert C. Roberts, Spirituality and Human Emotion. It’s because of this book that I’m a philosopher, believe it or not. Reading it, I saw that the philosophy needn’t be self-contained, but could be used to say something productive and interesting about matters of human concern. In my undergraduate years I spent a lot of time reading theology, and I learned most from the literature that was philosohpical in technique and in style. One book you’ve read more than once C. S. Lewis, The Voyage of the Dawn Treader. My out-and-out favourite of the Narnia Chronicles. One book you’d want on a desert island The Gospel according to John Following Jo’s example, I’ll treat different biblical books as different books. For me it’s a toss-up for different gospels, but John’s the one I’m least ‘at home’ with and I think it bears repeated readings, as is fitting on a desert island. One book that made you laugh Stanislaw Lem, The Cyberiad. It’s an intellectual laugh mostly, rather than slapstick, but there are laughs on each page of these wise, knowing, sensitive tales of the robot constructors Trurl and Klaupacius.

On the Interview

Sat, 22 Jul 2006 19:24:00 +1100

I’ve listened to the interview, and I’m pretty happy with how it went. The ABC team did a good job with the editing (I think the interview I recorded was a bit over 30 minutes). I never thought that I’d live to hear the day that someone explained the semantics of first degree entailment on national radio, and I was especially grateful that my little plug for Richard Sylvan and Bob Meyer didn’t end up on the cutting-room floor. For your edification, the transcript is of the interview here and the audio is an MP3 file available for the next four weeks. You can subscribe to a podcast of the program if you like to hear this kind of thing more regularly. If you’re visiting my website on following the link from the Philosopher’s Zone page and you want to know more about logic, there are a few things you could do. Download my little introductory article “Logic” in a nice volume called the Fundamentals of Philosophy edited by John Shand. My textbook, also imaginatively entitled Logic is an introduction to the subject, geared to philosophy students. You can order it from Amazon US or Amazon UK. If you’re interested in pluralism about logic, the topic I ended my discussion with Alan on, then you’ll want to have a look at my book Logical Pluralism co-authored with my friend JC Beall from the University of Connecticut.

On Politics

Wed, 19 Jul 2006 20:36:02 +1100

In the car this evening, Z (our five-and-a-bit year-old son) says, unprompted, Dad, I’ve been thinking … I reply “yes …” … why does Kim Beazely want to be Prime Minister? It’s nice to know that he’s keeping up with his world. It’s telling that we found it difficult to answer the question.

Interviewed again

Tue, 18 Jul 2006 14:58:15 +1100

This morning I trooped down to the ABC Melbourne studios in Southbank, to sit in a ‘Tardis’ booth to talk to Alan Saunders of The Philosopher’s Zone on logic, the liar paradox and other stuff I think about. Alan was feeling unwell and suffering from a croaky voice, so the interview that goes to air might be more of a monologue than what took place in the recording. Regardless, it was good fun to do the interview: Alan’s a good interviewer and he knows his stuff. Tune in or download the interview once it airs on July 22. (The ABC keeps these program files online for a month.) If you, like me, are interested in this kind of radio program, you can always subscribe to the podcast to get the latest program when it is released. Last month I was interviewed for The Age, this month, it’s radio. What is coming next?

Assorted Observations

Fri, 07 Jul 2006 23:23:59 +1100

It’s been difficult to get back into the swing of work, but I’m slowly clawing my way through. Jetlag has been much worse on my return than on the European end of the trip. I’ve also been concentrating on my parental duties, as my Significant Other is taking her conference trip. Playing with your son is much more fun than doing niggly administrative tasks. If you’re awaiting an email reply from me, it should be coming in the next few days. The delay is nothing personal – the queue is rather long right now. The promised observations: When you speak to a radio presenter on the telephone, it sounds eerily like you are talking to a radio. If you buy your coffee on campus, pricing varies from place-to-place. Now, despite the fact that Plush Fish (the sushi place), Chill Out (the rice paper rolls and juice place) and Café Suave (the place in the commerce courtyard, near the Ballieu Library) are run by the same people, the price you pay extra for your fairtrade coffee differs from place to-place. This week I have paid a 30 cent premium for faitrade at Plush Fish and Chill Out but only 10 cents at Café Suave.

Back home

Mon, 03 Jul 2006 22:01:12 +1100

I’m back in Melbourne, after my whirlwind jaunt in Nancy. I was in too much of a rush on Day 4 of the conference to post thoughts on Day 3 or Day 4. I promise I’ll do that once I recover from my jetlag and deal with my growing pile of mail, both snail and electronic. In other news, the nicest thing about reading this article on blogging in this morning’s newspaper was hearing Z’s laugh when he recognised that a picture of me was in the newspaper. “That’s you, Dad!” he chortled when he recognised me.

Here in Nancy, Day 2

Fri, 30 Jun 2006 17:29:03 +1100

Nancy Day 2 was a quiet as far as the official program went. Talks were scheduled in the morning leaving the afternoon free. Michael Lynch and I took in the Musée des Beaux-Arts and talked philosophy and much else with Peter van Inwagen, Scott Shalkowski and others into the evening and late in the night. The morning featured a well-put-together talk by Jonathan Lowe on the contrast between his robust essentialism and conceptualism. For Jonathan, conceptualism collapses into a global anti-realism because it really requires a kind of essentialism about concepts and agents themselves. I also attended a helpful presentation by Mathieu Marion on game semantics for logic. (For an introduction to game semantics, try the Stanford Encyclopedia entry by Wilfrid Hodges.) Mathieu contrasted the agonistic conception of game semantics according to which the two players (proponent and opponent, or Abelard and Eloise) are competing against one another. He proposed a cooperative understanding in which the players are building something together, like a proof. This strikes me as plausible. The devil, however, is in the detail. The distinctive feature of game semantics is that the proof is not a play of a game, or something you can look back on as having constructed.

Here in Nancy, Day 1

Thu, 29 Jun 2006 12:32:42 +1100

I’m conference blogging here in Nancy at the Realism/Anti-Realism conference. It’s been neet, chatching up with people I’ve not seen for a while. Today, after a not-completely-rested-night as I tried to sleep through a rather rowdy French community beeping car horns after their 3-1 victory in the round of 16, I went to five talks. Here’s the quick run-down. Jacques Dubucs talked about his feasible antirealism, according to which feasible computable functions should play a role in meaning theory rather than the computable functions of more traditional constructive logic. Dubucs and Mathieu Marion and Shahid Rahman are interested in the application of substructural logics (and game semantics) to these matters, and I should have a look at this more. There are interesting connections between feasibility and structural rules, worth pursuing. John Cogburn gave a nice paper on Moore’s paradox for anti-realists. The Moorean inference is the step from p to “I believe that p”. This inference is clearly wrong, but on semantically anti-realist lights (at least, for those kinds of constructivists for whom an argument is valid if warrant for the premises can be converted into warrant for the conclusion) then it seems valid enough. If I have warrant for p, then this warrant is the kind of thing that convinces me that p is true — I’ll believe it.

Off to France

Mon, 26 Jun 2006 10:47:18 +1100

Today I’m flying off to Nancy for (Anti)Réalismes Logique & Métaphysique. It should be an interesting meeting: the program is here. If I get time to edit my paper on the plane, I’ll try posting the text of the paper before I give it, internet access permitting.

Teaching in Semester 2, 2006

Sun, 25 Jun 2006 15:18:25 +1100

In Semester 2, which starts on July 31, I’ll be teaching an honours seminar 161-438 Logic and Philosophy, in which we cover proof theory and its applications to semantics.

Key Ideas in the theory of proofs #1: The Duality of Proofs and Counterexamples

Tue, 13 Jun 2006 11:05:52 +1100

Now that my undergraduate teaching is done for the semester, I can devote more time to research. I’m going to use this place to post some material from my proof theory project. My goal is to present some key ideas in an accessible way. Today’s idea is the duality between proofs and counterexamples. Proofs: You can think of a valid argument as underwritten by a proof. A proof leads from the premises to the conclusions, and shows how the conclusions follow from the premises. An invalid argument from premises P to conclusions C is one for which there is no proof leading from P to C. Different theories of proofs will characterise these in different ways (natural deduction, Hilbert-style axiom systems, Gentzen-style sequent systems, tableaux, etc.) but at heart, a proof is a formal structure that bears witness to a valid argument. Counterexamples: You can think of an invalid argument as witnessed by a counterexample. A counterexample is a way of making the premises true and the conclusions untrue, and it shows how the conclusions say something more than what is present in the premises. A valid argument from premises P to conclusions C is one for which there is no counterexample rendering P true and C not.

This football game is pretty tense...

Tue, 13 Jun 2006 01:00:51 +1100

When the team are 0–1 down at the eighty-fourth minute, you don’t think they’ll win 3–1. Update: I’m glad I don’t have to detail the match, and can point you to this excellent piece of writing instead. It’s a reflection on the match by Jeremy, a first-year Arts/Engineering student here at the University of Melbourne. He’s one of the contributors at the First_year@UniMelb blog.

Interviewed

Tue, 06 Jun 2006 17:05:05 +1100

This afternoon I was interviewed by Lisa Mitchell from The Age about academic blogging. She’s writing an article for the education supplement (published on Mondays). It’s a piece with a long lead time. I wonder if I’ll recognise anything I said in the article, and if she’ll quote me. (I blabbed on for 25 minutes. I hope there was some useful raw material there somewhere.)

End of Semester

Wed, 31 May 2006 10:22:50 +1100

We’ve finished lectures for the semester, and I’m officially recovering. It was fun teaching Logic and Non-Classical Logic, but it was also a hard slog. Next semester I have a light teaching load, with my honours logic seminar. The highlight last week was Richard Zach’s visit. He gave two nice talks. One on vagueness and logic – setting out different options for logic and semantics in the light of vagueness – and the other, on the epsilon calculus. (How Richard managed to get through the content of his big paper with Georg Moser in an hour with each of us following everything, I have no idea, but he did.) Now I have to slog through some marking, and some writing. This weekend, however, I’m in Adelaide for a little ``logic axis’’ meeting. I’ll give a talk on fixing up Prawitz’s proof theory for the positive fragment of the relevant logic R. I’m still working on the written version of the paper.

On Regret and Slingshots

Tue, 02 May 2006 23:18:45 +1100

Here’s a silly puzzle for you. Regret seems like a factive emotion. You can’t truly regret that you forgot to feed the cat today if, indeed, you did feed the cat. If you did feed the cat and you forgot that you did, then it seems to you like you’re regretting not feeding the cat, but you’re as mistaken about your regret as you are about feeding the cat. You think you didn’t feed the cat and you’re sorry that you didn’t, but that’s not regret. Or so some people think, anyway. Now, suppose that regret is factive like that. Then it’s plausible to think that the circumstance you’re regretting is in some way a ‘component’ of that regretting. (On this view, that’s why it’s factive: if the circumstance of my not-feeding-the-cat isn’t there, then I can’t be related to that circumstance in “regretful” manner.) If this is the case, then it’s plausible to think that disjunctive regrets are a strange sort of thing. Regretting that p or q might be a relationship to a situation in which p is true (when it’s the truth of p that makes the disjunction true) or one in which q is true (the other case).

On the Cable Guy Paradox

Thu, 20 Apr 2006 20:23:52 +1100

Have you ever waited for a tradesperson to come to your place? Alan Hájek uses this example in an article entitled “The Cable Guy Paradox.” The issue is straightforward. You and I are waiting for the cable guy who has given you a window between 8am and 4pm for his arrival. He will arrive after 8am, and before 4pm. We while away our time with a bet. Before 8am, we bet on whether he’ll come in the morning or the afternoon. It seems that there’s no reason to choose between the interval (8am,12noon) and (12noon,4pm). (Let’s assume, with Hájek that it’s probability zero that the moment of arrival is 12noon exactly.) It’s equally likely that he arrive in the morning and in the afternoon. Now, there seems to be an asymmetry. Hájek cites the ‘Avoid Certain Frustration Principle’: Suppose you now have a choice between two options. You should not choose one of these options if you are certain that a rational future self of yours will prefer that you had chosen the other one – unless both your options have this property. Hájek seems that using this principle, we should bet on the afternoon and not the morning, since whenever the cable guy arrives, it will be after 8am (we specified that in the setup of the problem), so the will be some interval of time (8 am, 8+ε am) during which I will take the probability of the morning arrival to be something smaller than the probability of the afternoon arrival, and I will regret my choice for the bet if I had chosen morning.

Masses of Formal Philosophy: Question 1

Wed, 19 Apr 2006 01:08:06 +1100

As I mentioned before, I’ve been thinking about Vincent Hendricks and John Symons’ five questions about Formal Philosophy. This seems like as good a place as any to answer them. So, today, I’ll have a crack at the most autobiographical of the questions: Why were you initially drawn to formal methods? I suppose the natural way to interpret this question is something like “why do formal methods rather than anything else in philosophy” but in my case I’d rather answer the related question “why, given that you’re interested in formal methods, apply them in philosophy rather than elsewhere.” I started off my academic life as an undergraduate student in mathematics, because I was good at mathematics and studying it more seemed like a good idea at the time. I enjoyed mathematics a great deal. At the University of Queensland, where I was studying, there was a special cohort of “Honours” students right from the first year. You were taught more research-oriented and rigourous subjects than were provided for the “Pass” students. This meant that we had a small cohort of students, who knew each other pretty well, studied together and learned a lot. I could see myself making an academic career in mathematics.

Happy 5 day!

Tue, 11 Apr 2006 11:34:44 +1100

Happy fifth birthday, Z! Today it’s unwrapping presents, a quiet family breakfast and lego-making, talking to grand-dad on the telephone, a little practice on the new bike, some school, and dinner together with more fiddling around with the new presents. The Party is on Saturday. (Wish us luck. We’ve not hosted a fifth birthday party before.)

Well, that was easy...

Sat, 08 Apr 2006 21:33:50 +1100

Sometimes technologies can be use for purposes the inventors didn’t really intend. I’m sure that when the designers of the iTunes music store set the thing up, they didn’t expect that you’d be able to use it to download esoteric papers in philosophical logic. But you can. All I had to do was submit the RSS feed of my papers to the iTunes store on this form, after tweaking it to include the information that the iTunes store is looking for. A day later, there it is. Now you can use iTunes to automatically download any papers I write. (Though, you can’t yet get them automatically downloaded onto your iPod for you to read, as the iPod doesn’t have a pdf reader – at least not yet.) This works since iTunes knows what to do with pdf files: if you buy tracks from the music store with lyrics or booklets, they can come as pdf files, which iTunes can store and hand over to your system to display. My feed that I use to serve up pointers to my recent publications has the same general structure as the feeds used in podcasting. The difference is that my publications feed just contained pointers to webpages on which you could download pdf files, while podcasting feeds contain enclosures of pdf files.

The Shifty Salesman

Wed, 05 Apr 2006 12:16:16 +1100

Su Rogerson has brought to my attention a lovely rendition of Curry’s paradox, due to Peter Geach, in his Reason and Argument (Blackwell, 1976), pages 93–95. The discussion takes place at a car yard. I’ll quote the discussion, interspersed with some commentary. (I’m quoting from a draft of Su Rogerson’s thesis, and not from the text. I’ll check this for correctness when I have access to the volume.) Customer: I’ll buy the car only if, if I buy the car, you’ll guarantee it. You might suspect that this claim of the customer’s is not the best starting point for a car purchase negotiation, and that rather too much is conceded straight away. But that’s not quite right. The customer has only conceded that the guarantee (conditional on the purchase) is necessary for the purchase, not sufficient. (The claim is an ‘only if’ not an ‘if’.) Salesman: But let’s suppose that if you buy the car, then I’ll guarantee it: what then? Customer: Then I’ll buy the car. That’s the crucial concession. It’s all downhill from here. Salesman: Can I hold you to both the things you’ve just said? Customer: Certainly. The customer is going to regret saying that. Salesman: Then you are logically committed to buying the car anyhow, even if I don’t guarantee it.

Enclosures

Sun, 02 Apr 2006 12:59:05 +1100

I’ve finally added enclosures to my RSS feed of papers. This means that if you subscribe to my papers page using that feed, and you’re using an RSS reader that handles enclosures (this feature is primarily used for podcasts), then you can get it to download my papers automatically. This has the side effect that you can subscribe to me in iTunes: That is weird. (I wonder if I should submit this podcast to the iTunes Music Store?)

2006 redesign in progress

Sat, 01 Apr 2006 22:29:33 +1100

I’m trying to neaten up this place a bit, and I need your help and advice. Have a look here and let me know what you think of it. I’m trying to make it easier to see what’s new, what’s changed, etc. In particular, you look down to find links to other new stuff on the site. This opens up the sidebars for four things. The most recent picture, my “recently browsed” linklist, my list of events (which I need to update) and finally, a very modest list of links to friends and colleagues. So, what do you think? Clearer? Simpler? Easier to understand? Or not? Does it break in your browser? Leave a comment and let me know.

AJL Papers

Thu, 30 Mar 2006 10:44:34 +1100

This morning I uploaded a few new papers to the Australasian Journal of Logic, one of the projects I (mostly!) enjoy working on. It’s an open access, fully refereed, online journal in logic. If you’ve not seen it before, browse around and check it out. Let me know what you think. One thing that an online journals make possible is the freedom from constraints of filling a fixed number of fixed-size volumes a year. Different ejournals do this differently. At the AJL, we publish one volume per year, but papers appear in the volume whenever they complete the refereeing and editing process. We close the volume at the end of the year, and start a new one in the next year. This means that wherever one paper is in the queue doesn’t block the progress of any other paper. We can publish Tony Roy’s massive 146 page Natural Derivations for Priest An Introduction to Non-Classical Logic without worrying that this will delay the other things in the queue until the next volume. Tony’s paper has meant that we’ve well-and-truly broken our pagecount record for the 2006 volume, and we’re only in March.

Spooky coincidence? I think not

Tue, 28 Mar 2006 20:42:07 +1100

Today in Logic I talked about vagueness and the sorites paradox. Then when I get back to the office, I find that this Saturday, Alan Saunders on The Philosopher’s Zone has a program on – you guessed it – the sorites paradox. Is that a spooky coincidence or what? No, it’s not spooky. It’s just a coincidence. Anyway, from the little outline, it’s clear that Saunders has a more adept journalistic sense of what can get a point across than me. His example (is Bruce Willis bald? Peter Garrett definitely is, and Billy Connolly definately isn’t, but what to say of Bruce?) is much nicer than my boring examples of strips of colour shading from red to yellow, or heaps of grains of sand or the other stock philosophical examples… Regardless, the class had a pretty energetic discussion, from people trying on different responses to the paradox, to other students pushing pretty hard on the “just stop doing logic if it produces problems like this” sort of line. I like these sort of questions – at least if they’re attempts to get to grips with the issues, as they seemed to be to me – because they’re a sign that when you do something like logic, it’s our job to address the philosophical or interpretive issues of what is going on.

Oh, and there's another paper, too

Tue, 28 Mar 2006 20:28:34 +1100

I forgot to mention that I have written another paper recently. It’s a proof theory paper, putting down some thoughts on modal proof theory that I formulated when giving the S5 paper around and about in the last six months or so. In “Comparing Modal Sequent Systems” I look at different ways to understand modal deduction. In particular, I argue that you can understand labelled proof systems – those in which proofs consist of statements annotated with labels, often thought of as denoting ‘possible worlds’ at least in modal proof theory – can be reconceived in such a way as to not really require talk of worlds. When you play close attention to the kind of work done by the labels, it can be understood instead as a different representation of a structural feature of modal deduction. The point in this paper is a technical one, but the moral is broader than that. The view I argue for in the ‘Invention’ paper feeds off this kind of point. Properly modal deduction involves doing new things in the structure of argument – you can do a kind of supposing (say, ‘hypothetical’ supposing), which has its own interesting behaviour. Or so I think, anyway.

Being a logician means sometimes having to say that you're sorry. Or at least, that you're wrong.

Tue, 28 Mar 2006 11:18:51 +1100

I’m sorry, I truly am. Back in the period 1990-1992, I wrote a couple of papers on the semantics of relevant logics. I thought they were pretty nifty, and I submitted them to a prestigious journal. They got accepted. The first of these papers is “Simplified Semantics for Relevant Logics (and some of their rivals).” Unfortunatley, there’s a hole in the middle of this paper. That’s the problem with being a logician, people can sometimes prove you wrong by presenting a counterexample to your claim. That’s the nice thing about playing a game with relatively clear rules. You can figure out what constitutes good play! Anyway, Tony Roy did a fantastic job of isolating the problem, which exists in not only my paper, but Graham Priest’s text An Introduction to Non-Classical Logic, which followed the simplified semantics rather closely. The problem is serious, for in my semantics for the relevant logic R, it turns out that disjunctive syllogism (the inference from A v B and ~A to B) is valid, and it shouldn’t be. Tony isolated the problem, and we came up with a fix. The result is the paper “On Permutation in Simplified Semantics.” Please take a look over it before we sent it off to a journal.

Dame Edna at the Commonwealth Games Closing Ceremony

Sun, 26 Mar 2006 22:11:01 +1100

This little spot called Melbourne Is the city of my birth It’s not as hot as Brisbane or as far away as Perth, It’s not as small as Adelaide, Compared to Canberra, it’s bliss, And if you’ve been to Melbourne, You can give Sydney a miss… As the world gets scarier, It’s a pretty decent area Melbourne The envy of the world. I think that was a knowing, self-reflective and ironic appropriation of Australian (and Melburnian) parochialism, but I’m not altogether sure about how the irony was received, given the insipid television commentary. Still, anything a 72 year old bloke in drag with purple hair who shot to fame (such fame as it was) knowingly making fun of Melbourne suburban life and other Australian attitudes can’t be all that bad.

Last Night at the MCG

Sat, 25 Mar 2006 08:40:25 +1100

Z and I – together with around 80,000 other people – filled the MCG to watch the track and field portion of Day 9 of the Commonwealth Games here. Some highlights: The completed renovations at the MCG make the huge stadium very easy to navigate. We got there early, so had no long queues to deal with (once we were at the ground itself – it was another matter on the train network). It’s easy to get in and out of the ground, the seats were comfortable, the view was good on Deck 2 of the Olympic Stand, and once the stadium filled up, you understood why the MCG is such a popular venue. It fits so many people, yet you feel very close to the action. The crowd was good-natured. Much alcohol was consumed by some of our neighbours (the trip to the bar was very short, and repeatedly taken), but any drunks were quiet and happy drunks, not loud and angry ones. We were, as a whole, terribly parochially Australian, but not without occasional flashes of appreciation for athletes of other countries. One crowd favourite in the latter category was Sapolai Yao, the 1.52 metre tall distance runner from PNG who earned the appreciation of the crowd by enthusiastically completing the 3km steeplechase, nearly managing to hold off being lapped by the lead Kenyan runners.

Marathon Effort

Sun, 19 Mar 2006 16:39:07 +1100

Z and I went in to the city to have a look at the Women’s and Men’s Marathon held here for the 2006 Commonwealth Games. The city was buzzing, the crowds friendly, and it was great to get a glimpse of the lead runners: especially Kerryn McCann and Hellen Cherono Koskei who raced neck-and-neck right into the stadium. No matter how you cut it, 42+ kilometres is a long way to run. I suppose the task is quite literally, a marathon effort. C arrived with lunch when the marathon had finished, and our afternoon was wrapped up with some great live music from Ravi Bandhu Vidyapathy and friends from Sri Lanka, at the Festival site down at Alexandra Gardens. Even for people who have a middling interest in sport, like me, the Commonweath Games has brought its own kind of spark into the city. Watch out for Friday night, when Z and I head down to the athletics at the MCG.

Greg Hjorth coming back to Melbourne

Fri, 10 Mar 2006 10:12:36 +1100

It’s a happy day for us logicians at Melbourne. Greg Hjorth is going to be joining the Department of Mathematics and Statistics later in the year, taking up a five-year, professorial fellowship. He’s a super-smart set theorist and model theorist, and I’m looking forward to having him around on campus.

Logics, Situations and Channels

Fri, 03 Mar 2006 00:00:00 +0000

The notion of that information is relative to a context is important in many different ways. The idea that the context is small – that is, not necessarily a consistent and complete possible world – plays a role not only in situation theory, but it is also an enlightening perspective from which to view other areas, such as modal logics, relevant logics, categorial grammar and much more. In this article we will consider these areas, and focus then on one further question: How can we account for information about one thing giving us information about something else? This is a question addressed by channel theory. We will look at channel theory and then see how the issues of information flow and conditionality play a role in each of the different domains we have examined.

Questions and Answers on Formal Philosophy

Fri, 03 Mar 2006 00:00:00 +0000

My entry in a book of interviews of philosophers who work on the more formal side of the discipline. It gives an account of how I got into this area, what I think logic is good for – when it comes to philosophy – and where I think we should head.

Relevant and Substructural Logics

Fri, 03 Mar 2006 00:00:00 +0000

An historical essay, sketching the development of relevant and substructural logics throughout the 20th Century and into the 21st.

Relevant Restricted Quantification

Fri, 03 Mar 2006 00:00:00 +0000

The paper reviews a number of approaches for handling restricted quantification in relevant logic, and proposes a novel one. This proceeds by introducing a novel kind of enthymematic conditional.

Masses of Formal Philosophy

Fri, 03 Mar 2006 09:35:09 +1100

Vincent Hendricks and John Symons are working on a sequel to their book Formal Philosophy, in which philosophers who use “formal methods” talked about their work, their motivations, and their take on the state of philosophy. For the sequel, they are opening things up for others to take an “interview” with five questions, to appear in the next book, Masses of Formal Philosophy. I’m not sure what the criteria Vincent and John are using for selecting answers for appearing in the book – but I am thinking about their five questions. Here they are: Why were you initially drawn to formal methods? What example(s) from your work (or the work of others) illustrates the role formal methods can play in philosophy? What is the proper role of philosophy in relation to other disciplines? What do you consider the most neglected topics and/or contributions in late 20th century philosophy? What are the most important open problems in philosophy and what are the prospects for progress? These look like interesting questions to think about. Maybe I’ll post some draft answers here first, for feedback, and if I’m happy with the result, I’ll post it off to Vincent and John for consideration. If you work in this area, you might consider doing the same thing.

Degrees of Truth, Degrees of Falsity

Sun, 19 Feb 2006 08:57:49 +1100

Toby Ord (a former student of mine, now taking Oxford by storm), has written up a nice short essay on degrees of truth and degrees of falsity. It shows how you can get a very nice little algebra if you extend the usual non-classical idea of a 4-valued logic in which truth and falsity are somewhat independent with the “fuzzy” idea of degrees of truth between zero and one. Both ideas have a heritage. The idea of considering the interval [0,1] as a lattice of truth values goes back to Łukasiewicz, and the four-valued algebra, now known as BN4, traces back at least to some early work by Mike Dunn. Toby considers nice properties of this little algebra. It seems to me that a good exercise for someone who likes fiddling with concrete algebras would be this: define a conditional → on the algebra such that when restricted to the fuzzy interval [0,1] it agrees with Łukasiewicz’s conditional. when restricted to the values t, b, n and f agrees with the usual BN4 conditional. has as many natural properties as possible. In particular, defining ‘A fuse B’ as ~(A → ~B) gives an associative and commutative operator, and fusion is connected with the conditional by means of the usual residuation postulates.

Fun with Playlists: Squeezing your music library onto a 2GB iPod

Sat, 04 Feb 2006 21:06:19 +1100

I’ve finally joined the ranks of the zombies with white earbuds. My trusty 2GB black iPod nano was acquired on the way home from our Sabbatical, in a duty free store at Changi Airport. I’ve had a little while to experiment with it, and so far, it’s been working better than I’d expected. I take it with me on my early morning walks, sometimes listening to spoken word things and sometimes listening to music. My taste in podcasts is pretty conservative – I seem to treat this as time-shifted radio, rather than anything more cutting-edge. However, it’s pretty mind-bending that you can get esoteric seminars off the web, so academic seminars I’m interested in but didn’t go to can be downloaded onto my computer and put on the iPod. I’ve been having even more fun seeing how to get my middling sized (20+GB) music collection into my very svelte (2GB) iPod. If you’re interested in how I do that, read on. It’s a long post, but I don’t feel like doing anything else this evening as I’ve sprained my ankle and I’m lying down giving my foot a rest. It’s just the time to write a long blog post you’ve been ruminating on for a week or so.

Phase Change

Wed, 01 Feb 2006 18:47:20 +1100

Today is the first day of the new phase of our lives – Zachary Luke Parker Restall had his first day of school today. Well done, Zachary!

Assorted crosscultural observations, upon visiting the supermarket

Wed, 04 Jan 2006 13:10:43 +1100

This morning found Z and me in the grocery store, doing yet more re-stocking of the pantry. Here are some hastily selected observations. Organic and fairtrade food has significantly higher market presence in the UK than in Australia. Our nearest Sainsbury’s had many more organic products than can be found in either of the two major chains here in Australia. In our local supermarket, the paltry selection of organic products is filed away in the small “health food” section, than distributed through the relevant parts of the shop. This makes comparison with other products difficult. There’s much more fresh food available in Australia, and correspondingly fewer pre-prepared meals. Don’t even get me started talking about that UK phenomenon Iceland. No, it does not feature Icelandic cuisine… A 2 Litre plastic milk bottle looks very odd once you’ve been buying milk in the UK for a few months.

Teaching in Semester 1, 2006

Tue, 03 Jan 2006 15:18:38 +1100

I’m back in the teaching saddle for 2006. In Semester 1, I teach two undergraduate courses. 161-115: Logic and 161-212: Non-Classical Logic.

Happy 2006

Sun, 01 Jan 2006 12:08:20 +1100

Happy New Year, my readers! We’ve landed in Melbourne (to a hot 8pm at around 42 degrees Celcius), and are slowly reconstructing our household bit by bit. The internet connection seems to work at least. We’ve managed to change lightbulbs, unpack bags, look at a bit of our large pile of mail, get the boy to bed (and even to sleep) and have some Kaya Toast for a midnight or New Year snack. See you later in 2006.

Logic

Sun, 01 Jan 2006 00:00:00 +0000

This is an introductory textbook in logic, in the series Fundamentals of Philosophy, published by Routledge. I use the text in my Introduction to Formal Logic class at Melbourne. It’s available at Amazon UK and Amazon US. The book was translated into Chinese in 2024. The book has a dedicated website at http://consequently.org/logic.

Logical Pluralism

Sun, 01 Jan 2006 00:00:00 +0000

This is our manifesto on logical pluralism. We argue that the notion of logical consequence doesn’t pin down one deductive consequence relation, but rather, there are many of them. In particular, we argue that broadly classical, intuitionistic and relevant accounts of deductive logic are genuine logical consequence relations. We should not search for One True Logic, since there are Many.

About to leave Oxford

Fri, 16 Dec 2005 12:22:40 +1100

We’re in the middle of packing up our Oxford home, to start the short trip back. We land in Melbourne on New Year’s Eve, after some stops in Köln (for the Christmas Markets), Augsburg (for Christmas with friends), Frankfurt Airport (to get on the plane), Singapore (to break the journey, and give us some summer in close proximity to air conditioning before landing in the Melbourne summer heat). There’s too much to say in the limited time I’ve got online here, but here’s just a taste of what’s been going on. Academic highlights in the last few weeks have involved: Cambridge. I zipped over to Cambridge on the bus (though Milton Keynes, which is a very strange city, at least when viewed from the window of a bus) for a CAMELEON workshop organised by Tom Forster. It was great to meet Tom, and also Randall Holmes, Michael Rathjen, Allen Mann (who all presented papers), and Martin Hyland, who gave me useful feedback on the paper I presented. Paris. My visit there was wonderful. The IHSPT is a wonderful place, with lots of nice people (a fantastic bunch of students who managed to make me eat my lunch or dinner very slowly by asking Interesting Questions and making me think Very Hard), together with a gorgeous city.

Logical Consequence

Mon, 12 Dec 2005 00:00:00 +0000

A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline).

Survived so far...

Fri, 18 Nov 2005 20:59:32 +1100

I’ve given my two talks here in St. Andrews, with really useful discussion in both of the talks. In the philosophy talk, questions ranged from matters of meaning theory, metaphysics and epistemology of possibility and necessity. Thanks especially to Daniel Nolan, Carrie Jenkins, Marcus Rossberg, Ole Thomassen Hjortland, Crispin Wright, Stephen Read and other people whose webpages I haven’t yet found. Then on the next day in my talk in computer science, I had a great discussion with Roy Dyckhoff and Robert Rothenberg on converting the sequent system in my paper on S5 into one with invertible rules, which can then be used as an implementation. It seems to work. (Photographic evidence is here.) Yesterday, I went to three other talks on truth and paradox, by Hartry, Graham and JC. Now that I’m thoroughly over-paradoxed, I’ve got the morning to recover before we attempt to solve the problems of vagueness. (This should be fun because I have no settled opinion on matters of vagueness, so I’ll look at this more dispassionately, and I’ll be free to heckle from the back of the audience.)

Time flies when...

Tue, 15 Nov 2005 21:57:33 +1100

… you’re on sabbatical. I’m now in St. Andrews visiting Arché for what can only be described as “November Madness.” I’m giving a seminar this afternoon in Arché/Philosophy (in the wonderful Edgecliffe building) and then tomorrow, I’m giving a seminar in computer science tomorrow. This means I’ll miss the Vagueness Seminar and a Philosophy Departmental Colloquium. However, I will be at the Revenge Workshop on Thursday, and the Vagueness Workshop for Friday and Saturday. That should be enough to keep me going.

Week 1

Tue, 11 Oct 2005 19:46:51 +1100

We’re now in the second day of “week 1” of Michaelmas, here in Oxford, and the philosophy faculty library is considerably busier than it was in week 0, let alone the negative weeks. It’s just as well that I’ve got some writing done already. Today I’m heading off to Nottingham to give two talks. The Philosophy Department will get the first outing of my fresh-off-the-presses paper Invention is the Mother of Necessity in which I try out a (new, I hope) semantically anti-realist account of metaphysical necessity and possibility. Have a look at the paper and let me know what you think, if you’re into that sort of thing. I’ll be presenting it also in Bristol, St. Andrews and here in Oxford, so it will probably see signifiant modifications and updates over the next months. The other talk is the formal partner of that one, on proof theory for S5. I’m still writing the notes up for this (we’ll see how much I can get written on the train today), and I’ll upload that when I can. Then, with these papers largely written, I can get back to the book! I’m looking forward to it.

Cut elimination generalised

Sat, 17 Sep 2005 12:04:13 +1100

This post is a change of scene for those of you that have come to expect gentle meditations on matters personal. I’m going to experiment with posting some of my work scribblings as an irregular record of thoughts I’m having while working on sabbatical. It may well confirm the impression that you might have that I’m crazy, but hey, I have tenure, so I won’t worry about that too much. Over the last couple of days I’ve been thinking about cut elimination arguments, and in particular, the general cut elimination argument that you can find in my book (look around here). It struck me last night – as I was trying to find the right mix of generality and clarity – that there’s nothing in Gentzen’s original argument that requires that formulas have just two distinct positions in a sequent (the traditional two positions are antecedent and consequent, or left and right, or if you like, premise and conclusion or input and output). Every presentation of cut elimination that I’m aware of (even for single sided systems which don’t have an explicit left or right to a sequent) has two “positions” in which a formula can reside. (In single sided systems this is represented by the dualising map on formulas).

Reflections on Iona, part 2

Mon, 05 Sep 2005 20:43:02 +1100

Here’s the second part of my reflections on our week on Iona. If you’re interested in that kind of news, read on. I haven’t mentioned the most significant thing about our time on Iona – the week was tempered by an event just before we left for the island. The story started on the Friday before we left Edinburgh on Saturday. It was my last day teaching at ESSLLI, and I was having lunch with Denis Bonnay and Benjamin Simmenauer, talking about logic, and the possibility of me visting Paris – just before giving the last lecture of my course. My phone rings. Inside the restaurant there is terrible reception, but I can see that it’s Christine calling. I apologise to Denis and Ben, and head out of the restaurant to hear Christine. She’s upset. She says “Zachary’s had an accident. We’re in the hospital.” Christine sounds like I feel – frantic and unsettled. She explains that Zachary fell while running down a hill in the Princes Street Gardens in the centre of Edinburgh. He split open his chin, there was a lot of blood, and after help from passers-by and a first aid officer from the Scottish National Gallery, they were in an ambulance to the children’s hospital.

Reflections on Iona, part 1

Sat, 03 Sep 2005 01:41:19 +1100

This is an extended post reflecting on the week in Iona. Read on if you want to find out what we did for a week on a small Hebridean island. It’s not immediately straightforward to get to Iona, when you start from our B&B in North Queensferry. Our day’s journey started early with a taxi ride to Edinburgh Waverley, and then a train to Glasgow Queen Street (not to be confused, as many people do, with Glasgow Central, an altogether different train station in an altogether different place). With our bags piled around us, we had a little while to collect supplies for the next train trip, and to play the game of “who else here is going to Iona.” The late-middle-aged single American women with loud voices and Celtic crosses were very easy to spot. As was John Bell (we’d met him on one of his visits to Australia). We were less sure of others, but it turned out that lots of people on the train were heading to retreat on Iona, like us. The rest of the trip was a long train journey west to Oban, a ferry from Oban to Craignure on Mull, a bus across Mull to Fionnphort and then a ferry to Iona.

Constant Domain Quantified Modal Logics without Boolean Negation

Fri, 02 Sep 2005 00:00:00 +0000

The paper examines what its title says. Constant domain modal frames seem to be the natural semantics for quantified relevant logics and their cousins. Kit Fine has shown us that things are not that simple, as the natural proof theory is not complete for the natural semantics. In this paper I explore the somewhat simpler case of one-place modal operators. The natural proofs work, but there are a few surprises, such as the need to use intuitionistic implication and its dual, subtraction, in the completeness proof. This paper is dedicated to the memory of Richard Sylvan, who contributed so much to the study of the semantics of relevant logics and their neighbours.

…Iona, Glasgow, Oxford.

Tue, 30 Aug 2005 12:25:14 +1100

Phew! We’re back in Oxford, and arrived in our house today. We’re in a cute little 2 bedroom house in the northern outskirts of Oxford, a quick busride down Banbury Road into Wolfson or the centre of town. I’m currently dealing with some of the backlog of email in a Starbucks (of all places!) with a wireless network. The week at Iona was great. I’ll write an extended post about it in a little while. For now I have lots of other things to do to make the cute little house a bit more like a home, and figuring out how to best use my worktime for the best mix of reading, writing, going to seminars, giving talks, etc., etc. (That and figuring out how I’ll live without internet access at home.) More later.

Denmark, Oxford, Edinburgh…

Thu, 18 Aug 2005 22:30:35 +1100

Too many things are happening for me to spend any time posting here until the whirl of activity settles down. Still, people have been wondering what’s going on, so here’s a bit of a run-down. After Logic Colloquium, I went back to Denmark to rejoin the family, to have a holiday in Blokhus with close friends. Much fun was had, involving sand dunes, beaches, rain, driving on the other side of the road, etc. From Blokhus we trained to Copenhagen, where we all had more holiday, involving Tivoli (in the rain), Danish icecream, cathedrals, public transport, and a flash designer hostel. (Photographic evidence of some of this can be found here.) From Copenhagen we flew to Heathrow (let me tell you, flying in to Heathrow from merely one timezone away is qualitatively different from flying in from the antipodes) to catch a bus to Oxford, to spy out the land. Unfortunately, spying out the land involved learning that our college had failed to find us accommodation for our stay, and that we were flung onto the open rental market. From there, a train to Edinburgh, where we’re staying in a B&B in North Queensferry with a wonderful host (thanks, Molly!) and a view over the water.

Logic Colloquium 2005 Day 4

Mon, 01 Aug 2005 14:11:49 +1100

Today is Day 4 of the colloquium. Yesterday was a day of rest, which for me involved a visit to the Byzantine and Christian Museum, in the morning, a lazy afternoon restructuring the order things in my ESSLLI course while staying in the shade, and visiting the Herakleidon Museum which featured an exhibit of Escher prints (the detail in some of the woodcuts was beyond belief), and then the obligatory visit to the Acroplolis as it got slightly cooler. Photographic evidence is found over there. Today features lots of philosophical logic and proof theory. I’ll post comments as we go, battery life and concentration permitting. First up, Michael Sheard, on “A transactional approach to the logic of truth”. The topic was theories of truth, involving consistent reductions of the class of T biconditionals T<A> iff A. The liar paradox causes problems, so Michael gave a nice overview of the kinds of consistent thoeries of truth that weaken the assumptions to restore consistency. He argued for what he called a “transactional approach,” to thinking about the issue which – sensibly, in my view – focusses on why one might be interested in using the concept of truth, in communication between agents, and in evaluation of what those agents say.

Logic Colloquium 2005 Day 3

Sat, 30 Jul 2005 16:20:50 +1100

It’s another busy day with lots of things to learn. Here’s my impressions so far, updated through the day. 9:20: Phokion Kolaitis is giving his second lecture — he’s good, and it’s been interesting to learn about dichotomy theorems in CSPs. Schaefer’s dichotomy results cast in terms of constraint satisfaction problems, are really interesting. The idea is that we can think of a class of constraints as expressed as the problem of whether or not there’s a homomorphism betwen your structure and another one (which is thought of as expressing the constraints). So, 3-colourability might be expressed as the existence or nonexistence of a homomorphism between your graph and the complete graph K3. If there is such a homomorphism, your graph is 3-colourable, if there isn’t, your graph is. Phokion’s talk yesterday explained why CSPs can be expressed as homomorphism existence fact. The idea is that conjunctive queries (predicate logic expressions using existential quantification and conjunction only) have exactly the right kind of preservation properties to express these homomorphism existence facts on relational structures. Well, Schaefer’s result is that the problem of whether or not (for a given boolean algebra B), the issue of whether or not you’ve got a homomorphism from your structure into B, is either solvable in polynomial time (like 2-colourability) or is NP-complete (like 3-colourability – K3 isn’t a boolean algebra, but 3-colourability apparently can be expressed as CSP(B) for some boolean algebra).

Logic Colloquium 2005 Day 2

Fri, 29 Jul 2005 16:18:47 +1100

9:25: There’s wireless access in the large lecture theatre, so I’ll take notes and upload comments occassionally through the day. At the moment, Phokion Kolaitis is giving his first short course lecture on constraint satisfaction problems. So far he’s talking about examples of CSPs (graph colourings and 3-SAT are his first examples) and he’s just looking now at homomorphisms of relational structures, and the Feder & Vardi claim (1993) that CSPs may be idenified with (or may be represented by) the problem of whether or not there’s a homomorphism between two relational structures. (This bit is new to me, so I’ll listen closely.) 11:20: Sergey Goncharov giving an invited lecture on ‘Isomorphisms and Definable Relations on Computable Models’. It’s always struck me as weird that infinite ordinals have something to do with the characterisation of computable structures (where being computable is an essentially finitary notion in one sense – any computation process can be finitely described), but then, ordinal notations are finite too. Sergey’s talk is about constucting interesting structures (boolean algebras, rings etc.) that are computable, but have the Scott Rank of ω1CK (that’s big). This generalises some earlier results of Makkai (who showed that there were arithmetical structures of that rank) and Knight and Sheard (who showed that there were computable structures with that high a rank).

Logic Colloquium 2005 Day 1

Fri, 29 Jul 2005 07:23:56 +1100

I’ll try this conference blogging thing, to see how it goes. Logic Colloquium has started, with the opening address by Charles Parsons from Harvard. His talk was on Paul Bernay’s later philosophy of mathematics — a subject of which I knew nothing. So I learned a bit. After Bernays’ collaboration with Hilbert at Göttingen, he left Germany because of Nazi persecution, and spent the rest of his working life in Switzerland. His philosophy of mathematics (according to Parsons) post war is characterised by his response to Gödel’s results and the failure of Hilbert’s program. For Bernays, mathematical truth is not necessarily a priori in any strong sense – mathematical claims are verified in the competition of different conflicting mathematical theories, in just the same way that scientific theories are verified in competition with other theories. The category of the a priori is relativised into the weaker catergory of the antecedent. There are always beliefs or theories antecedent to our views, which may be examined, clarified and accepted and rejected. But nothing makes those especially immune from criticism or revision. This might sound like a Quinean holism, but it’s not. It is motivated by a kind of neo Kantianism, and Bernays shares none of Quine’s empiricism.

Legroom

Thu, 28 Jul 2005 21:48:19 +1100

Remind me to check legroom on airlines before I book them for the first time – especially if I’m planning to write on the plane. Maersk Airlines has an interesting policy of offering seats in S/M/L sizes. S seats have a tight pitch of 29 inches (73-and-a-bit centimetres). You can upgrade to a roomier seet for a price–a fact that the stewards make clear repeatedly on the flight itself. I am very pleased that I got a seat in an otherwise empty row of three in an otherwise packed cabin. Another person asked to move into one of them, and between us, we had the middle seat to stretch our legs just a little and open my laptop. (This is also the one occasion in which a 12inch Powerbook has a distinct advantage over my 17inch model.)

Off to Athens...

Wed, 27 Jul 2005 18:57:19 +1100

I’m off to Athens tomorrow. Tonight I catch the train to København, where I stay overnight in a cheap hotel near the airport, to get my 7:30am flight. Logic Colloquium 2005 looks like it will be lots of fun. I’ve been busy writing slides, and I should use the time on the train to put the finishing touches on Day 2. Once they’re done, I’ll post them on the wiki page. If you’re coming to Athens, feel free to post things up on that wiki page.

Keep up with our travels: the map

Wed, 20 Jul 2005 17:49:12 +1100

I’ve been experimenting with a map using plazes, a neat little doodad that keeps track of where you are when you use a computer to access the net (if you explicitly log in yourself—it’s not spying on you without your consent). It powers the map on the main page here, and you can use it to generate a map of where you’ve been. Read on if you want to see our journey. This is the map: It’s kind-of-neat. (That Melbourne to Copenhagen flight is long).

In Århus

Wed, 20 Jul 2005 03:18:12 +1100

We’re settled in Århus now, while Christine does work with a colleague on a research project. This gives Z and me lots of time for lots of fun. Yesterday’s fun was Legoland (Vibeke and Christine wanted to come too, so we all made a day of it). Z loved the rides, and the displays, and the Lego for sale. We bought a set for him to grow into. There’s lots of fiddly pieces, but we’ve already managed to make one of the models together. (He does the putting together and I help find the pieces for him.) More later. I’m going to cook some dinner now.

Up, up and away

Wed, 13 Jul 2005 01:04:53 +1100

We fly later today! The big study leave adventure starts later this afternoon. The first few weeks alternate holidays and work for Christine and for me, and it gives us lots of time to enjoy the company of our young son. Zachary and I are especially looking forward to Legoland in Billund and exploring the surrounds of Århus. Our apartment apparently has an internet connection (I have no idea what kind, as of yet) so I’ll hopefully be able to post photos and comments here to keep you up to date from time to time.

Loud

Mon, 11 Jul 2005 17:03:52 +1100

Sometimes music is best heard at volume. A case in point? The second movement of Shostakovich’s String Quartet No. 8. I recommend the Fitzwilliam String Quartet recording. (The Emerson Quartet’s recording is available on iTunes, but the Fitzwilliams’ isn’t.) (Apologies to our next door neighbours. I’m enjoying my last bout of loud music at home before we travel and don’t have access to a decent sound system.) That is all.

AJL

Fri, 08 Jul 2005 23:42:31 +1100

The Australasian Journal of Logic is firing on all cylinders again. As the managing editor, there was a period in this last semester where I wasn’t managing very well, and things piled up and got the better of me for quite some time. To speak overly frankly for a moment, I got quite depressed over the state that things were in and over my own disorganisation. Unfortunately, being depressed is not a good condition in which to be motivated to do anything about that which you’re depressed about. Anyway, after a few solid days of working at it, sending things off, writing reports, collating information, copy-editing, typesetting, emailing, coding HTML, uploading, etc., I have something to show for it. And I’m feeling much better about it all, too. So, without further ado: Here’s the start of Volume 3: 2005. I’m quite proud of it. Thanks to the authors, and to my referees and readers. There are currently eighteen other papers in various stages of the submission pipeline, so you should expect us to be much more productive over the last half of 2005, when compared to the first.

On applescript and php, and an archive page

Sun, 26 Jun 2005 20:24:30 +1100

A half hour of php hacking has finally produced a working archive page for the photos I’ve been posting from my phonecam. My little Sony Ericsson T630 phone has a terribly lo-res camera, but it’s fun to take shots now and then when you wouldn’t normally take photos. Now, instead of sliding off the front page when I upload another one, you can see the old photos on the archive page. What makes the thing fun is that the process of uploading them is so easy. The camera has a bluetooth connection to the computer, so I can get photos from the camera while the camera is still in my pocket. (That’s easy.) Then I drag a photo to an icon on the dock on my computer and the little applescript I wrote pops up a dialogue box to enter a title, and the thing scales the image (the camera takes pictures at 288x352 and I scale them to 180x220 to fit in the sidebar) and uploads the result to the /phonecam directory on consequently.org. The next bit of magic is two php scripts. One on the main page takes the latest jpg file out of the /phonecam directory and puts it in the right sidebar.

Logic Colloquium 2005 and ESSLLI 2005

Fri, 24 Jun 2005 21:55:01 +1100

I’m teaching some classes on proof theory at Logic Colloquium 2005 and ESSLLI 2005 soon. Spurred on a little bit by discussion of commentary at conferences, I’ve wondered a little bit about how I can get interesting feedback going during/after the presentations. (The material I’ll be teaching is from the book, which is a Work very-much-in Progress so I’m keen to get feedback on how it comes across.) Obviously, I’ll point everyone to the wiki, and set up spaces for people to write comments after the talk. (That’s a given.) But I’m sure that lots of people will have wireless-equipped laptops on the scene when I’m there. Wikis aren’t the best for group communication in real-time, or joint simultaneous writing/editing of a document. It strikes me now (after only thinking about it for a teensy bit) that I could host a document in SubEthaEdit for all of the mac users to take notes together. That’s fine for collaborative notetaking. People could use that as a way to take notes, write up questions, etc. If you’re coming to LC2005 or to ESSLLI2005, do you think that would be useful? If so, bring your computer along! Do you have anything else to recommend?

Rearranging

Wed, 15 Jun 2005 01:02:48 +1100

I’m rearranging stuff around here, in preparation for the Big Trip. Let me know if anything broke in the transition.

The Flight of the Balloon

Mon, 13 Jun 2005 10:28:55 +1100

This Queen’s birthday holiday moning, cool and clear, was a perfect time for Zachary and me to fly our hot-air balloon. The flight was short but glorious. Flying a balloon like this is not completely straightforward. You need to unfurl the balloon carefully, and make sure no bit touches the little burner on the ground (fuelled with cotton wool soaked in Methylated Spirits). Zachary held the balloon carefully like so: Notice the building in the background. We’re in the middle of a park in Brunswick, and the air is still. The balloon inflated quite rapidly. I held the base steady, and Zachary held the top, letting go when it was obvious that the air was warm enough to keep the balloon inflated. The air inside warmed very quickly: We let go, and it was away – It gained altitude remarkably quickly. (It was tricky to get the shots I did with the camera while we figured out which way it was going.) The wind picked up and it drifted westwards as it gained height: It got further west than we expected, given the still air at ground level. Notice the building it’s near. It made a gentle landing: on the roof of the factory of “Valve Tech Engineering.

Teaching in Semester 2

Sat, 11 Jun 2005 22:52:09 +1100

Since I’m on study leave from July, I’m not doing any teaching in Semester 2 this year. Hooray! (I need a break.)

Skype

Sat, 11 Jun 2005 16:28:19 +1100

Next travel thing: I’ve signed up at skype.com and I have an account to make super-cheap voice calls home and almost anywhere else from the computer while overseas at least, when I’m connected to the internet. It looks very well executed. There are clients for Mac OS X, Windows and Linux, and at 1.7 euro cents a minute for calls to any landline (it’s a bit more expensive to call mobiles) in Western Europe, North America and Aus/NZ, it is a very good deal. Then there’s the fact that it’s free if you’re making a Skype-to-Skype call. If I were a telecommunications company making money from traditional telephony, I think I’d be worried. Anyway, if you’ve got a Skype account, let me know. Look me up in the Skype directory to get my username.

We're off in almost a month: I've got to get ready!

Fri, 10 Jun 2005 23:44:04 +1100

I’ve taught my last classes for 2005. I’m officially on study leave from July 1, until December 31. We’re off on our journey from July 13. First, Denmark, where I’ll be looking after Z, while Christine works with a colleague. (Z and I plan to have some fun in our time there.) Then I’m off to the Logic Colloquium 2005, in Athens, and ESSLLI2005 in Edinburgh, and a week’s retreat on the island of Iona, before heading down to Oxford, where we’ll both be Visiting Scholars at Wolfson. (You can keep track of our travels in the “Events” sidebar on this site.) It’s very exciting, but there’s heaps to do, and if I need advice on travel, I’ll post questions here. Here’s my first: I’m visiting the U.K. and Europe for six months, and I’ll want to use a mobile. Should I buy a prepaid sim card for my (triband) phone that I’ll be bringing with me? What should I do if I wanted to use my phone through Europe without paying too much in roaming fees? Keep in mind that I’ll be arriving first on the continent, and then going the UK. Do you have any advice? If you do, post in the comments on this entry.

Searching in Mac OS X Tiger

Sun, 01 May 2005 12:09:46 +1100

Like almost everyone else, I’ve installed Mac OS X 10.4 Tiger, and it’s a really very good upgrade to the operating system. The system-wide search will be really handy. It’s indexed all of my documents (which includes a 2.3GB library of PDF files of other people’s papers), so being able to find the 29 papers that mention hypersequents (together with the 18 emails in my mail client) is just something special. I’m having even more fun, however, with organising my files using tagging. I’ll talk about that one later.

Seminars at Otago

Tue, 05 Apr 2005 13:56:36 +1100

Tomorrow I’m presenting a paper at the staff seminar in the Philosophy Department at the University of Otago. Given Charles Pigden’s delightful account of seminars at Otago, I’m not quite sure what to expect. Wish me luck!

Logic Books at print.google.com

Tue, 22 Mar 2005 23:42:32 +1100

I think that Google’s new search-inside-the-book service will prove to be quite handy to a researcher like me. It’s not satisfactory to read through a book like that on the screen (especially when copyright restrictions do not permit you to keep going after a couple of pages of continuous reading), but it will be a boon to research to be able to search inside the book for everywhere lattices are mentioned. I like the idea so much I’m setting up a list of logic-related books on my wiki. Feel free to browse the list, and to add to it any other logic-related works you find on print.google.com.

University Library Proxy Bookmarklet

Sat, 12 Mar 2005 21:02:43 +1100

If you’re like me, you do research on the web, and you browse around different sites for journals, seeing what’s been published, and what you should read. Our library has online subscriptions for lots of these journals, so I can download the papers, file them, read them, etc. It’s all very nice. But it’s not always easy to get from the site to actually download the paper. Recently, our university introduced a proxy server as the means to keep track of whether someone attempting to download material under the banner of a University of Melbourne subscription is actually authorised to do so (i. e., is a member of staff or a student). So, if I’m at a page for a journal like this http://mind.oupjournals.org/current.dtl and I want to download a paper like this http://mind.oupjournals.org/cgi/content/abstract/114/453/1 I need to manually edit the address, by sticking “.mate.lib.unimelb.edu.au” at the end of the hostname, before the path. (Between “.org” and “/cgi” here). I am lazy and I don’t like having to type “.mate.lib.unimelb.edu.au” time and again when browsing the journals. So, I use a little link I made – ‘+mate.lib’ – in the bookmarks bar of my browser, and whenever I’m on a site, it takes me directly to the relevant page viewed through proxy site, and I can download at will.

Please Help Us #1: On Self-Saucing Pudding

Mon, 07 Mar 2005 19:16:53 +1100

Here’s one thing I find comments on a website really good for: trawling for information. My spouse and I were talking over dessert this evening, over some lemon delcious pudding that she had made. She asked the question she asks every couple of years (when a self saucing pudding is made): How do self-saucing puddings work? They really are miraculous, and we haven’t got much of an idea of how the chemistry of these things makes them work. A cursory Googling reveals nothing, and neither does a quick flick through On Food and Cooking. Can you help us?

Comments on Weblogs

Mon, 07 Mar 2005 16:27:50 +1100

At logicandlanguage.net the author talks a little bit about comments on weblogs. On why one might allow them, or do without them. As you can see, I leave comments open here. (Behind the scenes there is a hardworking comment spam filter. Without that, this place would be deluged with spam.) I can understand why one would want the place to yourself and do without comments. However, if you do, and if you want your readers to email you (as the author of logicandlanguage.net does) then please let the reader know your email address.

Get your kicks on Route 55

Thu, 03 Mar 2005 20:18:47 +1100

I sometimes catch the tram to get to or from the office, and usually it’s a not unpleasant experience. This evening was not one of the usual reasonably relaxing fifteen minute trips down Flemington Road and through Royal Park. It’s remarkable how a violent scuffle between passengers can change the feeling throughout the tram. There was pushing, shoving, curses and epithets, clutching at throats, and a tense standoff from the protagonists. Shocked stares, shy glances away, pleas of “calm down!”, “watch out!”, and “cool it!” from the onlookers. After the protagonists got off there were relieved sighs, a few tears, and a tram full of not-quite-returning-to-nomal people.

Łukasiewicz, Supervaluations and the Future