https://consequently.org/writing/ Recent content in Writings on consequently.org Hugo -- gohugo.io en-gb Greg Restall Wed, 10 Sep 2025 00:00:00 +0000 https://consequently.org/writing/reflections-on-mc/ Wed, 10 Sep 2025 00:00:00 +0000 https://consequently.org/writing/reflections-on-mc/ 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. https://consequently.org/writing/prm/ Fri, 29 Aug 2025 00:00:00 +0000 https://consequently.org/writing/prm/ 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. https://consequently.org/writing/what-can-we-mean/ Sat, 19 Apr 2025 00:00:00 +0000 https://consequently.org/writing/what-can-we-mean/ 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. https://consequently.org/writing/proofs-with-star-and-perp/ Tue, 25 Feb 2025 00:00:00 +0000 https://consequently.org/writing/proofs-with-star-and-perp/ 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. https://consequently.org/writing/mlce-ge2/ Mon, 20 Jan 2025 00:00:00 +0000 https://consequently.org/writing/mlce-ge2/ 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. https://consequently.org/writing/gen-inf-acc/ Thu, 10 Oct 2024 00:00:00 +0000 https://consequently.org/writing/gen-inf-acc/ 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. https://consequently.org/writing/qjrid/ Fri, 16 Aug 2024 00:00:00 +0000 https://consequently.org/writing/qjrid/ 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. https://consequently.org/writing/slintro/ Thu, 15 Aug 2024 00:00:00 +0000 https://consequently.org/writing/slintro/ 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.) https://consequently.org/writing/finitude-eternity-etc/ Sat, 27 Jul 2024 00:00:00 +0000 https://consequently.org/writing/finitude-eternity-etc/ 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. https://consequently.org/writing/logic_chinese_translation/ Wed, 01 May 2024 00:00:00 +0000 https://consequently.org/writing/logic_chinese_translation/ This is a translation into Chinese of my introductory textbook Logic. It is available from Huazhong University of Science & Technology Press. https://consequently.org/writing/collection-frames/ Mon, 18 Dec 2023 00:00:00 +0000 https://consequently.org/writing/collection-frames/ 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.) https://consequently.org/writing/psp/ Wed, 13 Dec 2023 00:00:00 +0000 https://consequently.org/writing/psp/ 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). https://consequently.org/writing/looking-at-logics/ Mon, 07 Aug 2023 00:00:00 +0000 https://consequently.org/writing/looking-at-logics/ 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. https://consequently.org/writing/contradictory-christ-review/ Mon, 07 Aug 2023 00:00:00 +0000 https://consequently.org/writing/contradictory-christ-review/ This is short review of Jc Beall’s The Contradictory Christ, OUP, 2021. https://consequently.org/writing/structural-rules-in-natural-deduction-with-alternatives/ Tue, 18 Jul 2023 00:00:00 +0000 https://consequently.org/writing/structural-rules-in-natural-deduction-with-alternatives/ 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. https://consequently.org/writing/logical_methods/ Tue, 03 Jan 2023 00:00:00 +0000 https://consequently.org/writing/logical_methods/ 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. https://consequently.org/writing/contradictions-in-theology/ Sat, 31 Dec 2022 00:00:00 +0000 https://consequently.org/writing/contradictions-in-theology/ 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. https://consequently.org/writing/pmpl-elements/ Fri, 25 Mar 2022 00:00:00 +0000 https://consequently.org/writing/pmpl-elements/ 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). https://consequently.org/writing/geometric-models/ Fri, 14 May 2021 00:00:00 +0000 https://consequently.org/writing/geometric-models/ 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. https://consequently.org/writing/speech-acts-for-classical-proofs/ Tue, 15 Sep 2020 00:00:00 +0000 https://consequently.org/writing/speech-acts-for-classical-proofs/ 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. https://consequently.org/writing/proofs-and-models-in-npt/ Thu, 02 Apr 2020 00:00:00 +0000 https://consequently.org/writing/proofs-and-models-in-npt/ 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?”. https://consequently.org/writing/two-negations/ Thu, 12 Dec 2019 00:00:00 +0000 https://consequently.org/writing/two-negations/ 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. https://consequently.org/writing/nap/ Mon, 02 Dec 2019 00:00:00 +0000 https://consequently.org/writing/nap/ 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. https://consequently.org/writing/generality-and-existence-1/ Fri, 01 Mar 2019 00:00:00 +0000 https://consequently.org/writing/generality-and-existence-1/ 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. https://consequently.org/writing/bradwardine-truth-tellers/ Wed, 21 Mar 2018 00:00:00 +0000 https://consequently.org/writing/bradwardine-truth-tellers/ 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. https://consequently.org/writing/fixed-point-models/ Tue, 28 Feb 2017 00:00:00 +0000 https://consequently.org/writing/fixed-point-models/ 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. https://consequently.org/writing/fde-symmetry-paradox/ Thu, 23 Feb 2017 00:00:00 +0000 https://consequently.org/writing/fde-symmetry-paradox/ 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. https://consequently.org/writing/proof-terms-for-classical-derivations/ Sun, 12 Feb 2017 00:00:00 +0000 https://consequently.org/writing/proof-terms-for-classical-derivations/ 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). https://consequently.org/writing/existence-definedness/ Mon, 03 Oct 2016 00:00:00 +0000 https://consequently.org/writing/existence-definedness/ 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. https://consequently.org/writing/advances-in-pts-review/ Mon, 16 May 2016 00:00:00 +0000 https://consequently.org/writing/advances-in-pts-review/ 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… https://consequently.org/writing/on-priest-on-nonmonotonic/ Tue, 08 Mar 2016 00:00:00 +0000 https://consequently.org/writing/on-priest-on-nonmonotonic/ 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. https://consequently.org/writing/three-cultures/ Thu, 29 Oct 2015 00:00:00 +0000 https://consequently.org/writing/three-cultures/ 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. https://consequently.org/writing/assertiondenialparadox/ Tue, 05 May 2015 00:00:00 +0000 https://consequently.org/writing/assertiondenialparadox/ 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. https://consequently.org/writing/npcfdsr/ Mon, 01 Dec 2014 00:00:00 +0000 https://consequently.org/writing/npcfdsr/ 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. https://consequently.org/writing/pluralism-and-proofs/ Sun, 03 Aug 2014 00:00:00 +0000 https://consequently.org/writing/pluralism-and-proofs/ 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. https://consequently.org/writing/adnct/ Wed, 31 Jul 2013 00:00:00 +0000 https://consequently.org/writing/adnct/ 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. https://consequently.org/writing/ncm-igpl-2013/ Fri, 15 Feb 2013 00:00:00 +0000 https://consequently.org/writing/ncm-igpl-2013/ 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. https://consequently.org/writing/new-waves-in-philosophical-logic/ Mon, 31 Dec 2012 00:00:00 +0000 https://consequently.org/writing/new-waves-in-philosophical-logic/ 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 https://consequently.org/writing/cfss2dml/ Thu, 22 Nov 2012 00:00:00 +0000 https://consequently.org/writing/cfss2dml/ 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.) https://consequently.org/writing/bradwardine-hypersequents/ Fri, 03 Aug 2012 00:00:00 +0000 https://consequently.org/writing/bradwardine-hypersequents/ 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.) https://consequently.org/writing/hlc/ Sat, 03 Mar 2012 00:00:00 +0000 https://consequently.org/writing/hlc/ 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. https://consequently.org/writing/interp-apply-ptml/ Sat, 03 Mar 2012 00:00:00 +0000 https://consequently.org/writing/interp-apply-ptml/ 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. https://consequently.org/writing/ternary-cond/ Thu, 02 Feb 2012 00:00:00 +0000 https://consequently.org/writing/ternary-cond/ 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. https://consequently.org/writing/molinism-trl/ Sun, 01 Jan 2012 00:00:00 +0000 https://consequently.org/writing/molinism-trl/ 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. https://consequently.org/writing/antirealist/ Thu, 13 Oct 2011 00:00:00 +0000 https://consequently.org/writing/antirealist/ 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. https://consequently.org/writing/logic_in_australasia/ Sat, 01 Jan 2011 00:00:00 +0000 https://consequently.org/writing/logic_in_australasia/ 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. https://consequently.org/writing/sitconchan/ Sat, 25 Dec 2010 00:00:00 +0000 https://consequently.org/writing/sitconchan/ 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. https://consequently.org/writing/alwaysmore/ Tue, 16 Nov 2010 00:00:00 +0000 https://consequently.org/writing/alwaysmore/ 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. https://consequently.org/writing/dlot/ Sun, 03 Oct 2010 00:00:00 +0000 https://consequently.org/writing/dlot/ 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. https://consequently.org/writing/ptm-context/ Sun, 03 Oct 2010 00:00:00 +0000 https://consequently.org/writing/ptm-context/ 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. https://consequently.org/writing/barriers/ Thu, 15 Jul 2010 00:00:00 +0000 https://consequently.org/writing/barriers/ 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. https://consequently.org/writing/relevant_agents/ Wed, 03 Mar 2010 00:00:00 +0000 https://consequently.org/writing/relevant_agents/ 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. https://consequently.org/writing/on_t_and_u/ Tue, 16 Feb 2010 00:00:00 +0000 https://consequently.org/writing/on_t_and_u/ 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. https://consequently.org/writing/stp/ Mon, 18 Jan 2010 00:00:00 +0000 https://consequently.org/writing/stp/ 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. https://consequently.org/writing/mfsa/ Thu, 31 Dec 2009 00:00:00 +0000 https://consequently.org/writing/mfsa/ 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. https://consequently.org/writing/permutation/ Tue, 03 Nov 2009 00:00:00 +0000 https://consequently.org/writing/permutation/ 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. https://consequently.org/writing/apriori/ Fri, 17 Jul 2009 00:00:00 +0000 https://consequently.org/writing/apriori/ 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. https://consequently.org/writing/tvpt/ Fri, 01 May 2009 00:00:00 +0000 https://consequently.org/writing/tvpt/ 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. https://consequently.org/writing/peer-instruction/ Mon, 20 Apr 2009 00:00:00 +0000 https://consequently.org/writing/peer-instruction/ We explain how Eric Mazur’s technique of Peer Instruction may be used to teach philosophy, logic and critical thinking — to good effect. https://consequently.org/writing/yo-lo-appendix/ Thu, 01 Jan 2009 00:00:00 +0000 https://consequently.org/writing/yo-lo-appendix/ 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. https://consequently.org/writing/bradwardine-liars/ Wed, 03 Dec 2008 00:00:00 +0000 https://consequently.org/writing/bradwardine-liars/ 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. https://consequently.org/writing/ten2008/ Fri, 03 Oct 2008 00:00:00 +0000 https://consequently.org/writing/ten2008/ 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. https://consequently.org/writing/bradwardine/ Mon, 15 Sep 2008 00:00:00 +0000 https://consequently.org/writing/bradwardine/ 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. https://consequently.org/writing/pa/ Mon, 03 Mar 2008 00:00:00 +0000 https://consequently.org/writing/pa/ 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. https://consequently.org/writing/acdei/ Mon, 03 Mar 2008 00:00:00 +0000 https://consequently.org/writing/acdei/ 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. https://consequently.org/writing/costing/ Mon, 03 Mar 2008 00:00:00 +0000 https://consequently.org/writing/costing/ 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. https://consequently.org/writing/envelopes/ Mon, 03 Mar 2008 00:00:00 +0000 https://consequently.org/writing/envelopes/ 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. https://consequently.org/writing/universal/ Sat, 01 Dec 2007 00:00:00 +0000 https://consequently.org/writing/universal/ 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… https://consequently.org/writing/invention/ Wed, 03 Oct 2007 00:00:00 +0000 https://consequently.org/writing/invention/ 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. https://consequently.org/writing/ptm-second-order/ Sat, 03 Mar 2007 00:00:00 +0000 https://consequently.org/writing/ptm-second-order/ 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. https://consequently.org/writing/s5nets/ Sat, 03 Mar 2007 00:00:00 +0000 https://consequently.org/writing/s5nets/ 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. https://consequently.org/writing/symbolic-logic/ Sat, 03 Mar 2007 00:00:00 +0000 https://consequently.org/writing/symbolic-logic/ 1007 words on symbolic logic – concentrating on the history of 20th Century logic, aimed at an audience of social scientists https://consequently.org/writing/channels/ Fri, 03 Mar 2006 00:00:00 +0000 https://consequently.org/writing/channels/ 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. https://consequently.org/writing/masses/ Fri, 03 Mar 2006 00:00:00 +0000 https://consequently.org/writing/masses/ 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. https://consequently.org/writing/hpplssl/ Fri, 03 Mar 2006 00:00:00 +0000 https://consequently.org/writing/hpplssl/ An historical essay, sketching the development of relevant and substructural logics throughout the 20th Century and into the 21st. https://consequently.org/writing/rrq/ Fri, 03 Mar 2006 00:00:00 +0000 https://consequently.org/writing/rrq/ 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. https://consequently.org/writing/logic/ Sun, 01 Jan 2006 00:00:00 +0000 https://consequently.org/writing/logic/ 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. https://consequently.org/writing/pluralism/ Sun, 01 Jan 2006 00:00:00 +0000 https://consequently.org/writing/pluralism/ 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. https://consequently.org/writing/logical_consequence/ Mon, 12 Dec 2005 00:00:00 +0000 https://consequently.org/writing/logical_consequence/ 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). https://consequently.org/writing/cdqml/ Fri, 02 Sep 2005 00:00:00 +0000 https://consequently.org/writing/cdqml/ 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. https://consequently.org/writing/lsf/ Thu, 03 Mar 2005 00:00:00 +0000 https://consequently.org/writing/lsf/ In this paper I consider an interpretation of future contingents which motivates a unification of a Łukasiewicz style logic with the more classical supervaluational semantics. This in turn motivates a new non-classical logic modelling what is “made true by history up until now.” I give a simple Hilbert-style proof theory, and a soundness and completeness argument for the proof theory with respect to the intended models. This paper is available at http://www.units.it/~episteme/L&PS_Vol3No1/contents_L&PS_Vol3No1.htm. https://consequently.org/writing/dmap-belnap-lambert/ Thu, 03 Mar 2005 00:00:00 +0000 https://consequently.org/writing/dmap-belnap-lambert/ Two short entries on two great philosophers of logic (and philosophical logicians) of late 20th and early 21st Century American philosophy, Nuel Belnap and Karel Lambert. https://consequently.org/writing/minitrue/ Thu, 03 Mar 2005 00:00:00 +0000 https://consequently.org/writing/minitrue/ Minimalists about truth say that the important properties of the truth predicate are revealed in the class of T-biconditionals. Most minimalists demur from taking all of the T-biconditionals of the form “<p> is true if and only if p”, to be true, because to do so leads to paradox. But exactly which biconditionals turn out to be true? I take a leaf out of the epistemic account of vagueness to show how the minimalist can avoid giving a comprehensive answer to that question. I also show that this response is entailed by taking minimalism seriously, and that objections to this position may be usefully aided and abetted by Gupta and Belnap’s revision theory of truth. https://consequently.org/writing/mf/ Thu, 03 Mar 2005 00:00:00 +0000 https://consequently.org/writing/mf/ In this paper we introduce a distinct meta-ethical position, fictionalism about morality. We clarify and defend the position, showing that it is a way to save the “moral phenomena” while agreeing that there is no genuine objective prescriptivity to be described by moral terms. In particular, we distiguish moral fictionalism from moral quasi-realism, and we show that fictionalism many all of the virtues of quasi-realism but few of the vices. https://consequently.org/writing/multipleconclusions/ Thu, 03 Mar 2005 00:00:00 +0000 https://consequently.org/writing/multipleconclusions/ I argue for the following four theses. (1) Denial is not to be analysed as the assertion of a negation. (2) Given the concepts of assertion and denial, we have the resources to analyse logical consequence as relating arguments with multiple premises and multiple conclusions. Gentzen’s multiple conclusion calculus can be understood in a straightforward, motivated, non-question-begging way. (3) If a broadly anti-realist or inferentialist justification of a logical system works, it works just as well for classical logic as it does for intuitionistic logic. The special case for an anti-realist justification of intuitionistic logic over and above a justification of classical logic relies on an unjustified assumption about the shape of proofs. Finally, (4) this picture of logical consequence provides a relatively neutral shared vocabulary which can help us understand and adjudicate debates between proponents of classical and non-classical logics. This paper has now been reprinted in Analysis and Metaphysics, 6, 2007, 14-34. https://consequently.org/writing/notevery/ Thu, 03 Mar 2005 00:00:00 +0000 https://consequently.org/writing/notevery/ According to the “knowability thesis,” every truth is knowable. Fitch’s paradox refutes the knowability thesis by showing that if we are not omniscient, then not only are some truths not known, but there are some truths that are not knowable. In this paper, I propose a weakening of the knowability thesis (which I call the “conjunctive knowability thesis”) to the effect that for every truth p there is a collection of truths such that (i) each of them is knowable and (ii) their conjunction is equivalent to p. I show that the conjunctive knowability thesis avoids triviality arguments against it, and that it fares very differently depending on one other issue connecting knowledge and possibility. If some things are knowable but false, then the conjunctive knowability thesis is trivially true. On the other hand, if knowability entails truth, the conjunctive knowability thesis is coherent, but only if the logic of possibility is quite weak. Update, April 2005: I’ve added a reference (thanks to Joe Salerno) to a recent paper of Risto Hilpinen, in which he motivates a conjunctive knowability thesis on the grounds of a Peircean pragmatism. https://consequently.org/writing/gndl/ Thu, 03 Mar 2005 00:00:00 +0000 https://consequently.org/writing/gndl/ In this paper, we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness. We show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic, and we indicate how proofs in this system may be labelled with terms exhibiting a kind of Curry-Howard isomorphism. This natural deduction system is inspired both by Shoesmith and Smiley’s multiple conclusion systems for classical logic and Girard’s proofnets for linear logic. This paper is joint work with Francesco Paoli. https://consequently.org/writing/facts/ Sun, 03 Oct 2004 00:00:00 +0000 https://consequently.org/writing/facts/ Stephen Neale, in Facing Facts takes theories of facts, truthmakers, and non-extensional connectives to be threatened by triviality in the face of powerful “slingshot” arguments. In this paper I rehearse the most powerful of these arguments, and then show that friends of facts have resources sufficient to not only resist slingshot arguments but also to be untroubled by them. If a fact theory is provided with a model, then the fact theorist can be sure that this theory is secure from triviality arguments. https://consequently.org/writing/lnclem/ Tue, 20 Jul 2004 00:00:00 +0000 https://consequently.org/writing/lnclem/ There is widespread agreement that the law of non-contradiction is an important logical principle. There is less agreement on exactly what the law amounts to. This unclarity is brought to light by the emergence of paraconsistent logics in which contradictions are tolerated (in the sense that not everything need follow from a contradiction, and that there are “worlds” in which contradictions are true) but in which the statement ~(A & ~A) (it is not the case that A and not-A) is still provable. This paper attempts to clarify the connection between different readings of the law of non-contradiction, the duality between the law of non-contradiction and the law of the excluded middle, and connections with logical consequence in general. https://consequently.org/writing/warrantpres/ Wed, 03 Mar 2004 00:00:00 +0000 https://consequently.org/writing/warrantpres/ I defend logical pluralism against the charge that One True Logic is motivated by considerations of warrant preservation. On the way I attempt to clarify just a little the connections between deductive validity and epistemology. https://consequently.org/writing/routes/ Wed, 03 Mar 2004 00:00:00 +0000 https://consequently.org/writing/routes/ It is well known that contraction-related principles trivialise naïve class theory. It is less well known that many other principles unrelated to contraction also render the theory trivial. This paper provides a characterisation of a large class of formulas which do the job. This class includes all properly implication formulas known in the literature, and adds countably many more. https://consequently.org/writing/logicians/ Mon, 05 Jan 2004 00:00:00 +0000 https://consequently.org/writing/logicians/ Bernhard Bolzano (1781-1848) Bolzano was a philosopher, mathematician and logician who had important things to contribute in each of these fields. He’s probably best known for what he has done in mathematics: the precise definition of continuity when it comes to real-valued functions. But for my money (and in my discipline) Bolzano was worth much more than this. He was the first philosopher to give a precise analysis of logical consequence in terms we would recognise today. (This was important for his project of understanding mathematics, and making it clearer, because mathematicians were getting into knots considering infinite numbers, infinitesimal numbers, and many other seemingly paradoxical things. The 1800s involved mathematicians painstakingly going over old reasoning step-by-step to see if (and how) the reasoning worked, and to see where it might go wrong. You can really only do this if you have a good understanding of how reasoning works. This is what Bolzano was trying to supply.) Anyway, he argued that you can tell that an argument from premises to a conclusion is logically valid if and only if it never proceeds from truth to falsity no matter how you change the non-logical vocabulary in the argument. So Sally is coming to the party. https://consequently.org/writing/modelling/ Thu, 03 Jul 2003 00:00:00 +0000 https://consequently.org/writing/modelling/ Published in a special issue of Logique et Analyse, on Truthmakers, edited by Peter Forrest and Drew Khlentzos. This expands on my paper “Truthmakers, Entailment and Necessity" by showing that the theses on truthmaking given in that paper are mutually consistent. It does this by providing a model in which each thesis is true. This model is an independently motivated model of a weak quantified relevant logic with an existence predicate. https://consequently.org/writing/whatisfbp/ Mon, 03 Mar 2003 00:00:00 +0000 https://consequently.org/writing/whatisfbp/ Mark Balaguer has, in Platonism and Anti-Platonism in Mathematics, given us an intriguing new brand of platonism, which he calls, plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that none of Balaguer’s attempts to characterise full-blooded platonism succeed. They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it. EXCERPT: “Just What is Full-Blooded Platonism?” https://consequently.org/writing/logic_chapter/ Mon, 03 Mar 2003 00:00:00 +0000 https://consequently.org/writing/logic_chapter/ An introduction to the formal and philosophical logic, for a general introductory handbook to philosophy. I introduce the concepts of deductive and inductive validity, formal languages, proofs, models, soundness and completeness, and many other things besides. https://consequently.org/writing/strenge/ Sun, 03 Mar 2002 00:00:00 +0000 https://consequently.org/writing/strenge/ We consider the virtues of relevant arithmetic couched in the logic E of entailment as opposed to R of relevant implication. The move to a stronger logic allows us to construct a complete system of true arithmetic, in which whenever an entailment A → B is not true (an example: 0=2 → 0=1 is not provable) then its negation ~(A → B) is true. https://consequently.org/writing/carnap/ Sun, 03 Mar 2002 00:00:00 +0000 https://consequently.org/writing/carnap/ In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap’s Principle of Tolerance from a pluralism which maintains that there are different, equally “good” logical consequence relations on the one language. I will argue that this second form of pluralism does more justice to the contemporary state of logical theory and practice than does Carnap’s more moderate pluralism. https://consequently.org/writing/pev/ Sun, 03 Mar 2002 00:00:00 +0000 https://consequently.org/writing/pev/ Paraconsistent logics are, by definition, inconsistency tolerant: In a paraconsistent logic, inconsistencies need not entail everything. However, there is more than one way a body of information can be inconsistent. In this paper I distinguish contradictions from other inconsistencies, and I show that several different logics are, in an important sense, “paraconsistent” in virtue of being inconsistency tolerant without thereby being contradiction tolerant. For example, even though no inconsistencies are tolerated by intuitionistic propositional logic, some inconsistencies are tolerated by intuitionistic predicate logic. In this way, intuitionistic predicate logic is, in a mild sense, paraconsistent. So too are orthologic and quantum propositional logic and other formal systems. Given this fact, a widespread view that traditional paraconsistent logics are especially repugnant because they countenance inconsistencies is undercut. Many well-understood nonclassical logics countenance inconsistencies as well. (This paper was published in 2004, despite the apparent date of 2002 in the citation. The tale is told in more detail here.) https://consequently.org/writing/rle/ Sun, 03 Mar 2002 00:00:00 +0000 https://consequently.org/writing/rle/ A revision of the Handbook of Philosophical Logic essay on relevant logics, updating it to include work on display logic, relevant predication, Urquhart’s undecidability and complexity results, and connections with other substructural logics. https://consequently.org/writing/conlogtr/ Wed, 03 Oct 2001 00:00:00 +0000 https://consequently.org/writing/conlogtr/ Shapiro and Taschek have argued that simply using intuitionistic logic and its Heyting semantics, one can show that there are no gaps in warranted assertibility. That is, given that a discourse is faithfully modelled using Heyting”s semantics for the logical constants, that if a statement S is not warrantedly assertible, then its negation ~S is. Tennant has argued for this conclusion on similar grounds. I show that these arguments fail, albeit in illuminating ways. I will show that an appeal to constructive logic does not commit one to this strong epistemological thesis, but that appeals to semantics of intuitionistic logic nonetheless do give us certain conclusions about the connections between warranted assertibility and truth. https://consequently.org/writing/defplur/ Tue, 03 Jul 2001 00:00:00 +0000 https://consequently.org/writing/defplur/ A defence of logical pluralism against a number of objections, primarily from Graham Priest in his article “Logic: One or Many” in the same volume. https://consequently.org/writing/logical_pluralism/ Wed, 20 Dec 2000 00:00:00 +0000 https://consequently.org/writing/logical_pluralism/ This is our article 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. A bigger version of the argument (with much more detail) is found in our book of the same name. https://consequently.org/writing/defdneg/ Tue, 03 Oct 2000 00:00:00 +0000 https://consequently.org/writing/defdneg/ In his paper “Generalised Ortho Negation” J. Michael Dunn mentions a claim of mine to the effect that there is no condition on `perp frames’ equivalent to the holding of double negation elimination (from ~~A to infer A). That claim of mine was wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses. (Ed Mares has pointed out that there’s some overlap between this paper and his earlier “A Star-Free Semantics for R” (JSL 1995), which I’d read long before writing this one. The particular modelling condition for DNE is still original to me, however.) https://consequently.org/writing/isl/ Sat, 01 Jan 2000 00:00:00 +0000 https://consequently.org/writing/isl/ This book is, as the title says, an introduction to substructural logics. Think of it as a book which tries to do for substructural logics what Hughes and Cresswell’s New Introduction did for modal logics. This book was published by Routledge in January 2000. It’s available at Amazon and many other bookstores. https://consequently.org/writing/negrl/ Wed, 03 Mar 1999 00:00:00 +0000 https://consequently.org/writing/negrl/ Negation raises three thorny problems for anyone seeking to interpret relevant logics. The frame semantics for negation in relevant logics involves a point shift operator *. Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ~AvB to B to be invalid, because the corresponding relevant conditional A&(~AvB)→B is not a theorem. Yet we often make the inference from A and ~AvB to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x makes A true if and only if x doesn’t make A true. Problem number three involves deciding which is the real negation. How can we decide between orthodox negation and the new, Boolean negation. In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow us to give principled answers to each of these questions. https://consequently.org/writing/display/ Tue, 03 Nov 1998 00:00:00 +0000 https://consequently.org/writing/display/ Many logics in the relevant family can be given a proof theory in the style of Belnap’s display logic. However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to show decidability for a large range of substructural logics. https://consequently.org/writing/desecsed/ Tue, 03 Mar 1998 00:00:00 +0000 https://consequently.org/writing/desecsed/ In classical and intuitionistic arithmetics, any formula implies a true equation, and a false equation implies anything. In weaker logics fewer implications hold. In this paper we rehearse known results about the relevant arithmetic R#, and we show that in linear arithmetic LL# by contrast false equations never imply true ones. As a result, linear arithmetic is desecsed. A formula A which entails 0=0 is a secondary equation; one entailed by 0≠0 is a secondary unequation. A system of formal arithmetic is secsed if every extensional formula is either a secondary equation or a secondary unequation. We are indebted to the program MaGIC for the simple countermodel SZ7, on which 0=1 is not a secondary formula. This is a small but significant success for automated reasoning. https://consequently.org/writing/logiclaws/ Tue, 03 Mar 1998 00:00:00 +0000 https://consequently.org/writing/logiclaws/ This is an introductory essay on the notion of a “Logical Law.” In it, I show that there are three important different questions one can ask about logical laws. Firstly, what it means to be a logical law. Secondly, what makes something a logical law, and thirdly, what are the logical laws. Each of these questions are answered differently by different people. I sketch the important differences in views, and point the way ahead for logical research. https://consequently.org/writing/cantbe/ Tue, 03 Mar 1998 00:00:00 +0000 https://consequently.org/writing/cantbe/ I show that the believer in possible worlds can have impossible worlds for free, as classes of possible worlds. These do exactly the job of ways that things cannot be, and they model the simple paraconsistent logic LP. This motivates a semantics for paraconsistent logic that even a classical logician can love. https://consequently.org/writing/extendingj/ Thu, 03 Apr 1997 00:00:00 +0000 https://consequently.org/writing/extendingj/ Ideas on the extension of intuitionistic propositional and predicate logic with a ‘subtraction’ connective, Galois connected with disjunction, dual to the implication connective, Galois connected with conjunction. Presented to an audience at Victoria University of Wellington, July 1997. I like this material, but it does not contain any ideas not accessible elsewhere, so it won’t be published anywhere other than here. https://consequently.org/writing/combipn/ Mon, 03 Mar 1997 00:00:00 +0000 https://consequently.org/writing/combipn/ Combining non-classical (or ‘sub-classical’) logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility (in the presence of conjunction and disjunction), and then we combine the resulting systems with intuitionistic logic. We will find that some of Marcus Kracht’s results on the undecidability of classical modal logics generalise to a non-classical setting. We will also see conditions under which intuitionistic logic can be combined with a non-intuitionistic negation without corrupting the intuitionistic fragment of the logic. https://consequently.org/writing/plog/ Mon, 03 Mar 1997 00:00:00 +0000 https://consequently.org/writing/plog/ I respond to an interesting argument of Hartley Slater to the effect that there is no such thing as paraconsistent logic. Slater argues that since paraconsistent logics involve interpreting a sentence and its negation as both true at points in a model structure, it is not really negation that is being modelled, since negation is meant to be a contradictory forming operator. I sketch how different paraconsistentists can respond to his argument, and I then defend my own response, that although contradictions are indeed never true (and cannot be true) it does not follow that a semantics ought not evaluate them as true in certain models. https://consequently.org/writing/dggl/ Tue, 05 Mar 1996 00:00:00 +0000 https://consequently.org/writing/dggl/ This paper is a revised version of a talk given at the Logic and Logical Philosophy conference in Poland in September 1995. In it, I sketch the connections between Nuel Belnap’s Display Logic and J. Michael Dunn’s Gaggle Theory. https://consequently.org/writing/ten/ Sun, 03 Mar 1996 00:00:00 +0000 https://consequently.org/writing/ten/ A truthmaker for a proposition p is an object such that necessarily, if it exists, then p is true. In this paper, I show that the truthmaker thesis (that every truth has a truthmaker) and the disjunction thesis (that a truthmaker for a disjunction must be a truthmaker for one of the disjuncts) are jointly incompatible with the view that the if in the truthmaking clause is a merely material conditional. I then defend the disjunction thesis, and go on to argue that a relevant conditional will serve well in place of the material conditional in defining what it is to be a truthmaker. Conversely, I argue that a semantics involving truthmakers could shed some light on the interpretation of relevant implication. https://consequently.org/writing/rbr/ Sun, 08 Oct 1995 00:00:00 +0000 https://consequently.org/writing/rbr/ In this paper we consider the implications for belief revision of weakening the logic under which belief sets are taken to be closed. A widely held view is that the usual belief revision functions are highly classical, especially in being driven by consistency. We show that, on the contrary, the standard representation theorems still hold for paraconsistent belief revision. Then we give conditions under which consistency is preserved by revisions, and we show that this modelling allows for the gradual revision of inconsistency. https://consequently.org/writing/arithluk/ Tue, 03 Oct 1995 00:00:00 +0000 https://consequently.org/writing/arithluk/ Peano arithmetic formulated in Łukasiewicz’s infinitely valued logic collapses into classical Peano arithmetic. However, not all additions to the language need also be classical. The way is open for the addition of a real truth predicate satisfying the T-scheme into the language. However, such an addition is not pleasing. The resulting theory is omega-inconsistent. This paper consists of the proofs and interpretations of these two results. https://consequently.org/writing/modalitiessl/ Mon, 03 Apr 1995 00:00:00 +0000 https://consequently.org/writing/modalitiessl/ This paper generalises Girard’s results which embed intuitionistic logic into linear logic by showing how arbitrary substructural logics can be embedded into weaker substructural logics, using a single modality which ’encodes’ the new structural rules. https://consequently.org/writing/fourvalued/ Fri, 03 Mar 1995 00:00:00 +0000 https://consequently.org/writing/fourvalued/ This paper gives an outline of three different approaches to the four-valued semantics for relevant logics (and other non-classical logics in their vicinity). The first approach borrows from the ‘Australian Plan’ semantics, which uses a unary operator ‘*’ for the evaluation of negation. This approach can model anything that the two-valued account can, but at the cost of relying on insights from the Australian Plan. The second approach is natural, well motivated, independent of the Australian Plan, and it provides a semantics for the contraction-free relevant logic RW. Unfortunately, its approach seems to model little else. The third approach seems to capture a wide range of formal systems, but at the time of writing, lacks a completeness proof. https://consequently.org/writing/infflow/ Fri, 03 Mar 1995 00:00:00 +0000 https://consequently.org/writing/infflow/ John Perry, one of the two founders of the field of situation semantics, indicated in an interview in 1986 that there is some kind of connection between relevant logic and situation semantics. I do know that a lot of ideas that seemed off the wall when I first encountered them years ago now seem pretty sensible. One example that our commentators don’t mention is relevance logic; there are a lot of themes in that literature that bear on the themes we mention. In 1992, in Entailment volume 2, Nuel Belnap and J. Michael Dunn hinted at similar ideas. Referring to situation semantics, they wrote … we do not mean to claim too much here. The Barwise-Perry semantics is clearly independent and its application to natural-language constructions is rich and novel. But we like to think that at least first degree (relevant) entailments have a home there. In this paper I show that these hints and gestures are true. And perhaps truer than those that made them thought at the time. In this paper I introduce the semantics of relevant logics, then I will sketch the parts of situation theory relevant to our enterprise. Finally, I bring the two together in what is hopefully, a harmonious way. https://consequently.org/writing/nggl/ Fri, 03 Mar 1995 00:00:00 +0000 https://consequently.org/writing/nggl/ I introduce Nietzsche’s critique of religious belief, for an audience of thinking Christians. I show that his criticism cannot and ought not be simply shrugged off, but rather, it can form the basis of a useful self-critique for the religious believer. I also argue that any response to a Nietzschean critique of religious belief and practice must itself take the form of an embodied believing life, rather than a merely theoretical response. https://consequently.org/writing/usl/ Thu, 03 Mar 1994 00:00:00 +0000 https://consequently.org/writing/usl/ I defend the extension of the lambek calculus with a distributive extensional conjunction and disjunction. I show how it independently arises in linguistics, information flow and relevant logics, and relation algebra. I give the logic a cut-free Gentzenisation and show that it is decidable. https://consequently.org/writing/subint/ Thu, 03 Mar 1994 00:00:00 +0000 https://consequently.org/writing/subint/ Once the Kripke semantics for normal modal logics were introduced, a whole family of modal logics other than the Lewis systems S1 to S5 were discovered. These logics were obtained by changing the semantics in natural ways. The same can be said of the Kripke-style semantics for relevant logics: a whole range of logics other than the standard systems R, E and T were unearthed once a semantics was given. In a similar way, weakening the structural rules of the Gentzen formulation of classical logic gives rise to other “substructural” logics such as linear logic. This process of “strategic weakening” is becoming popular today, with the discovery of applications of these logics to areas such as linguistics and the theory of computation. This paper examines what the process of weakening does to the Kripke-style semantics of intuitionistic logic, introducing the family of subintuitionistic logics. https://consequently.org/writing/onlogics/ Wed, 05 Jan 1994 00:00:00 +0000 https://consequently.org/writing/onlogics/ My Ph.D. Thesis, completed in January 1994. I was supervised by Prof. Graham Priest, at the University of Queensland. The thesis is 292 pages of work on logics without the contraction rule. https://consequently.org/writing/dlpsr/ Wed, 03 Mar 1993 00:00:00 +0000 https://consequently.org/writing/dlpsr/ I argue that the extant reasons for sticking to classical logic in the face of the paradoxes of self reference are not good reasons. The paradoxes are really difficult and we should use all of the weapons at our disposal. https://consequently.org/writing/reallycf/ Wed, 03 Mar 1993 00:00:00 +0000 https://consequently.org/writing/reallycf/ I show that any finitely valued logic of a simple kind fails to support naïve comprehension, if it has a conditional. I then go on to show how some infinitely valued logics also fail to be robustly contraction free. Then I make a bold conjecture that robust contraction freedom is sufficient to support naïve set theory. This conjecture was later proved to be wrong by two graduate students from Monash, Sam Buchart and Su Rogerson, in some delightful work in 1997, which has since also been published in Studia Logica. https://consequently.org/writing/simpsem/ Wed, 03 Mar 1993 00:00:00 +0000 https://consequently.org/writing/simpsem/ I show how Priest and Sylvan’s simplified semantics extends from basic relevant logics to a large class of stronger logics. The completeness proof is a little tricky, given the different behaviour of the normal world in the models. This is the first paper from my Ph.D. thesis. https://consequently.org/writing/nstlp/ Tue, 03 Mar 1992 00:00:00 +0000 https://consequently.org/writing/nstlp/ My first publication. It stems from work I did in my Honours year (1989) with Graham Priest, on paraconsistent logic. I explain a particularly simple yet powerful technique for constructing models of naïve set theory in the paraconsistent logic LP. This can be used to show the consistency of the theory, and to construct models invalidating some of the axioms of ZFC.