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

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

Not that there’s anything wrong with that.

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

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

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

The view from my desk.

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

- “Three Cultures—or: what place for logic in the humanities?” unpublished essay. Abstract pdf
- “Generality and Existence I: Quantification and Free Logic,” article in progress. Abstract pdf
- “Assertion, Denial, Accepting, Rejecting, Symmetry and Paradox,” pages 310-321 in
*Foundations of Logical Consequence*, edited by Colin R. Caret and Ole T. Hjortland, Oxford University Press, 2015 Abstract pdf - “Normal Proofs, Cut Free Derivations and Structural Rules,”
*Studia Logica*102:6 (2014) 1143–1166. Abstract pdf - “Pluralism and Proofs,”
*Erkenntnis*79:2 (2014) 279–291. Abstract pdf

*Generality and Existence 3: Substitution and Identity*, Arché Super Special Seminar and HPLM Seminar (3 December 2015). coming soon*Generality and Existence 2: Modality and Quantifiers*, Arché Logic Group Seminar (2 December 2015). coming soon*Generality and Existence 1: Quantification and Free Logic*, Arché Workshop on Inferentialism (25 November 2015).*Fixed Point Models for Theories of Properties and Classes*, 15th Congress on Logic, Methodology, and Philosophy of Science, University of Helsinki (5 August 2015).*Merely Verbal Disputes and Coordinating on Logical Constants*, Australasian Association for Philosophy Conference, Macquarie University (6 July 2015).

- PHIL20030: Meaning, Possibility and Paradox, July–October 2015. completed
- PHIL40013: Uncertainty, Vagueness and Disagreement, July–October 2015. completed

I’m *Greg Restall*, and this is my personal website. I teach philosophy and logic as Professor of Philosophy at the University of Melbourne. ¶ Start at the home page of this site—a compendium of recent additions around here—and go from there to learn more about who I am and what I do. ¶ This is my personal site on the web. Nothing here is *in any way* endorsed by the University of Melbourne.

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