## Recent News

### Language, Logic and Existential Commitment

#### 18 November 2015

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

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

Not that there’s anything wrong with that.

### Logic at Melbourne finally has a presence on the web

#### 29 October 2015

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.

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

#### 24 October 2015

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.

### Another Day at the Office

#### 4 June 2015

The view from my desk.

#### 30 May 2015

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.

## Recent Writing

• “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