WORKSHOP 3
Saturday, April 20, 2013 at UCLA
Speakers:, Aldo Antonelli (UC Davis), Mark Balaguer (Cal State LA), and William Tait (University of Chicago)

Location: UCLA's Royce Hall, Room 314 | maps & directions

Speaker: Mark Balaguer (Cal State LA)

Title: Modal Nothingism

Abstract: I develop and defend a view of modal discourse that's compatible with a sober materialistic metaphysics and that can be used in conjunction with anti-platonistic philosophies of mathematics like fictionalism.

Speaker: William Tait (University of Chicago)

Title: In Defense of the Ideal.

Abstract: A brand of skepticism about mathematics rests on a view (a ‘view from inside’) that meaningful discourse about an alleged domain requires some form of mental representation of the objects of that domain, a requirement that discourse about ordinary things—tables, chairs, people, living things in general—satisfies but discourse about ideal things—mathematical structures and their elements—do not. Part of my argument against this skepticism is to note that, by its own criterion, the meaningfulness of discourse about ordinary things is undermined as well. (This is itself a skepticism with ancient lineage, but one which tends to be ignored by the modern skeptic about ideal things.) The remainder of the argument outlines a more adequate point of view about meaning (a ‘view from outside’) from which the meaningfulness of neither discourse is challenged.

Speaker: Aldo Antonelli (UC Davis)

Title: On the General Interpretation of First-Order Quantifiers

Abstract: In his 1950 dissertation, Leon Henkin showed how to provide higher-order quantifiers with  non-standard, or "general" interpretations, on which, for instance, second-order quantifiers are taken to range over collections of subsets of the domain that may fall short of the full power-set. In contrast, first-order quantifiers are usually regarded as immune to this sort of non-standard interpretations, since the semantics for first-order quantifiers is ordinarily taken to be determined by the selection of a first-order domain of objects.

The asymmetry is particularly evident from the point of view of the modern theory of generalized quantifiers, according to which a first-order quantifier is construed as a predicate of subsets of the domain. For example, the first-order existential quantifier is taken to denote the collection of all non-empty subsets, the quantifier "there are exactly k" is taken to denote the collection of all k-membered subsets, etc. But the generalized conception still views first-order quantifiers as predicates over the full power-set, while the possibility that they, similarly to their second-order counterparts, might denote arbitrary collections of subsets has gone mostly unnoticed.

This talk introduces a Henkin-style semantics for arbitrary first-order quantifiers, exploring some of the resulting properties, and emphasizing the effects of imposing various further closure conditions on the second-order component of the interpretation. Among other results, we show by a model-theoretic argument that in certain cases the notion of validity relative to models satisfying the closure conditions is axiomatizable. Finally, although the talk is mainly devoted to laying the technical groundwork, he will touch upon some of the philosophical insights that can be gained from the consideration of non-standard interpretations, especially as regards issues of semantic determinacy of first-order quantifiers and their role in expressing existence claims and ontological commitment.