Provisional Programme:
Thursday 19 June

14.00 Jc Beall (University of Connecticut): The knower as spandrel of truth
I am a glut theorist who thinks that gluts arise as spandrels — or unintended side-effects — of our transparent truth predicate (our see-through device), which we introduced for familiar practical reasons. But is this so for *all* gluts — that all are due to the truth predicate’s introduction into our language? In previous work, I have backed away from this stronger claim, maintaining a weaker claim that gluts are due, one and all, to ‘semantic notions’. In this paper, I explore the stronger (and I think enticing) position, namely, that all gluts, one and all, are due to the truth predicate. In doing so, I meet some concerns (about symmetry, simplicity) and possible objections voiced by Toby Meadows about the ’simply semantic’ glut (or, if you want, gap) picture.
15.45 Tea/coffee

16.15 Greg Restall (University of Melbourne): Modal Definedness
The distinction between defined terms and undefined terms provides a metaphysically “light” way to give a semantics for free logic. Singular terms may be undefined, and if they are undefined, they are not appropriate substitution instances for inference rules for the quantifiers. Sol Feferman, in “Definedness” (Erkenntnis, 1995), provides an elegant system for an extensional negative free logic with undefined terms: it is a very natural model for mathematical reasoning in which we allow undefined terms (like 1/0) and we keep track of the behaviour of such terms by talking of when they are defined and when they are not.
Free logics also see use when it comes to quantified modal logic. It is very tempting to conceive of the domain of quantification as varying from world to world, that what exists in one world might fail to exist in another. This seems to be a very different motivation for free logics—terms which denote in this world and which do not denote in another are not undefined at that world. They are defined all too well—defined to denote something that fails to exist at that world.
In this talk, I will explain the motivations for these two different approaches to free logic, and show that a sequent calculus for Feferman’s own system (with a metaphysically ‘light’ interpretation, that eschews all talk of an outer domain of quantification of non-existent objects) can, with one small change, be naturally extended into a modal hypersequent calculus for a quantified modal logic with a non-constant domain. This, too, has a metaphysically ‘light’ interpretation. The result is a natural proof-theoretical account of a modal logic with varying domain, in which the Barcan formula not only fails, but fails straightforwardly, in a well motivated way.
After introducing the modal system, I will consider what this might mean, about the nature of quantification, and the role of defining rules in characterising logical constants.
Friday 20 June
09.00 Oystein Linnebo (University of Oslo): Indefinite extensibility
According to Russell, Dummett, and many others, various concepts that play an important role in the foundations of mathematics and semantics (such as set, ordinal, cardinal, proposition, property) are indefinitely extensible. By this they mean, very roughly, that whenever we have a “definite totality” of instances of the concept, we can define a further instance of the concept. This talk will canvass some attempts to provide a precise analysis of the concept of indefinite extensibility and put it to philosophical and mathematical use.
10.45 Tea/coffee

11.15 Gabriel Uzquiano (Arché and University of Southern California): On Bernays’ Generalization of Cantor’s Theorem
Cantor’s theorem states that there is no one-to-one correspondence between the members of a set a and the subsets of a. Paul Bernays showed how to encode the claim that there is no one-to-one correspondence between the members of a class A and the subclasses of A by means of a sentence of the language of class theory. Moreover, he proved his generalization of Cantor’s theorem by means of a diagonal argument: given a one-to-one assignment of subclasses of A to members of A, he defined a subclass of A, which, on pain of contradiction, is not assigned to any member of A. It follows from Bernays’ observation that if one assigns a member of A to every subclass of A, then the assignment is not one-one. Unfortunately, familiar arguments for this claim fail to provide an explicit characterization of two different subclasses of A to which one and the same member of A is assigned by the assignment. George Boolos showed how to specify explicit counterexamples to the claim that a function from the power set of a set a onto the set a is one-one. Similar constructions turn out to be available in the case of classes, but they are sensitive to the presence of global choice and impredicative class comprehension. We explore some ramifications of this observation for traditional philosophical puzzles raised by the likes of Russell’s paradox of propositions in Appendix B of The Principles of Mathematics and Kaplan’s paradox.