Modal Definedness

June 19, 2014

A presentation in a workshop in the Philosophy of Logic at Arché.

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.