First-order modal logic in the necessary framework of objects

Abstract

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument.

Keywords:

• First-order modal logic
• necessitism
• metaphysical universality
• two-cardinal theorems
• Kreisel’s principle

Acknowledgements

Thanks to Jeremy Goodman and Timothy Williamson for comments on a draft, Øystein Linnebo, Beau Mount and Sam Roberts for discussions of Kreisel’s principle, and audiences in Cambridge, Oxford, Paris and Montreal – especially my commentators, Michael Gabbay in Cambridge and Beau Mount in Oxford – for questions and comments.

Additional information

Funding

This research was supported by an AHRC/Scatcherd European Scholarship.