The Logic and Metaphysics Workshop will meet on April 23rd from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Melvin Fitting (CUNY).
Title: Quantifiers and Modal Logic
Abstract: In classical logic the move from propositional to quantificational is profound but essentially takes one route, following a direction we are all familiar with. In modal logic, such a move shoots off in many directions at once. One can quantify over things or over intensions. Quantifier domains can be the same from possible world to possible world, shrink or grow as one moves from a possible world to an accessible one, or follow no pattern whatsoever. A long time ago, Kripke showed us how shrinking or growing domains related to validity of the Barcan and the converse Barcan formulas, bringing some semantic order into the situation. But when it comes to proof theory things get somewhat strange. Nested sequents for shrinking or growing domains, or for constant domains or completely varying domains, are relatively straightforward. But axiomatically some oddities are quickly apparent. A simple combination of propositional modal axioms and rules with standard quantificational axioms and rules proves the converse Barcan formula, making it impossible to investigate its absence. Kripke showed how one could avoid this, at the cost of using a somewhat unusual axiomatization of the quantifiers. But things can be complicated and even here an error crept into Kripke’s work that wasn’t pointed out until 20 years later, by Fine. Justification logic was started by Artemov with a system related to propositional S4, called LP. This was extended to a quantified version by Artemov and Yavorskaya, for which a semantics was supplied by Fitting. Recently Artemov and Yavorskaya introduced what they called bounding modalities, by transferring ideas back from quantified LP to S4. In this paper we continue the investigation of bounding modalities, but for axiomatic K since modal details aren’t that important for what I’m interested in. We wind up with axiomatic systems allowing for a monotonic domain condition, an anti-monotonic one, neither, or both. We provide corresponding semantics and give direct soundness and completeness proofs. Unlike in Kripke’s treatment, the heavy lifting is done through generalization of the modal operator, instead of restriction on quantification. (This talk continues one given earlier in the semester in Artemov’s seminar. There are differences, but if you happened to hear that talk, you could easily skip this one since the differences are not great.)
