Modal Correspondence Theory for Possibility Semantics
Kentarô Yamamoto · eScholarship (California Digital Library) · 2018
MODAL CORRESPONDENCE THEORY FOR POSSIBILITY SEMANTICS KENTARO YAMAMOTO 1. I NTRODUCTION Possibility semantics [12] (based on [13]) is a generalization of standard Kripke se- mantics that makes use of a concept of possibility frames. 1 Like Kripke frames, possibility frames have a set of states and binary accessibility relations for modalities. In addition, possibility frames have a refinement relation, which is a partial order between states. Some states in a possibility model may only partially determine the atomic proposi- tions, in contrast to worlds in Kripke models, which completely determine each atomic proposition. Consequently, possibility frames have a close connection with intuitionistic modal frames, but the former yield classical modal logic. As is the case for intuitionistic modal semantics, a key issue for possibility semantics is the interaction between the refinement and accessibility relations. In this setting, modal axioms express properties not only of the accessibility relation but also of the interaction between accessibility and refinement. While standard Kripke frames are semantically equivalent to complete, atomic and completely additive Boolean algebras with operators (BAOs), possibility frames are se- mantically equivalent to complete and completely additive, but not necessarily atomic, BAOs. As shown in [12], for any complete and completely additive BAO, there exists a possibility frame that validates the same modal formulae as the BAO does, and vice versa, just as there exists such a modally equivalent Kripke frame for any complete, atomic and completely additive BAO. It follows from this and other results [14] that more normal modal logics are sound and complete with respect to some class of possi- bility frames than with respect to some class of Kripke frames. For other recent results on possibility semantics and related work, see [3, 4, 10, 11]. In the present paper, we show how correspondence theory, as studied for standard Kripke semantics [2], can be extended to the more general setting of possibility se- mantics. In Section 2, we introduce possibility semantics briefly, referring to [12] for a more detailed account of the semantics. We define key concepts such as possibility frames, possibility models and the standard translation. In Section 3, we study syntac- tic sufficient conditions for local correspondence. In particular, we prove the analogue of Sahlqvist’s Theorem for possibility semantics, namely, that every Sahlqvist formula locally corresponds to a first-order formula with respect to possibility frames. This ex- tends a result in [12] which states that Lemmon-Scott formulae ◊ a ¯ ¯ b p → ¯ c ◊ d ¯ p have Date: June 3, 2016. I wish to give special thanks to Wesley Holliday for his extensive and helpful comments and discussion. I also wish to thank James Walsh and Matthew Harrison-Trainor, who read a draft of the present paper and gave me useful comments. I am grateful for useful comments on the paper from the participants of the 50th MLG meeting in Kyoto, Japan, in January 2016. While completing this work, I also benefited from a talk by Alessandra Palmigiano at the Berkeley-Stanford Circle in Logic and Philosophy in May 2016. Finally, I gratefully acknowledge financial support from the Takenaka Scholarship Foundation. What we call “possibility frames” in the present paper are essentially the “full possibility frames” of [12].