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Possibility Frames and Forcing for Modal Logic

2025; Volume: 22; Issue: 2 Linguagem: Inglês

10.26686/ajl.v22i2.5680

1448-5052

Wesley H. Holliday,

Advanced Algebra and Logic

Possibility Frames and Forcing for Modal Logic ∗ Wesley H. Holliday Department of Philosophy & Group in Logic and the Methodology of Science University of California, Berkeley December 30, 2015 Abstract This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humberstone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. The analogues of classical Kripke frames, i.e., full world frames, are full possibility frames, in which propositional variables may be interpreted as any regular open sets. We develop the beginnings of duality theory, definability/correspondence theory, and complete- ness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] established a duality between the category of Kripke frames with p-morphisms and the category of complete (C), atomic (A), and completely additive (V) BAOs with com- plete BAO-homomorphisms (these categories being dually equivalent), we establish a duality between the category of full possibility frames with possibility morphisms and the category of CV-BAOs, i.e., allow- ing non-atomic BAOs, with complete BAO-homomorphisms (the latter category being dually equivalent to a reflective subcategory of the former). This parallels the connection between forcing posets and Boolean-valued models in set theory, but now with accessibility relations and modal operators involved. Generalizing further, we introduce a class of principal possibility frames that capture the generality of V-BAOs. If we do not require a full or principal frame, then every BAO has an equivalent possibility frame, whose possibilities are proper filters in the BAO. With this filter representation, which does not require the ultrafilter axiom, we are lead to a notion of filter-descriptive possibility frames. Whereas Goldblatt [1974] showed that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of descriptive world frames with p-morphisms, we show that the category of BAOs with ∗ For helpful comments or discussion, I wish to give special thanks to Johan van Benthem, Guram Bezhanishvili, Nick Bezhanishvili, Matthew Harrison-Trainor, and Tadeusz Litak. I also wish to thank Ivano Ciardelli, Josh Dever, Davide Grossi, Lloyd Humberstone, Thomas Icard, Hans Kamp, Larry Moss, Lawrence Valby, Yanjing Wang, and Dag Westerstahl, as well as the participants in my Fall 2014 or Spring 2015 graduate seminars at UC Berkeley: Russell Buehler, Sophia Dandelet, Matthew Harrison-Trainor, Alex Kocurek, Alex Kruckman, James Moody, James Walsh, and Kentaro Yamamoto. I am also thankful for feedback I received when presenting some of this material at the following venues: the Modal Logic Workshop on Consistency and Structure at Carnegie Mellon University in April 2014; my course at the 3rd East-Asian School on Logic, Language and Computation (EASSLLC) at Tsinghua University in July 2014; the Advances in Modal Logic conference at the University of Groningen in August 2014 (see Holliday 2014); the Workshop on the Future of Logic in honor of Johan van Benthem in Amsterdam in September 2014; the Hans Kamp Seminar in Logic and Language at the University of Texas at Austin in April 2015; the 4th CSLI Workshop on Logic, Rationality and Intelligent Interaction at Stanford University in May 2015; and the New Mexico State University Mathematics Colloquium in November 2015. Finally, I wish to gratefully acknowledge an HRF grant from UC Berkeley that allowed me to complete this work in Fall 2015.