The last talk of our Lunch Seminar series for this semester will take place on Monday, May 30th at 13:00 and will be given by Rineke Verbrugge ( University of Gronigen)
Title: Every formula of provability logic is either almost always valid or almost always invalid
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. For modal logics, limit behavior for models and frames may differ. In 1994, Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5. They also proposed zero-one laws for the corresponding classes of frames, but their zero-one law for K-frames has since been disproved.
In this talk, we prove zero-one laws for provability logic with respect to both model and frame validity. Moreover, we axiomatize validity in almost all irreflexive transitive finite models and in almost all irreflexive transitive finite frames, leading to two different axiom systems. In the proofs, we use a combinatorial result by Kleitman and Rothschild about the structure of almost all finite partial orders. On the way, we also show that a previous result by Halpern and Kapron about the axiomatization of almost sure frame validity for S4 is not correct.
The talk is based on the following paper: Verbrugge, R. (2021). Zero-one laws for provability logic: Axiomatizing validity in almost all models and almost all frames. In L. Libkin, ed., Proceedings 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (pp. 1-13). IEEE Press.
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