Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility

Abstract

Abstract.Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.

Decidability of Admissibility: On a Problem by Friedman and Its Solution by Rybakov

“…Rules, however, are not. We refer to Fedorishin and Ivanov (2003) and Goudsmit (2016) for a full argument on this and point to Rybakov, Kiyatkin, and Oner (1999) for an argument in the modal case. In the next section, we inspect a weakening of the notion of exactness that can be safely restricted to the finite.…”