On the Finite Model Property for Kripke Models

Abstract

This work is a sequel to Q4]. The familiarity with the results and terminology of T4] is presupposed. It is a well-known fact that the intuitionistic prepositional logic (abbreviated as LJ) has not a finite characteristic model. But, Jaskowski proved that there is a monotonic descending sequence of finite models which converges to LJ. Using the notation in Q4], we can restate this result as follows; there are finite (pseudo-Boolean) models Pi(i € /) such that LJ^Cr\ieIPi. Now, let's consider the following problem. Let L be any intermediate logic. Are there finite pseudo-Boolean models Pi(i €E /) such that ZOCAie/ Pi? If this problem is solved affirmatively for a logic Z,, we say L has the finite model property, following Harrop's terminology Ql]. In Ql], it is proved that if a logic L is finitely axiomatizable and has the finite model property then L is decidable. It is an interesting problem whether all intermediate logics are decidable. But we don't know even whether all intermediate logics have the finite model property. In this paper, we show that this problem can be reduced to the following problem presented in \j£}: Has any intermediate logic a characteristic Kripke model ? We extend the result to that for logics between Johansson's minimal logic and the classical prepositional logic, and show that the logics whobe decidability are not known in £5] are decidable. Now, we state our main theorem.