Finite Model Property of Normal Modal Logics and Constant Formulas: an Example

Abstract

We consider the class of propositional normal modal logics. The two main concepts related to this class and analyzed in the paper are the finite model property and constant formula. A propositional normal modal logic has the finite model property, if it can defined as the set of formulas true in frames of some set. All “natural” propositional normal modal logics turned out to have the finite model property. In the 60 years it has been observed that in some cases adding to the axiomatics constant axiom remains Kripke completeness, and hence the finite model property. Note (folklore) that using the deduction theorem it can be shown that here as logic can take the minimal normal modal propositional logic ${\bf K}$. Under constant formula, the constraction of which does not use variables, that is, the basic formula is the constant $\bot$ (false). (Note that in the absence in language the constant can be considered constant formula is a formula that is equivalent to any of substitutional instant; that is, say, the formula $p\wedge\neg p$. The main result of the paper is the definition of a normal modal propositional logic $L$ and a constant formula $\varphi$, such that the result of adding to the logic $L$ axiom $\varphi$ does not have the finite model property. The paper concludes with a short list of open problems