A Modal Logic of a Truth Definition for Finite Models *

Abstract

The property of being true in almost all finite, initial segments of the standard model of arithmetic is ${\sum }_2^0$ –complete. Thus, it admits a kind of a truth definition. We define such an arithmetical predicate. Then, we define its modal logic SL and prove a completeness theorem with respect to finite models semantics. The proof that SL is the modal logic of the approximate truth definition for finite arithmetical models is based on an extension of SL by a fixed-point construction.