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 for arithmetic is a Σ 0 2-complete property. Thus, it admits a kind of a weak 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 a weak truth definition for finite arithmetical models is based on an extension of SL by a fixpoint construction.

The Modal Logics of Kripke–feferman Truth

“…7 Nonetheless, already the T-sentences for atomic sentences in the language without the truth predicate-which are required to provide nontrivial axiomatizations of the truth predicate-have an impact on the modal logic of the theory and this impact cannot be immediately read off the axioms of the truth theory. In fact, we will show that such T-sentences force the modal logic of theories of Kripke-Feferman truth to comprise the axiom ✷ϕ ∧ ¬✷¬ϕ → ϕ (faith ✷ ) 5 Similarly, Czarnecki and Zdanowski [11] show the modal logic KDDc to be maximal. 6 We highlight, however, that our results will also apply to base theories weaker than I∆ 0 + Exp, such as Buss's S 1 2 or I∆ 0 + Ω 1 .…”

“…A saturated set is, roughly, a nontrivial set of formulas closed under the particular logic whose completeness is at stake. 11 The detailed proof of the next claim is provided in Appendix B.…”

“…Here * is a function that maps the propositional variable of the modal language to sentences of the language of the truth theory; I is the truth-interpretation, that is, a translation that commutes with the logical connectives and translates the modal operator ✷ by the truth predicate: I * (✷ϕ) = T I * (ϕ) . Some initial research in this direction has already been undertaken by Czarnecki and Zdanowski [11] and Standefer [41], who study theories of truth inspired by revision theoretic approaches [19]. 3 Czarnecki and Zdanowski, and Standefer determine, albeit in slightly different guise, the modal logic of nearly stable truth, that is, the modal logic of the axiomatic truth theory Friedman-Sheard [17,21] to be the modal logic KDD c .…”