First-order justification logic with constant domain semantics

Justification logic is a term used to identify a relatively new family of modal-like logics. There is an established literature about propositional justification logic, but incursions on the first-order case are scarce. In this paper we present a constant domain semantics for the first-order logic of proofs with the Barcan Formula (FOLPb); then we prove Soundness and Completeness Theorems. A monotonic semantics for a version of this logic without the Barcan Formula is already in the literature, but constant domains require substantial new machinery, which may prove useful in other contexts as well. Although we work mainly with one system, we also indicate how to generalize these results for the quantified version of JT45, the justification counterpart of the modal logic S5. We believe our methods are more generally applicable, but initially examining specific cases should make the work easier to follow.

A proof-theoretic study of the correspondence of classical logic and modal logic

It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints' result to the basic modal logic S4; we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof.