On semantics of first-order justification logic with binding modalities

TL;DR

This paper introduces a first-order justification logic with binding modalities, providing a new semantics based on variable valuations, and proves its soundness and completeness.

Contribution

It develops the first formal semantics for first-order justification logic with binding modalities using variable valuations, avoiding constants.

Findings

Established soundness of the logic.

Proved completeness of the logic.

Provided a new semantics for formulas with free variables.

Abstract

We introduce the first order logic of proofs $FOLP^\Box$ in the joint language combining justification terms and binding modalities. The main issue is Kripke--style semantics for this logic. We describe models for $FOLP^\Box$ in terms of valuations of individual variables instead of introducing constants to the language. This approach requires a new format of the evidence function. This allows us to assign semantic meaning to formulas that contain free variables. The main results are soundness and completeness of $FOLP^\Box$ with respect to the described semantics.