On Intermediate Justification Logics

TL;DR

This paper develops a unified semantic framework and completeness theorems for a broad class of intermediate justification logics, extending classical justification logic concepts to intermediate propositional logics.

Contribution

It introduces new semantics combining Heyting algebras or Kripke frames with justification logic models, and proves unified completeness and realization theorems for these logics.

Findings

Unified semantics for intermediate justification logics

Completeness theorems established for these logics

Realization theorems extended to a large class of intermediate logics

Abstract

We study abstract intermediate justification logics, that is arbitrary intermediate propositional logics extended with a subset of specific axioms of (classical) justification logics. For these, we introduce various semantics by combining either Heyting algebras or Kripke frames with the usual semantic machinery used by Mkrtychev's, Fitting's or Lehmann's and Studer's models for classical justification logics. We prove unified completeness theorems for all intermediate justification logics and their corresponding semantics using a respective propositional completeness theorem of the underlying intermediate logic. Further, by a modification of a method of Fitting, we prove unified realization theorems for a large class of intermediate justification logics and accompanying intermediate modal logics.