Justification logics are modal-like logics that provide a framework for reasoning about justifications. This paper introduces labeled sequent calculi for justification logics, as well as for combined modal-justification logics. Using a method due to Sara Negri, we internalize the Kripke-style semantics of justification and modal-justification logics, known as Fitting models, within the syntax of the sequent calculus to produce labeled sequent calculi. We show that all rules of these systems are invertible and the structural rules (weakening and contraction) and the (...) cut rule are admissible. Soundness and completeness are established as well. The analyticity for some of our labeled sequent calculi are shown by proving that they enjoy the subformula, sublabel and subterm properties. We also present an analytic labeled sequent calculus for S4LPN based on Artemov–Fitting models. (shrink)

We present tableau proof systems for the annotated version of propositional justification logics, that is, justification logics which are formulated using annotated application operators. We show that the tableau systems are sound and complete with respect to Mkrtychev models, and some tableau systems are analytic and provide a decision procedure for the annotated justification logics. We further show Craig’s interpolation property and Beth’s definability theorem for some annotated justification logics.

Justification Logic provides an axiomatic description of justifications and delegates the question of their nature to semantics. In this note, we address the conceptual issue of the logical type of justifications: we argue that justifications in the logical setting are naturally interpreted as sets of formulas which leads to a class of epistemic models that we call modular models . We show that Fitting models for Justification Logic naturally encode modular models and can be regarded as convenient pre-models of the (...) former. (shrink)

In this paper a complete proper subclass of Hilbert-style S4 proofs, named non-circular, will be determined. This study originates from an investigation into the formal connection between S4, as Logic of Provability and Logic of Knowledge, and Artemov's innovative Logic of Proofs, LP, which later developed into Logic of Justification. The main result concerning the formal connection is the realization theorem , which states that S4 theorems are precisely the formulas which can be converted to LP theorems with proper justificational (...) objects substituting for modal knowledge operators. We extend this result by showing that on the proof level, non-circular proofs are exactly the class of S4 proofs which can be realized to LP proofs. In turn, this study provides an alternative algorithm to achieve the realization theorem, and a novel logical system, called S4ΔS4Δ, is introduced, which, under an adequate interpretation, is worth studying for its own sake. (shrink)

We present algebraic semantics for the classical logic of proofs based on Boolean algebras. We also extend the language of the logic of proofs in order to have a Boolean structure on proof terms and equality predicate on terms. Moreover, the completeness theorem and certain generalizations of Stone’s representation theorem are obtained for all proposed algebras.