Section: New Results

Justification Logic for Constructive Modal Logic

Participants : Lutz Straßburger, Sonia Marin.

Justification logic is a family of modal logics generalizing the Logic of Proofs $LP$, introduced by Artemov in  [45]. The original motivation, which was inspired by works of Kolmogorov and Gödel in the 1930's, was to give a classical provability semantics to intuitionistic propositional logic. The language of the Logic of Proofs can be seen as a modal language where occurrences of the $\square$-modality are replaced with terms, also known as proof polynomials, evidence terms, or justification terms, depending on the setting. The intended meaning of the formula `$t:A$' is `$t$ is a proof of $A$' or, more generally, the reason for the validity of $A$. Thus, the justification language is viewed as a refinement of the modal language, with one provability construct $\square$ replaced with an infinite family of specific proofs. In a joint work with Roman Kuznets (TU Wien), we add a second type of terms, which we call witness terms and denote by Greek letters. Thus, a formula $\diamond A$ is to be realized by `$\mu :A$'. The intuitive understanding of these terms is based on the view of $\diamond$ modality as representing consistency (with $\square$ still read as provability). The term $\mu$ justifying the consistency of a formula is viewed as an abstract witnessing model for the formula. We keep these witnesses abstract so as not to rely on any specific semantics. All the operations on witness terms that we employ to ensure the realization theorem for $CK$, $CD$, $CT$, and $CS4$. This work has been presented at the IMLA 2017 workshop [40]