Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.

Justification Logic: Reasoning with Reasons

We present two finitary cut-free sequent calculi for the modal μ-calculus. One is a variant of Kozen's axiomatisation in which cut is replaced by a strengthening of the induction rule for greatest fixed point. The second calculus derives annotated sequents in the style of Stirling's ‘tableau proof system with names’ (2014) and features a generalisation of the ν-regeneration rule that allows discharging open assumptions. Soundness and completeness for the two calculi is proved by establishing a sequence of embeddings between proof systems, starting at Stirling's tableau-proofs and ending at the original axiomatisation of the μ-calculus due to Kozen. As a corollary we obtain a new, constructive, proof of completeness for Kozen's axiomatisation which avoids the usual detour through automata and games.

Cut-free completeness for modal mu-calculus

This chapter is dedicated to the memory of George Boolos. From the start of the subject until his death on 27 May 1996 he was the prime inspirer of the work in the logic of provability.

The logic of the provability

We introduce the logics GLP(\Lambda), a generalization of Japaridze's polymodal provability logic GLP(\omega) where \Lambda is any linearly ordered set representing a hierarchy of provability operators of increasing strength. 
We shall provide a reduction of these logics to GLP(\omega) yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP(\Lambda) and the decidability of GLP(\Lambda) for recursive orderings \Lambda. Further, we give a restricted axiomatization of the variable-free fragment of GLP(\Lambda).

On provability logics with linearly ordered modalities

Spectral Properties of a Fourth-Order Differential Operator with Integrable Coefficients. Proceedings of the Steklov Institute of Mathematics

Sequent calculus for the provability logic GL is considered, in which provability is based on the notion of a circular proof. Unlike ordinary derivations, circular proofs are represented by graphs allowed to contain cycles, rather than by finite trees. Using this notion, we obtain a syntactic proof of the Lyndon interpolation property for GL.

Circular proofs for the Gödel-Löb provability logic

We present a sequent-style proof system for provability logic GL that admits so-called circular proofs. For these proofs, the graph underlying a proof is not a finite tree but is allowed to contain cycles. As an application, we establish Lindon interpolation for GL syntactically.

Circular Proofs for G\"odel-L\"ob Logic

We study interpolation properties of provability logics. We prove the Lyndon interpolation for GL and the uniform interpolation for GLP.

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Interpolation properties for provability logics GL and GLP

We present a proof system for the provability logic GLP in the formalism of nested sequents and prove the cut elimination theorem for it. As an application, we obtain the reduction of GLP to its important fragment called J syntactically.

Nested Sequents for Provability Logic GLP

We present a new justification logic corresponding to the Godel-Lob provability logic and prove the realization theorem connecting these two systems in such a way that all the realizations provided in the theorem are normal. Bibliography: 9 titles.

https://iopscience.iop.org/article/10.1070/SM8667/pdf

Daniyar S. Shamkanov

Papers

Circular proofs for the Gödel-Löb provability logic

Circular Proofs for G\"odel-L\"ob Logic

Interpolation properties for provability logics GL and GLP

Nested Sequents for Provability Logic GLP

A realization theorem for the Gödel-Löb provability logic