In this paper, we introduce the probabilistic justification logic PJ, a logic in which we can reason about the probability of justification statements. We present its syntax and semantics, and establish a strong completeness theorem. Moreover, we investigate the relationship between PJ and the logic of uncertain justifications.

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We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as a corollary.

This paper presents a new model construction for a natural cut-free infinitary version K + ω (µ) of the propositional modal µ-calculus. Based on that the completeness of K + ω (µ) and the related system K ω (µ) can be established directly-no detour, for example through automata theory, is needed. As a side result we also obtain a finite, cut-free sound and complete system for the propositional modal µ-calculus.

ABSTRACT. Justification logics are epistemic logics that explicitly include justifications for the agents' knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting's semantics for the Logic of Proofs LP. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya's minimal bimodal explicit evidence logic, which is a two-agent version of LP.We discuss the relationship of our logic to the multi-agent modal logic S4 with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic.

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We present a probabilistic justification logic, PPJ, as a framework for uncertain reasoning about rational belief, degrees of belief and justifications. We establish soundness and strong completeness for PPJ with respect to the class of so-called measurable Kripke-like models and show that the satisfiability problem is decidable. We discuss how PPJ provides insight into the well-known lottery paradox.

Abstract. Justification logic is an epistemic framework that provides a way to express explicit justifications for the agent's belief. In this paper, we present OPAL, a dynamic justification logic that includes term operators to reflect public announcements on the level of justifications. We create dynamic epistemic semantics for OPAL. We also elaborate on the relationship of dynamic justification logics to Gerbrandy-Groeneveld's PAL by providing a partial realization theorem.