Recently, much work has been carried out to study simplicial interpretations of modal logic. While notions of (distributed) knowledge have been well investigated in this context, it has been open how to model belief in simplicial models. We introduce polychromatic simplicial complexes, which naturally impose a plausibility relation on states. From this, we can define various notions of belief. We further explore how these complexes support a somebody-knows modality.

Abstract We provide a simple cut elimination proof for the interpretability logic of $$\textsf{IL}$$ IL . To achieve this, we introduce a traditional Gentzen-style sequent calculus for $$\textsf{IL}$$ IL and a non-wellfounded version of it. The non-wellfounded calculus makes it possible to avoid diagonal formulas. Hence, we can give a simple argument based on a general proof-theoretic method for calculi of this kind. Our results provide a useful basis for further research; in particular, they will allow us to establish uniform interpolation for $$\textsf{IL}$$ IL .

Abstract Faroldi argues that deontic modals are hyperintensional and thus traditional modal logic cannot provide an appropriate formalization of deontic situations. To overcome this issue, we introduce novel justification logics as hyperintensional analogues to non-normal modal logics. We establish soundness and completeness with respect to various models and we study the problem of realization.

Simplicial complexes are a successful model for distributed computing. They have recently been observed to provide an interesting model for epistemic multi-agent logic where the agents' local states are the main building blocks (instead of the global states). A natural generalization is to study epistemic logic on semi-simplicial sets. However, finding the appropriate modal logic for semi-simplicial models has been an open question. We answer this by introducing the logic of synergistic knowledge and establishing its soundness and completeness.

Abstract Different notions of the consistency of obligations collapse in standard deontic logic. In justification logics, which feature explicit reasons for obligations, the situation is different. Their strength depends on a constant specification and on the available set of operations for combining different reasons. We present different consistency principles in justification logic and compare their logical strength. We propose a novel semantics for which justification logics with the explicit version of axiom D, $\textbf {jd}$, are complete for arbitrary constant specifications. Consistency is sometimes formulated in terms of permission. We therefore study permission in the context of justification logic, introducing a notion of free-choice permission for the first time. We then discuss the philosophical implications with regard to some deontic paradoxes.

Abstract We introduce the probabilistic two-agent justification logic $\textsf {IPJ}$, a logic in which we can reason about agents that perform interactive proofs. In order to study the growth rate of the probabilities in $\textsf {IPJ}$, we present a new method of parametrizing $\textsf {IPJ}$ over certain negligible functions. Further, our approach leads to a new notion of zero-knowledge proofs.

Abstract In traditional justification logic, evidence terms have the syntactic form of polynomials, but they are not equipped with the corresponding algebraic structure. We present a novel semantic approach to justification logic that models evidence by a semiring. Hence justification terms can be interpreted as polynomial functions on that semiring. This provides an adequate semantics for evidence terms and clarifies the role of variables in justification logic. Moreover, the algebraic structure makes it possible to compute with evidence. Depending on the chosen semiring this can be used to model trust, probabilities, cost, etc. Last but not least the semiring approach seems promising for obtaining a realization procedure for modal fixed point logics.

Abstract Blockchains are distributed data structures that are used to achieve consensus in systems for cryptocurrencies (like Bitcoin) or smart contracts (like Ethereum). Although blockchains gained a lot of popularity recently, there are only few logic-based models for blockchains available. We introduce $\mathsf{BCL}$, a dynamic logic to reason about blockchain updates, and show that $\mathsf{BCL}$ is sound and complete with respect to a simple blockchain model.

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

Abstract In this paper we provide a strongly complete axiomatization of a temporal epistemic logic in which non-rigid sets of agents are allowed. Using this framework, we prove a number of properties of the blockchain protocol with respect to the given set of axioms and premises.

Abstract Justification logics are modal logics that include justifications for the agent's knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications.

Up to now there was no interpretation available for λ‐calculi featuring overloading and late‐binding, although these are two of the main principles of any object‐oriented programming language. In this paper we provide a new semantics for a stratified version of Castagna's λ{}, a λ‐calculus combining overloading with late‐binding. The model‐construction is carried out in EETJ + (Tot) + (NF‐I), a system of explicit mathematics. We will prove the soundness of our model with respect to subtyping, type‐checking and reductions. Furthermore, we show that our semantics yields a solution to the problem of loss of information in the context of type‐dependent computations.