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We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes…
Recently an extension to higher-order logic -- called DHOL -- was introduced, enriching the language with dependent types, and creating a powerful extensional type theory. In this paper we propose two ways how choice can be added to DHOL.…
We investigate some finitely-valued generalizations of propositional dynamic logic with tests. We start by introducing the (n+1)-valued Kripke models and a corresponding language based on a modal extension of {\L}ukasiewicz many-valued…
Sequential propositional logic deviates from ordinary propositional logic by taking into account that during the sequential evaluation of a propositional statement,atomic propositions may yield different Boolean values at repeated…
In a previous paper, a tableau calculus has been presented, which constitute a decision procedure for hybrid logic with the converse and global modalities and a restricted use of the binder. This work extends such a calculus to multi-modal…
This paper introduces a logical system, called BV, which extends multiplicative linear logic by a non-commutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved…
In this chapter we present a case study, drawn from our research work, on the application of a fully automated theorem prover together with an automatic counter-example generator in the investigation of a class of algebraic structures. We…
A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic. The proofs of soundness and completeness are constructive and…
In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation. A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripke-style semantic founded…
System I is a proof language for a fragment of propositional logic where isomorphic propositions, such as $A\wedge B$ and $B\wedge A$, or $A\Rightarrow(B\wedge C)$ and $(A\Rightarrow B)\wedge(A\Rightarrow C)$ are made equal. System I enjoys…
We extend dynamic logic of propositional assignments by adding an operator of parallel composition that is inspired by separation logics. We provide an axiomatisation via reduction axioms, thereby establishing decidability. We also prove…
Structures based on polarities have been used to provide relational semantics for propositional logics that are modelled algebraically by non-distributive lattices with additional operators. This article develops a first order notion of…
Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with…
We introduce a natural Turing-complete extension of first-order logic FO. The extension adds two novel features to FO. The first one of these is the capacity to add new points to models and new tuples to relations. The second one is the…
A detailed exposition of foundations of a logic-algebraic model for reasoning with knowledge bases specified by propositional (Boolean) logic is presented. The model is conceived from the logical translation of usual derivatives on…
Formally specifying, let alone verifying, properties of systems involving multiple programming languages is inherently challenging. We introduce Heterogeneous Dynamic Logic (HDL), a framework for combining reasoning principles from distinct…
Game semantics extends the Curry-Howard isomorphism to a three-way correspondence: proofs, programs, strategies. But the universe of strategies goes beyond intuitionistic logics and lambda calculus, to capture stateful programs. In this…
Not all logical systems can be captured using algebras. We see this in classical logic (formalized by Boolean algebras) and many-valued logics (like Lukasiewicz logic with MV-algebras). Even quantum mechanics, initially formalized with…
Non-additive uncertainty theories, typically possibility theory, belief functions and imprecise probabilities share a common feature with modal logic: the duality properties between possibility and necessity measures, belief and…
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…