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We define a Kripke semantics for Intuitionistic Higher-Order Logic with constraints formulated within Church's Theory of Types via the addition of a new constraint base type. We then define an executable fragment, hoHH ( C ) , of the logic: a higher-order logic programming language with typed λ -abstraction, implication and universal quantification in goals and constraints, and give a modified model theory for this fragment. Both formal systems are shown sound and complete for their respective semantics. We also solve the impredicativity problem in λ Prolog semantics, namely how to give a definition of truth without appealing to induction on subformula structure. In the last section we give a simple semantics-based conservative extension proof that the language hoHH ( C ) satisfies a uniformity property along the lines of [39] . [ABSTRACT FROM AUTHOR]

We present a sound and complete graph calculus for modalities. This calculus is a general framework for expressing modal formulas and frame properties, with a rich repertoire of relations, and reasoning about them in a uniform manner. The calculus employs graphical interpretations of logical operators and builds graphical objects that represent conditions on Kripke structures. [ABSTRACT FROM AUTHOR]

In this work, we combine features from Justification Logics and Logics of Plausibility-Based Beliefs to build a logic for Multi-Agent Systems where each agent can explicitly state his justification for believing in a given sentence. Our logic is a normal modal logic based on the standard Kripke semantics, where we provide a semantic definition for the evidence terms and define the notion of plausible evidence for an agent, based on plausibility relations in the model. As we deal with beliefs, justifications can be faulty and unreliable. In our logic, agents can disagree not only over whether a sentence is true or false, but also on whether some evidence is a valid justification for a sentence or not. After defining our logic and its semantics, we provide a strongly complete axiomatic system for it, show that it has the finite model property, analyze the complexity of its Model-Checking Problem and show that its Satisfiability Problem has the same complexity as the one from basic modal logics. Thus, this logic seems to be a good first step for the development of a dynamic logic that can model the processes of argumentation and debate in Multi-Agent Systems. [ABSTRACT FROM AUTHOR]

By means of a countermodel we show that the homotopy equivalence of the fibers of a Kan fibration over a connected base cannot be proved constructively. [ABSTRACT FROM AUTHOR]