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We introduce a logic for a class of probabilistic Kripke structures that we call type structures, as they are inspired by Harsanyi type spaces. The latter structures are used in theoretical economics and game theory. A strong completeness theorem for an associated infinitary propositional logic with probabilistic operators was proved by Meier. By simplifying Meier’s proof, we prove that our logic is strongly complete with respect to the class of type structures. In order to do that, we define a canonical model (in the sense of modal logics), which turns out to be a terminal object in a suitable category. Furthermore, we extend some standard model-theoretic constructions to type structures and we prove analogues of first-order results for those constructions.

We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous...

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

We introduce a natural deduction system for the until-free subsystem of the branching time logic Although we work with labelled formulas, our system differs conceptually from the usual labelled deduction systems because we have no relational formulas. Moreover, no deduction rule embodies semantic features such as properties of accessibility relation or similar algebraic properties. We provide a suitable semantics for our system and prove that it is sound and weakly complete with respect to such semantics.

We study a predicate extension of an unbounded real valued propositional logic that has been recently introduced. The latter, in turn, can be regarded as an extension of both the abelian logic and of the propositional continuous logic. Among other results, we prove that our predicate extension satisfies the property of weak completeness (the equivalence between satisfiability and consistency) and, under an additional assumption on the set of premisses, the property of strong completeness (the equivalence between logical consequence and provability). Eventually we discuss some topological properties of the space of types in our logic.

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

For arbitrary finite group $G$ and countable Dedekind domain $R$ such that the residue field $R/P$ is finite for every maximal $R$ -ideal $P$ , we show that the localizations at every maximal ideal of two $RG$ -lattices are isomorphic if and only if the two lattices satisfy the same first order sentences. Then we investigate generalizations of the above results to arbitrary $R$ -torsion-free $RG$ -modules and we apply the previous results to show the decidability of the theory of ${\vec Z}C(2)^2$ -lattices. Eventually, we show that ${\vec Z} [i] C(2)^2$ -lattices have undecidable theory.

We describe the constructive content of proofs in a fragment of propositional Infinitary Linear Logic in terms of strategies for a suitable class of games. Such strategies interpret linear proofs as parallel algorithms as long as the asymmetry of the connectives ? and ! allows it.

Aim of this work is to investigate from a proof-theoretic viewpoint a propositional and a predicate sequent calculus with an ω–type schema of inference that naturally interpret the propositional and the predicate until–free fragments of Linear Time Logic LTL respectively. The two calculi are based on a natural extension of ordinary sequents and of standard modal rules. We examine the pure propositional case (no extralogical axioms), the propositional and the first order predicate cases (both with a possibly infinite set of extralogical axioms). For each system we provide a syntactic proof of cut elimination and a proof of completeness.