Stefano Baratella

An approach to infinitary temporal proof theory

Abstract.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.

A proof-theoretic investigation of a logic of positions

Abstract We introduce an extension of natural deduction that is suitable for dealing with modal operators and induction. We provide a proof reduction system and we prove a strong normalization theorem for an intuitionistic calculus. As a consequence we obtain a purely syntactic proof of consistency. We also present a classical calculus and we relate provability in the two calculi by means of an adequate formula translation.

A theory of sets with the negation of the axiom of infinity

In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in view of a non-standard development. MSC: 03E30, 03E35.

The Real truth

We study a real valued propositional logic with unbounded positive and negative truth values that we call -valued logic. Such a logic is semantically equivalent to continuous propositional logic, with a different choice of connectives. After presenting the deduction machinery and the semantics of -valued logic, we prove a completeness theorem for finite theories. Then we define unital and Archimedean theories, in accordance with the theory of Riesz spaces. In the unital setting, we prove the equivalence of consistency and satisfiability and an approximated completeness theorem similar to the one that holds for continuous propositional logic. Eventually, among unital theories, we characterize Archimedean theories as those for which strong completeness holds. We also point out that -valued logic provides alternative calculi for Łukasiewicz logic and for propositional continuous logic.

The theory of {vec Z}C(2)^2-lattices is decidable

Abstract. For arbitrary finite group

Continuous propositional modal logic

G

Non Standard Regular Finite Set Theory

and countable Dedekind domain

A note on infinitary continuous logic

R

An infinitary variant of Metric Temporal Logic over dense time domains

such that the residue field

Consequences of neocompact quantifier elimination

R/P