Pierangelo Miglioli

Some results on intermediate constructive logics.

Some techniques for the study of intermediate constructive logics are illustrated. In particular a general characterization is given of maximal constructive logics from which a new proof of the maximality of MV (Med- vedevs logic of finite problems ) can be obtained. Some semantical notions are also introduced, allowing a new characterization of MV, from which a new proof of a conjecture of Friedmans and a new family of principles valid in MV can be extracted.

An improved refutation system for intuitionistic predicate logic

In this paper a refutation calculus for intuitionistic predicate logic is presented where the necessity of duplicating formulas to which rules are applied is analyzed. In line with the semantics of intuitionistic logic in terms of Kripke models a new signFCbeside the SignsT andF is added which reduces the size of the proofs and the involved nondeterminism. The resulting calculus is proved to be correct and complete. An extension of it for Kuroda logic is given.

Counting the maximal intermediate constructive logics

A proof is given that the set of maximal intermediate propositional logics with the disjunction property and the set of maximal intermediate predicate logics with the disjunction property and the explicit definability property have the power of continuum. To prove our results, we introduce various notions which might be interesting by themselves. In particular, we illustrate a method to generate wide sets of pairwise “constructively incompatible constructive logics”. We use a notion of “semiconstructive” logic and define wide sets of “constructive” logics by representing the “constructive” logics as “limits” of decreasing sequences of “semiconstructive” logics. Also, we introduce some generalizations of the usual filtration techniques for propositional logics. For instance, “fitrations over rank formulas” are used to show that any two different logics belonging to a suitable uncountable set of “constructive” logics are “constructively incompatible”.

Refutation Systems for Propositional Modal Logics

Now we are working on extensions to first order modal logics and to modal logics with intuitionistic basis. A comparison with the work of Wallen [Wal] is planned.

A method to single out maximal propositional logics with the disjunction property I

This is the second part of a paper devoted to the study of the maximal intermediate propositional logics with the disjunction property (we will simply call maximal constructive logics), whose first part has appeared in this journal with the title “A method to single out maximal propositional logics with the disjunction property I”. In the first part we have explained the general results upon which a method to single out maximal constructive logics is based and have illustrated such a method by exhibiting the Kripke semantics of maximal constructive logics extending the logic ST of Scott, for which, in turn, a semantical characterization in terms of Kripke frames has been given. In the present part we complete the illustration of the method of the first part, having in mind some aspects which might be interesting for a classification of the maximal constructive logics, and an application of the heuristic content of the method to detect the nonmaximality of apparently maximal constructive logics. Thus, on the one hand we introduce the logic AST (“anti” ST), which is compared with ST and is seen as a logic “alternative” (or even “opposite”) to it, in a sense which will be precisely explained. We provide a Kripke semantics for AST and (without exhibiting them) show that (near the ones including ST and the ones including AST) there are maximal constructive logics which neither are extensions of ST nor are extensions of AST. Finally, we give a further application of the results of the first part by exhibiting the Kripke semantics of a maximal constructive logic extending AST. On the other hand, we compare the maximal constructive logics presented in both parts of the paper with a constructive logic introduced by Maksimova (1986), which has been conjectured to be maximal by Chagrov and Zacharyashchev (199 1); from this comparison a disproof of the conjecture arises.

Generalized Tableau Systems for Intemediate Propositional Logics

Given an intermediate propositional logic L (obtained by adding to intuitionistic logic INT a single axiom-scheme), a pseudo tableau system for L can be given starting from any intuitionistic tableau system and adding a rule which allows to insert in any line of a proof table suitable T-signed instances of the axiom-scheme. In this paper we study some sufficient conditions from which, given a well formed formula H, the search for these instances can be restricted to a suitable finite set of formulae related to H. We illustrate our techniques by means of some known logics, namely, the logic D of Dummett, the logics PR k (k≥1) of Nagata, the logics FIN m (m≥1), the logics G n (n≥1) of Gabbay and de Jongh, and the logic KP of Kreisel and Putnam

Constructive Theories with Abstract Data Types for Program Synthesis

The research explained in this paper originates from program synthesis in the frame of intuitionistic logic [6] and has been furtherly developed as a study involving, on the one hand, constructive proofs as programs [12], on the other hand the possibility of providing axiomatizations of mathematical structures (abstract data types) compatible with constructive logical principles [3].

An infinite class of maximal intermediate propositional logics with the disjunction property

SummaryInfinitely many intermediate propositional logics with the disjunction property are defined, each logic being characterized both in terms of a finite axiomatization and in terms of a Kripke semantics with the finite model property. The completeness theorems are used to prove that any two logics are constructively incompatible. As a consequence, one deduces that there are infinitely many maximal intermediate propositional logics with the disjunction property.

Program Specification and Synthesis in Constructive Formal Systems

Constructive mathematics has been proposed by many authors as a theoretical basis for program synthesis, and various implementations of this idea have been developed. However, the main problem in implementation is how to build a real environment for software development. In this paper, we present the main features of a logical system we are studying which provides specification tools and a deductive system for deriving programs from their specifications. Our aim is to use this system as a starting point for a real programming environment.

On Canonicity and Strong Completeness Conditions in Intermediate Propositional Logics

By using algebraic-categorical tools, we establish four criteria in order to disprove canonicity, strong completeness, w-canonicity and strong w-completeness, respectively, of an intermediate propositional logic. We then apply the second criterion in order to get the following result: all the logics defined by extra-intuitionistic one-variable schemata, except four of them, are not strongly complete. We also apply the fourth criterion in order to prove that the Gabbay-de Jongh logic D1 is not strongly w-complete.