Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.

Recently an expansion of Ł∏ logic with fixed points has been considered [23]. In the present work we study the algebraic semantics of this logic, namely μŁΠ algebras, from algebraic, model theoretic and computational standpoints. We provide a characterisation of free μŁΠ algebras as a family of particular functions from [0,1]n to [0,1]. We show that the first-order theory of linearly ordered μŁΠ algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered Ł∏ algebras. (...) Furthermore, we give a functional representation of any Ł∏ algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of μŁΠ algebras is in PSPACE. (shrink)

Łu logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of Łu logic by adding a new connective which expresses multiplication. The resulting logic, PŁ, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of PŁ logic is introduced and developed too.

An explicit categorical equivalence is defined between a proper subvariety of the class of \-algebras, as defined by Di Nola and Dvurečenskij, to be called \-algebras, and the category of semi-low \-rings. This categorical representation is done using the prime spectrum of the \-algebras, through the equivalence between \-algebras and \-groups established by Mundici, from the perspective of the Dubuc–Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low \-rings associated to Boolean algebras are (...) characterized. (shrink)

Abstract.This paper is devoted to a logical and algebraic treatment of conditional probability. The main ideas are the use of non-standard probabilities and of some kind of standard part function in order to deal with the case where the conditioning event has probability zero, and the use of a many-valued modal logic in order to deal probability of an event φ as the truth value of the sentence φ is probable, along the lines of Hájek’s book [H98] and of [EGH96]. (...) To this purpose, we introduce a probabilistic many-valued logic, called FP(SŁΠ), which is sound and complete with respect a class of structures having a non-standard extension [0,1]⋆ of [0,1] as set of truth values. We also prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over FP(SŁΠ) whose proper axioms reflect the assessment itself. (shrink)

A logic for classical conditional events was investigated by Dubois and Prade. In their approach, the truth value of a conditional event may be undetermined. In this paper we extend the treatment to many-valued events. Then we support the thesis that probability over partially undetermined events is a conditional probability, and we interpret it in terms of bets in the style of de Finetti. Finally, we show that the whole investigation can be carried out in a logical and algebraic setting, (...) and we find a logical characterization of coherence for assessments of partially undetermined events. (shrink)

An algebraic setting for the validity of Pavelka style completeness for some natural expansions of Łukasiewicz logic by new connectives and rational constants is given. This algebraic approach is based on the fact that the standard MV-algebra on the real segment [0, 1] is an injective MV-algebra. In particular the logics associated with MV-algebras with product and with divisible MV-algebras are considered.

We study a system, μŁΠ, obtained by an expansion of ŁΠ logic with fixed points connectives. The first main result of the paper is that μŁΠ is standard complete, i.e., complete with regard to the unit interval of real numbers endowed with a suitable structure. We also prove that the class of algebras which forms algebraic semantics for this logic is generated, as a variety, by its linearly ordered members and that they are precisely the interval algebras of real closed (...) fields. This correspondence is extended to a categorical equivalence between the whole category of those algebras and another category naturally arising from real closed fields. Finally, we show that this logic enjoys implicative interpolation. (shrink)

In many-valued logics with the unit interval as the set of truth values, from the standard negation and the product (or, more generally, from any strict Frank t-norm) all measurable logical functions can be derived, provided that also operations with countable arity are allowed. The question remained open whether there are other t-norms with this property or whether all strict t-norms possess this property. We give a full solution to this problem (in the case of strict t-norms), together with convenient (...) sufficient conditions. We list several families of strict t-norms having this property and provide also counterexamples (the Hamacher product is one of them). Finally, we discuss the consequences of these results for the characterization of tribes based on strict t-norms. (shrink)

It is a well-known fact that MV-algebras, the algebraic counterpart of Łukasiewicz logic, correspond to a certain type of partial algebras: lattice-ordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓ-groups in a straightforward manner. In this paper, we consider several logics differing from Łukasiewicz logics in that they contain further connectives: the PŁ-, PŁ'-, PŁ'△-, and ŁΠ-logics. For all their algebraic counterparts, we (...) characterise the corresponding type of partial algebras. We moreover consider the representing f-rings. All in all, we get three-fold correspondences: the total algebras - the partial algebras - the representing rings. (shrink)

In this paper we solve the problem how to axiomatize a system of quantum computational gates known as the Poincaré irreversible quantum computational system. A Hilbert-style calculus is introduced obtaining a strong completeness theorem.

ABSTRACTWe introduce the variety of Many-Valued-Weak rigs. We provide an axiomatisation and establish, in this context, basic properties about ideals, homomorphisms, quotients and radicals. This new class contains the class of product MV-algebras presented by Di Nola and Dvurečenskij in 2001 and by Montagna in 2005. The main result is the compactness of the prime spectrum of this new class, endowed with the co-Zariski topology as defined by Dubuc and Poveda in 2010.