92 found
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  1.  13
    Distinguished Algebraic Semantics for T -Norm Based Fuzzy Logics: Methods and Algebraic Equivalencies.Petr Cintula, Francesc Esteva, Joan Gispert, Lluís Godo, Franco Montagna & Carles Noguera - 2009 - Annals of Pure and Applied Logic 160 (1):53-81.
    This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit (...)
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  2.  52
    Substructural Fuzzy Logics.George Metcalfe & Franco Montagna - 2007 - Journal of Symbolic Logic 72 (3):834 - 864.
    Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are (...)
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  3.  23
    A Proof of Standard Completeness for Esteva and Godo's Logic MTL.Sándor Jenei & Franco Montagna - 2002 - Studia Logica 70 (2):183-192.
    In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo''s logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.
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  4.  18
    On the Standard and Rational Completeness of Some Axiomatic Extensions of the Monoidal T-Norm Logic.Francesc Esteva, Joan Gispert, Lluís Godo & Franco Montagna - 2002 - Studia Logica 71 (2):199 - 226.
    The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...)
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  5.  8
    A Categorical Equivalence for Product Algebras.Franco Montagna & Sara Ugolini - 2015 - Studia Logica 103 (2):345-373.
    In this paper we provide a categorical equivalence for the category \ of product algebras, with morphisms the homomorphisms. The equivalence is shown with respect to a category whose objects are triplets consisting of a Boolean algebra B, a cancellative hoop C and a map \ from B × C into C satisfying suitable properties. To every product algebra P, the equivalence associates the triplet consisting of the maximum boolean subalgebra B, the maximum cancellative subhoop C, of P, and the (...)
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  6. Provability in Finite Subtheories of Pa and Relative Interpretability: A Modal Investigation.Franco Montagna - 1987 - Journal of Symbolic Logic 52 (2):494-511.
  7.  19
    Three Complexity Problems in Quantified Fuzzy Logic.Franco Montagna - 2001 - Studia Logica 68 (1):143-152.
    We prove that the sets of standard tautologies of predicate Product Logic and of predicate Basic Logic, as well as the set of standard-satisfiable formulas of predicate Basic Logic are not arithmetical, thus finding a rather satisfactory solution to three problems proposed by Hájek in [H01].
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  8.  3
    Interpolation and Beth’s Property in Propositional Many-Valued Logics: A Semantic Investigation.Franco Montagna - 2006 - Annals of Pure and Applied Logic 141 (1):148-179.
    In this paper we give a rather detailed algebraic investigation of interpolation and Beth’s property in propositional many-valued logics extending Hájek’s Basic Logic [P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, 1998], and we connect such properties with amalgamation and strong amalgamation in the corresponding varieties of algebras. It turns out that, while the most interesting extensions of in the language of have deductive interpolation, very few of them have Beth’s property or Craig interpolation. Thus in the last part of the (...)
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  9.  9
    The LPi and LPifrac{1}{2} Logics: Two Complete Fuzzy Systems Joining Łukasiewicz and Product Logics. [REVIEW]Francesc Esteva, Lluís Godo & Franco Montagna - 2001 - Archive for Mathematical Logic 40 (1):39-67.
    In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called $L\Pi$ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from $L \Pi$ by the adding of a constant symbol and of a defining axiom for $\frac{1}{2}$ , called $L \Pi\frac{1}{2}$ . We show that $L \Pi \frac{1}{2}$ contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and Titani's (...)
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  10.  2
    On the Standard and Rational Completeness of Some Axiomatic Extensions of the Monoidal T-Norm Logic.Francesc Esteva, Joan Gispert, Lluís Godo & Franco Montagna - 2002 - Studia Logica 71 (2):199-226.
    The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0, 1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness of (...)
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  11.  25
    Supersound Many-Valued Logics and Dedekind-MacNeille Completions.Matteo Bianchi & Franco Montagna - 2009 - Archive for Mathematical Logic 48 (8):719-736.
    In Hájek et al. (J Symb Logic 65(2):669–682, 2000) the authors introduce the concept of supersound logic, proving that first-order Gödel logic enjoys this property, whilst first-order Łukasiewicz and product logics do not; in Hájek and Shepherdson (Ann Pure Appl Logic 109(1–2):65–69, 2001) this result is improved showing that, among the logics given by continuous t-norms, Gödel logic is the only one that is supersound. In this paper we will generalize the previous results. Two conditions will be presented: the first (...)
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  12.  14
    On the Predicate Logics of Continuous T-Norm BL-Algebras.Franco Montagna - 2004 - Archive for Mathematical Logic 44 (1):97-114.
  13.  40
    Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL∀.Franco Montagna & Hiroakira Ono - 2002 - Studia Logica 71 (2):227-245.
    The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the (...)
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  14. Some Properties of the Syntactic P-Recursion Categories Generated by Consistent, Recursively Enumerable Extensions of Peano Arithmetic.Robert A. di Paola & Franco Montagna - 1991 - Journal of Symbolic Logic 56 (2):643-660.
  15.  3
    A Note on the First‐Order Logic of Complete BL‐Chains.Petr Hájek & Franco Montagna - 2008 - Mathematical Logic Quarterly 54 (4):435-446.
    In [10] it is claimed that the set of predicate tautologies of all complete BL-chains and the set of all standard tautologies coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL-chain B satisfies all standard BL-tautologies iff for any transfinite sequence of elements of B, the condition ∧i ∈ I = 2 holds in B.
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  16.  8
    The Predicate Modal Logic of Provability.Franco Montagna - 1984 - Notre Dame Journal of Formal Logic 25 (2):179-189.
  17.  12
    An Algebraic Approach to Propositional Fuzzy Logic.Franco Montagna - 2000 - Journal of Logic, Language and Information 9 (1):91-124.
    We investigate the variety corresponding to a logic, which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative axiomatization of such variety. We also investigate the variety, called the variety of algebras, corresponding to the logic obtained from by the adding of a constant and of a defining axiom for one half. We also connect algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and prove that (...)
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  18.  6
    Kripke-Style Semantics for Many-Valued Logics.Franco Montagna & Lorenzo Sacchetti - 2003 - Mathematical Logic Quarterly 49 (6):629.
    This paper deals with Kripke-style semantics for many-valued logics. We introduce various types of Kripke semantics, and we connect them with algebraic semantics. As for modal logics, we relate the axioms of logics extending MTL to properties of the Kripke frames in which they are valid. We show that in the propositional case most logics are complete but not strongly complete with respect to the corresponding class of complete Kripke frames, whereas in the predicate case there are important many-valued logics (...)
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  19.  32
    On the Proof of Solovay's Theorem.Dick Jongh, Marc Jumelet & Franco Montagna - 1991 - Studia Logica 50 (1):51 - 69.
    Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in (...)
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  20.  30
    A Logical and Algebraic Treatment of Conditional Probability.Tommaso Flaminio & Franco Montagna - 2004 - Archive for Mathematical Logic 44 (2):245-262.
  21.  13
    Relatively Precomplete Numerations and Arithmetic.Franco Montagna - 1982 - Journal of Philosophical Logic 11 (4):419 - 430.
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  22.  5
    On the Proof of Solovay's Theorem.Dick de Jongh, Marc Jumelet & Franco Montagna - 1991 - Studia Logica 50 (1):51-69.
    Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular $\text{I}\Delta _{0}+\text{EXP}$ . The method is adapted (...)
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  23.  14
    Equational Characterization of the Subvarieties of BL Generated by T-Norm Algebras.Fransesc Esteva, Lluís Godo & Franco Montagna - 2004 - Studia Logica 76 (2):161 - 200.
    In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, the (...)
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  24.  5
    A Minimal Predicative Set Theory.Franco Montagna & Antonella Mancini - 1994 - Notre Dame Journal of Formal Logic 35 (2):186-203.
  25.  5
    Generic Generalized Rosser Fixed Points.Dick H. J. de Jongh & Franco Montagna - 1987 - Studia Logica 46 (2):193-203.
    To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical completeness theorem with respect to PA is obtained for LR.
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  26.  19
    N-Contractive BL-Logics.Matteo Bianchi & Franco Montagna - 2011 - Archive for Mathematical Logic 50 (3-4):257-285.
    In the field of many-valued logics, Hájek’s Basic Logic BL was introduced in Hájek (Metamathematics of fuzzy logic, trends in logic. Kluwer Academic Publishers, Berlin, 1998). In this paper we will study four families of n-contractive (i.e. that satisfy the axiom ${\phi^n\rightarrow\phi^{n+1}}$ , for some ${n\in\mathbb{N}^+}$ ) axiomatic extensions of BL and their corresponding varieties: BL n , SBL n , BL n and SBL n . Concerning BL n we have that every BL n -chain is isomorphic to an (...)
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  27.  14
    Partially Undetermined Many-Valued Events and Their Conditional Probability.Franco Montagna - 2012 - Journal of Philosophical Logic 41 (3):563-593.
    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, (...)
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  28.  4
    The Logic of Π1-Conservativity.Petr Hájek & Franco Montagna - 1990 - Archive for Mathematical Logic 30 (2):113-123.
    We show that the modal prepositional logicILM (interpretability logic with Montagna's principle), which has been shown sound and complete as the interpretability logic of Peano arithmetic PA (by Berarducci and Savrukov), is sound and complete as the logic ofπ 1-conservativity over eachbE 1-sound axiomatized theory containingI⌆ 1 (PA with induction restricted tobE 1-formulas). Furthermore, we extend this result to a systemILMR obtained fromILM by adding witness comparisons in the style of Guaspari's and Solovay's logicR (this will be done in a (...)
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  29.  8
    Δ-Core Fuzzy Logics with Propositional Quantifiers, Quantifier Elimination and Uniform Craig Interpolation.Franco Montagna - 2012 - Studia Logica 100 (1-2):289-317.
    In this paper we investigate the connections between quantifier elimination, decidability and Uniform Craig Interpolation in Δ-core fuzzy logics added with propositional quantifiers. As a consequence, we are able to prove that several propositional fuzzy logics have a conservative extension which is a Δ-core fuzzy logic and has Uniform Craig Interpolation.
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  30.  12
    Completeness with Respect to a Chain and Universal Models in Fuzzy Logic.Franco Montagna - 2011 - Archive for Mathematical Logic 50 (1-2):161-183.
    In this paper we investigate fuzzy propositional and first order logics which are complete or strongly complete with respect to a single chain, and we relate this properties with the existence of a universal chain for the logic.
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  31.  20
    On the Algebraization of a Feferman's Predicate.Franco Montagna - 1978 - Studia Logica 37 (3):221 - 236.
    This paper is devoted to the algebraization of an arithmetical predicate introduced by S. Feferman. To this purpose we investigate the equational class of Boolean algebras enriched with an operation (g=rtail), which translates such predicate, and an operation τ, which translates the usual predicate Theor. We deduce from the identities of this equational class some properties of (g=rtail) and some ties between (g=rtail) and τ; among these properties, let us point out a fixed-point theorem for a sufficiently large class of (...)
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  32.  11
    Quantifier Elimination and Other Model-Theoretic Properties of BL-Algebras.Tommaso Cortonesi, Enrico Marchioni & Franco Montagna - 2011 - Notre Dame Journal of Formal Logic 52 (4):339-379.
    This work presents a model-theoretic approach to the study of first-order theories of classes of BL-chains. Among other facts, we present several classes of BL-algebras, generating the whole variety of BL-algebras, whose first-order theory has quantifier elimination. Model-completeness and decision problems are also investigated. Then we investigate classes of BL-algebras having (or not having) the amalgamation property or the joint embedding property and we relate the above properties to the existence of ultrahomogeneous models.
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  33.  8
    The Logic ofII 1-Conservativity Continued.Petr Hájek & Franco Montagna - 1992 - Archive for Mathematical Logic 32 (1):57-63.
    It is shown that the propositional modal logic IRM (interpretability logic with Montagna's principle and with witness comparisons in the style of Guaspari's and Solovay's logicR) is sound and complete as the logic ofII 1-conservativity over each∑ 1-sound axiomatized theory containingI∑ 1. The exact statement of the result uses the notion of standard proof predicate. This paper is an immediate continuation of our paper [HM]. Knowledge of [HM] is presupposed. We define a modal logic, called IRM, which includes both ILM (...)
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  34.  6
    Complexity of T-Tautologies.Matthias Baaz, Petr Hájek, Franco Montagna & Helmut Veith - 2001 - Annals of Pure and Applied Logic 113 (1-3):3-11.
    A t-tautology is a propositional formula which is a tautology in all fuzzy logics defined by continuous triangular norms. In this paper we show that the problem of recognizing t-tautologies is coNP complete, and thus decidable.
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  35.  17
    Provable Fixed Points.Dick De Jongh & Franco Montagna - 1988 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 34 (3):229-250.
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  36.  18
    The Undecidability of the First-Order Theory of Diagonalizable Algebras.Franco Montagna - 1980 - Studia Logica 39 (4):355 - 359.
    The undecidability of the first-order theory of diagonalizable algebras is shown here.
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  37.  77
    Learning to Coordinate; a Recursion Theoretic Perspective.Franco Montagna & Daniel Osherson - 1999 - Synthese 118 (3):363-382.
    We consider two players each of whom attempts to predict the behavior of the other, using no more than the history of earlier predictions. Behaviors are limited to a pair of options, conventionally denoted by 0, 1. Such players face the problem of learning to coordinate choices. The present paper formulates their situation recursion theoretically, and investigates the prospects for success. A pair of players build up a matrix with two rows and infinitely many columns, and are said to “learn” (...)
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  38.  18
    Much Shorter Proofs: A Bimodal Investigation.Alessandra Carbone & Franco Montagna - 1990 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (1):47-66.
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  39.  22
    Interpretations of the First-Order Theory of Diagonalizable Algebras in Peano Arithmetic.Franco Montagna - 1980 - Studia Logica 39 (4):347 - 354.
    For every sequence |p n } n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula 0 A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, 0 A is a theorem ofPA.
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  40.  17
    Some Properties of the Syntactic P-Recursion Categories Generated by Consistent, Recursively Enumerable Extensions of Peano Arithmetic.Robert A. Paola & Franco Montagna - 1991 - Journal of Symbolic Logic 56 (2):643 - 660.
  41.  15
    Much Shorter Proofs.Dick de Jongh & Franco Montagna - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (3):247-260.
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  42.  8
    Provable Fixed Points.Dick De Jongh & Franco Montagna - 1988 - Mathematical Logic Quarterly 34 (3):229-250.
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  43.  39
    On First-Order Theories with Provability Operator.Sergei Artëmov & Franco Montagna - 1994 - Journal of Symbolic Logic 59 (4):1139-1153.
    In this paper the modal operator "x is provable in Peano Arithmetic" is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.
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  44.  22
    Archimedean Classes in Integral Commutative Residuated Chains.Rostislav Horčík & Franco Montagna - 2009 - Mathematical Logic Quarterly 55 (3):320-336.
    This paper investigates a quasi-variety of representable integral commutative residuated lattices axiomatized by the quasi-identity resulting from the well-known Wajsberg identity → q ≤ → p if it is written as a quasi-identity, i. e., → q ≈ 1 ⇒ → p ≈ 1. We prove that this quasi-identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi-variety is in fact a variety and provide an axiomatization. The obtained results shed some light (...)
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  45.  15
    For Everyn, Then-Freely Generated Algebra is Not Functionally Free in the Equational Class of Diagonalizable Algebras.Franco Montagna - 1975 - Studia Logica 34 (4):315 - 319.
    This paper is devoted to the algebraization of theories in which, as in Peano arithmetic, there is a formula, Theor(x), numerating the set of theorems, and satisfying Hilbert-Bernays derivability conditions. In particular, we study the diagonalizable algebras, which are been introduced by R. Magari in [6], [7]. We prove that for every natural number n, the n-freely generated algebra $\germ{J}_{n}$ is not functionally free in the equational class of diagonalizable algebras; we also prove that the diagonalizable algebra of Peano arithmetic (...)
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  46.  9
    Logic and Probabilistic Systems.Franco Montagna, Giulia Simi & Andrea Sorbi - 1996 - Archive for Mathematical Logic 35 (4):225-261.
    Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set ℐ of the sentences that get limit probability 1 is a Π3—theory, in fact ℐ can be a Π3—complete set. We prove incompleteness results for this setting, by showing for instance that for every k > 0 there are true Π3—sentences that get limit probability less than 1/2k. No set (...)
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  47.  17
    Rosser Orderings in Bimodal Logics.Alessandra Carbone & Franco Montagna - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (4):343-358.
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  48.  16
    Iterated Extensional Rosser's Fixed Points and Hyperhyperdiagonalizable Algebras.Franco Montagna - 1987 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (4):293-303.
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  49.  15
    A Completeness Result for Fixed-Point Algebras.Franco Montagna - 1984 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 30 (32-34):525-532.
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  50.  9
    Much Shorter Proofs: A Bimodal Investigation.Alessandra Carbone & Franco Montagna - 1990 - Mathematical Logic Quarterly 36 (1):47-66.
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