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  1. Sándor Jenei & Franco Montagna (forthcoming). A Classification of Certain Group-Like FL_e-Chains. Synthese:1-27.
    Classification of certain group-like FL $_e$ -chains is given: We define absorbent-continuity of FL $_e$ -algebras, along with the notion of subreal chains, and classify absorbent-continuous, group-like FL $_e$ -algebras over subreal chains: The algebra is determined by its negative cone, and the negative cone can only be chosen from a certain subclass of BL-chains, namely, one with components which are either cancellative (that is, those components are negative cones of totally ordered Abelian groups) or two-element MV-algebras, and with no (...)
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  2. Franco Montagna & Duccio Pianigiani (2013). A Short Note on Essentially Σ1 Sentences. Logica Universalis 7 (1):103-111.
    Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ1 (i.e., it is Σ1 under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form ${\square{B}}$ . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ1 formulas of languages including witness comparisons (...)
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  3. Franco Montagna (2012). Δ-Core Fuzzy Logics with Propositional Quantifiers, Quantifier Elimination and Uniform Craig Interpolation. 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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  4. Franco Montagna (2012). Partially Undetermined Many-Valued Events and Their Conditional Probability. 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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  5. Matteo Bianchi & Franco Montagna (2011). N-Contractive BL-Logics. 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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  6. Franco Montagna (2011). Completeness with Respect to a Chain and Universal Models in Fuzzy Logic. 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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  7. Tommaso Cortonesi, Enrico Marchioni & Franco Montagna (2010). Quantifier Elimination and Other Model-Theoretic Properties of BL-Algebras. 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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  8. Matteo Bianchi & Franco Montagna (2009). Supersound Many-Valued Logics and Dedekind-MacNeille Completions. 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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  9. Petr Cintula, Francesc Esteva, Joan Gispert, Lluís Godo, Franco Montagna & Carles Noguera (2009). Distinguished Algebraic Semantics for T-Norm Based Fuzzy Logics: Methods and Algebraic Equivalencies. Annals of Pure and Applied Logic 160 (1):53-81.
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  10. Rostislav Horčík & Franco Montagna (2009). Archimedean Classes in Integral Commutative Residuated Chains. Mathematical Logic Quarterly 55 (3):320-336.
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  11. Petr Hájek & Franco Montagna (2008). A Note on the First‐Order Logic of Complete BL‐Chains. Mathematical Logic Quarterly 54 (4):435-446.
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  12. George Metcalfe & Franco Montagna (2007). Substructural Fuzzy Logics. 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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  13. Franco Montagna (2006). Interpolation and Beth's Property in Propositional Many-Valued Logics: A Semantic Investigation. Annals of Pure and Applied Logic 141 (1):148-179.
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  14. Tommaso Flaminio & Franco Montagna (2005). A Logical and Algebraic Treatment of Conditional Probability. Archive for Mathematical Logic 44 (2):245-262.
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  15. Franco Montagna (2005). On the Predicate Logics of Continuous T-Norm BL-Algebras. Archive for Mathematical Logic 44 (1):97-114.
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  16. Fransesc Esteva, Lluís Godo & Franco Montagna (2004). Equational Characterization of the Subvarieties of BL Generated by T-Norm Algebras. 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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  17. Franco Montagna (2004). Francesc Esteva Lluıs Godo Franco Montagna. Studia Logica 76:155-194.
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  18. Francesc Esteva, Joan Gispert, Lluís Godo & Franco Montagna (2002). On the Standard and Rational Completeness of Some Axiomatic Extensions of the Monoidal T-Norm Logic. 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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  19. Sándor Jenei & Franco Montagna (2002). A Proof of Standard Completeness for Esteva and Godo's Logic MTL. 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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  20. Franco Montagna, G. Michele Pinna & B. P. Tiezzi (2002). Investigations on Fragments of First Order Branching Temporal Logic. Mathematical Logic Quarterly 48 (1):51-62.
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  21. Franco Montagna & Hiroakira Ono (2002). Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL∀. 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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  22. Franco Montagna, G. Michele Pinna & B. P. Tiezzi (2002). Investigations on Fragments of First Order Branching Temporal Logic. Mathematical Logic Quarterly 48 (1):51-62.
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  23. Matthias Baaz, Petr Hájek, Franco Montagna & Helmut Veith (2001). Complexity of T-Tautologies. Annals of Pure and Applied Logic 113 (1-3):3-11.
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  24. Francesc Esteva, Lluís Godo & Franco Montagna (2001). The LPi and LPifrac{1}{2} Logics: Two Complete Fuzzy Systems Joining Łukasiewicz and Product Logics. [REVIEW] 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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  25. Franco Montagna (2001). Three Complexity Problems in Quantified Fuzzy Logic. 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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  26. Franco Montagna (2000). An Algebraic Approach to Propositional Fuzzy Logic. Journal of Logic, Language and Information 9 (1):91-124.
    We investigate the variety corresponding to a logic (introduced in Esteva and Godo, 1998, and called there), which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative (and slightly simpler) 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 (...)
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  27. Franco Montagna (2000). Review: Petr Hájek, Lluis Godo, Francesc Esteva, A Complete Many-Valued Logic with Product-Conjunction. [REVIEW] Bulletin of Symbolic Logic 6 (3):346-347.
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  28. Duccio Luchi & Franco Montagna (1999). An Operational Logic of Proofs with Positive and Negative Information. Studia Logica 63 (1):7-25.
    The logic of proofs was introduced by Artemov in order to analize the formalization of the concept of proof rather than the concept of provability. In this context, some operations on proofs play a very important role. In this paper, we investigate some very natural operations, paying attention not only to positive information, but also to negative information (i.e. information saying that something cannot be a proof). We give a formalization for a fragment of such a logic of proofs, and (...)
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  29. Franco Montagna & Daniel Osherson (1999). Learning to Coordinate; a Recursion Theoretic Perspective. 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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  30. Franco Montagna & Giulia Simi (1999). Paradigms in Measure Theoretic Learning and in Informant Learning. Studia Logica 62 (2):243-268.
    We investigate many paradigms of identifications for classes of languages (namely: consistent learning, EX learning, learning with finitely many errors, behaviorally correct learning, and behaviorally correct learning with finitely many errors) in a measure-theoretic context, and we relate such paradigms to their analogues in learning on informants. Roughly speaking, the results say that most paradigms in measure-theoretic learning wrt some classes of distributions (called canonical) are equivalent to the corresponding paradigms for identification on informants.
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  31. Franco Montagna (1997). Review: Albert Visser, A Course on Bimodal Provability Logic. [REVIEW] Journal of Symbolic Logic 62 (2):686-687.
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  32. Franco Montagna (1996). An Algebraic Treatment of Quantifier-Free Systems of Arithmetic. Archive for Mathematical Logic 35 (4):209-224.
    By algebraic means, we give an equational axiomatization of the equational fragments of various systems of arithmetic. We also introduce a faithful semantics according to which, for every reasonable system T for arithmetic, there is a model where exactly the theorems of T are true.
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  33. Franco Montagna, Giulia Simi & Andrea Sorbi (1996). Logic and Probabilistic Systems. 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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  34. Sergei Artëmov & Franco Montagna (1994). On First-Order Theories with Provability Operator. 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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  35. Sandra Fontani, Franco Montagna & Andrea Sorbi (1994). A Note on Relative Efficiency of Axiom Systems. Mathematical Logic Quarterly 40 (2):261-272.
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  36. Franco Montagna & Antonella Mancini (1994). A Minimal Predicative Set Theory. Notre Dame Journal of Formal Logic 35 (2):186-203.
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  37. Franco Montagna (1993). Gleit Zachary and Goldfarb Warren. Characters and Fixed Points in Provability Logic. Notre Dame Journal of Formal Logic., Pp. 26–36. [REVIEW] Journal of Symbolic Logic 58 (2):715-715.
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  38. Franco Montagna (1993). Review: Frans Voorbraak, A Simplification of the Completeness Proofs for Guaspari and Solovay's R. [REVIEW] Journal of Symbolic Logic 58 (2):717-719.
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  39. Franco Montagna (1993). Review: Lisa Reidhaar-Olson, A New Proof of the Fixed-Point Theorem of Provability Logic. [REVIEW] Journal of Symbolic Logic 58 (2):714-715.
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  40. Franco Montagna (1993). Review: Zachary Gleit, Warren Goldfarb, Characters and Fixed Points in Provability Logic. [REVIEW] Journal of Symbolic Logic 58 (2):715-715.
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  41. Petr Hájek & Franco Montagna (1992). The Logic ofII 1-Conservativity Continued. 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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  42. Franco Montagna (1992). Polynomially and Superexponentially Shorter Proofs in Fragments of Arithmetic. Journal of Symbolic Logic 57 (3):844-863.
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  43. Dick de Jongh, Marc Jumelet & Franco Montagna (1991). On the Proof of Solovay's Theorem. Studia Logica 50 (1):51-69.
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  44. Dick de Jongh & Franco Montagna (1991). Rosser Orderings and Free Variables. Studia Logica 50 (1):71-80.
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  45. Robert A. di Paola & Franco Montagna (1991). Some Properties of the Syntactic P-Recursion Categories Generated by Consistent, Recursively Enumerable Extensions of Peano Arithmetic. Journal of Symbolic Logic 56 (2):643-660.
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  46. Dick Jongh, Marc Jumelet & Franco Montagna (1991). On the Proof of Solovay's Theorem. 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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  47. Dick Jongh & Franco Montagna (1991). Rosser Orderings and Free Variables. Studia Logica 50 (1):71 - 80.
    It is shown that for arithmetical interpretations that may include free variables it is not the Guaspari-Solovay system R that is arithmetically complete, but their system R –. This result is then applied to obtain the nonvalidity of some rules under arithmetical interpretations including free variables, and to show that some principles concerning Rosser orderings with free variables cannot be decided, even if one restricts oneself to usual proof predicates.
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  48. Robert A. Paola & Franco Montagna (1991). Some Properties of the Syntactic P-Recursion Categories Generated by Consistent, Recursively Enumerable Extensions of Peano Arithmetic. Journal of Symbolic Logic 56 (2):643 - 660.
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  49. Alessandra Carbone & Franco Montagna (1990). Much Shorter Proofs: A Bimodal Investigation. Mathematical Logic Quarterly 36 (1):47-66.
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  50. Petr Hájek & Franco Montagna (1990). The Logic of Π1-Conservativity. 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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