44 found
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  1.  47
    Averaging the Truth-Value in Łukasiewicz Logic.Daniele Mundici - 1995 - Studia Logica 55 (1):113 - 127.
    Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.
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  2.  22
    Coherence of de Finetti Coherence.Daniele Mundici - 2017 - Synthese 194 (10):4055-4063.
    We prove that de Finetti coherence is preserved under taking products of coherent books on two sets of independent events. This establishes a desirable closure property of coherence: were it not the case it would raise a question mark over the utility of de Finetti’s notion of coherence.
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  3.  33
    A Constructive Proof of McNaughton's Theorem in Infinite-Valued Logic.Daniele Mundici - 1994 - Journal of Symbolic Logic 59 (2):596-602.
    We give a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz. For the proof we only use Minkowski's convex body theorem and the rudiments of piecewise linear topology.
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  4.  36
    An Elementary Proof of Chang's Completeness Theorem for the Infinite-Valued Calculus of Lukasiewicz.Roberto Cignoli & Daniele Mundici - 1997 - Studia Logica 58 (1):79-97.
    The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.
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  5.  30
    An Elementary Presentation of the Equivalence Between MV-Algebras and L-Groups with Strong Unit.Roberto Cignoli & Daniele Mundici - 1998 - Studia Logica 61 (1):49-64.
    Aim of this paper is to provide a self-contained presentation of the natural equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.
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  6.  6
    Geometry of Robinson Consistency in Łukasiewicz Logic.Manuela Busaniche & Daniele Mundici - 2007 - Annals of Pure and Applied Logic 147 (1):1-22.
    We establish the Robinson joint consistency theorem for the infinite-valued propositional logic of Łukasiewicz. As a corollary we easily obtain the amalgamation property for MV-algebras—the algebras of Łukasiewicz logic: all pre-existing proofs of this latter result make essential use of the Pierce amalgamation theorem for abelian lattice-ordered groups together with the categorical equivalence Γ between these groups and MV-algebras. Our main tools are elementary and geometric.
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  7.  8
    Interpretation of De Finetti Coherence Criterion in Łukasiewicz Logic.Daniele Mundici - 2009 - Annals of Pure and Applied Logic 161 (2):235-245.
    De Finetti gave a natural definition of “coherent probability assessment” β:E→[0,1] of a set E={X1,…,Xm} of “events” occurring in an arbitrary set of “possible worlds”. In the particular case of yes–no events, , Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued Łukasiewicz propositional logic, and show: a possible world of is a valuation satisfying Θ, β is coherent iff it is a convex combination of valuations (...)
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  8.  77
    Many-Valued Points and Equality.Costas Drossos & Daniele Mundici - 2000 - Synthese 125 (1-2):77-95.
    In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we (...)
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  9.  24
    Generalized Quantifiers in Natural Language.Daniele Mundici, Johan van Benthem & Alice ter Meulen - 1987 - Journal of Symbolic Logic 52 (3):876.
  10.  31
    A Characterization of the Free N-Generated MV-Algebra.Daniele Mundici - 2004 - Archive for Mathematical Logic 45 (2):239-247.
    An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function f is a (...)
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  11.  14
    Decidable and Undecidable Prime Theories in Infinite-Valued Logic.Daniele Mundici & Giovanni Panti - 2001 - Annals of Pure and Applied Logic 108 (1-3):269-278.
    In classical propositional logic, a theory T is prime iff it is complete. In Łukasiewicz infinite-valued logic the two notions split, completeness being stronger than primeness. Using toric desingularization algorithms and the fine structure of prime ideal spaces of free ℓ -groups, in this paper we shall characterize prime theories in infinite-valued logic. We will show that recursively enumerable prime theories over a finite number of variables are decidable, and we will exhibit an example of an undecidable r.e. prime theory (...)
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  12.  35
    Consequence and Interpolation in Łukasiewicz Logic.Daniele Mundici - 2011 - Studia Logica 99 (1-3):269-278.
    Building on Wójcicki’s work on infinite-valued Łukasiewicz logic Ł ∞ , we give a self-contained proof of the deductive interpolation theorem for Ł ∞ . This paper aims at introducing the reader to the geometry of Łukasiewicz logic.
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  13.  30
    Applications of Many-Sorted Robinson Consistency Theorem.Daniele Mundici - 1981 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 27 (11-12):181-188.
  14.  32
    Paper Machines.Daniele Mundici & Wilfried Seig - 1995 - Philosophia Mathematica 3 (1):5-30.
    Machines were introduced as calculating devices to simulate operations carried out by human computers following fixed algorithms. The mathematical study of (paper) machines is the topic of our essay. The first three sections provide necessary logical background, examine the analyses of effective calculability given in the thirties, and describe results that are central to recursion theory, reinforcing the conceptual analyses. In the final section we pursue our investigation in a quite different way and focus on principles that govern the operations (...)
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  15.  45
    Many-Valued Logic and Cognition: Foreword.Shier Ju & Daniele Mundici - 2008 - Studia Logica 90 (1):1-2.
  16.  10
    Compactness, Interpolation and Friedman's Third Problem.Daniele Mundici - 1982 - Annals of Pure and Applied Logic 22 (2):197.
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  17.  18
    Applications of Many‐Sorted Robinson Consistency Theorem.Daniele Mundici - 1981 - Mathematical Logic Quarterly 27 (11‐12):181-188.
  18.  45
    Foreword.Daniele Mundici - 1998 - Studia Logica 61 (1):1-1.
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  19.  36
    Preface.MariaLuisa Dalla Chiara & Daniele Mundici - 1999 - Studia Logica 62 (2):117-120.
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  20.  9
    Tautologies with a Unique Craig Interpolant, Uniform Vs. Nonuniform Complexity.Daniele Mundici - 1984 - Annals of Pure and Applied Logic 27 (3):265-273.
    If S ⊆{0,1}; * and S ′ = {0,1} * \sb S are both recognized within a certain nondeterministic time bound T then, in not much more time, one can write down tautologies A n → A′ n with unique interpolants I n that define S ∩{0,1} n ; hence, if one can rapidly find unique interpolants, then one can recognize S within deterministic time T p for some fixed p \s>0. In general, complexity measures for the problem of finding (...)
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  21.  20
    An Algebraic Result About Soft Model Theoretical Equivalence Relations with an Application to H. Friedman's Fourth Problem.Daniele Mundici - 1981 - Journal of Symbolic Logic 46 (3):523-530.
    We prove the following algebraic characterization of elementary equivalence: $\equiv$ restricted to countable structures of finite type is minimal among the equivalence relations, other than isomorphism, which are preserved under reduct and renaming and which have the Robinson property; the latter is a faithful adaptation for equivalence relations of the familiar model theoretical notion. We apply this result to Friedman's fourth problem by proving that if L = L ωω (Q i ) i ∈ ω 1 is an (ω 1 (...)
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  22.  22
    Computability Theory.Daniele Mundici & Wilfried Sieg - unknown
    Daniele Mundici and Wilfred Sieg. Computability Theory.
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  23.  17
    Inverse Topological Systems and Compactness in Abstract Model Theory.Daniele Mundici - 1986 - Journal of Symbolic Logic 51 (3):785-794.
    Given an abstract logic L = L(Q i ) i ∈ I generated by a set of quantifiers Q i , one can construct for each type τ a topological space S τ exactly as one constructs the Stone space for τ in first-order logic. Letting T be an arbitrary directed set of types, the set $S_T = \{(S_\tau, \pi^\tau_\sigma)\mid\sigma, \tau \in T, \sigma \subset \tau\}$ is an inverse topological system whose bonding mappings π τ σ are naturally determined by (...)
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  24.  16
    Van Benthem J. F. A. K.. The Logic of Time. A Model-Theoretic Investigation Into the Varieties of Temporal Ontology and Temporal Discourse. Synthese Library, Vol. 156. D. Reidel Publishing Company, Dordrecht, Boston, and London, 1983, Xvi + 260 Pp. [REVIEW]Daniele Mundici - 1987 - Journal of Symbolic Logic 52 (3):874-878.
  25.  11
    De Finetti Coherence and the Product Law for Independent Events.Daniele Mundici - 2019 - Synthese 196 (1):265-271.
    In an earlier paper the present author proved that de Finetti coherence is preserved under taking products of coherent books on two finite sets of independent events. Conversely, in this note it is proved that product is the only coherence preserving operation on coherent books. Our proof shows that the traditional definition of stochastically independent classes of events actually follows from the combination of two more basic notions: boolean algebraic independence and de Finetti coherent betting system.
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  26.  17
    Finite Axiomatizability in Łukasiewicz Logic.Daniele Mundici - 2011 - Annals of Pure and Applied Logic 162 (12):1035-1047.
    We classify every finitely axiomatizable theory in infinite-valued propositional Łukasiewicz logic by an abstract simplicial complex equipped with a weight function . Using the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, we then construct a Turing computable one–one correspondence between equivalence classes of weighted abstract simplicial complexes, and equivalence classes of finitely axiomatizable theories, two theories being equivalent if their Lindenbaum algebras are isomorphic. We discuss the relationship between our classification and Markov’s undecidability theorem for PL-homeomorphism of (...)
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  27.  15
    Foreword.Evandro Agazzi, Itala M. L. D'Ottaviano & Daniele Mundici - 2011 - Manuscrito 34 (1):09-17.
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  28.  29
    Mathematics Studies Machines.Daniele Mundici & Wilfried Sieg - unknown
    Machines were introduced as calculating devices to simulate operations carried out by human computors following fixed algorithms: this is true for the early mechanical calculators devised by Pascal and Leibniz, for the analytical engine built by Babbage, and the theoretical machines introduced by Turing. The distinguishing feature of the latter is their universality: They are claimed to be able to capture any algorithm whatsoever and, conversely, any procedure they can carry out is evidently algorithmic. The study of such "paper machines" (...)
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  29.  21
    Foreword.Lev Beklemishev, Guram Bezhanishvili, Daniele Mundici & Yde Venema - 2012 - Studia Logica 100 (1-2):1-7.
  30.  12
    Preface.Maurice Boffa, Annalisa Marcja & Daniele Mundici - 1997 - Annals of Pure and Applied Logic 88 (2-3):93.
  31.  26
    A Group-Theoretical Invariant for Elementary Equivalence and its Role in Representations of Elementary Classes.Daniele Mundici - 1981 - Studia Logica 40 (3):253 - 267.
    There is a natural map which assigns to every modelU of typeτ, (U ε Stτ) a groupG (U) in such a way that elementarily equivalent models are mapped into isomorphic groups.G(U) is a subset of a collection whose members are called Fraisse arrows (they are decreasing sequences of sets of partial isomorphisms) and which arise in connection with the Fraisse characterization of elementary equivalence. LetEC λ U be defined as {U εStr τ: ℬ ≡U and |ℬ|=λ; thenEG λ U can (...)
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  32.  25
    Foreword.Silvio Ghilardi & Daniele Mundici - 2003 - Studia Logica 73 (1):1-1.
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  33.  8
    Germinal Theories in Łukasiewicz Logic.Leonardo Manuel Cabrer & Daniele Mundici - 2017 - Annals of Pure and Applied Logic 168 (5):1132-1151.
  34.  23
    Foreword: Logics of Uncertainty. [REVIEW]Daniele Mundici - 2000 - Journal of Logic, Language and Information 9 (1):1-3.
  35.  11
    Universal Properties of Łukasiewicz Consequence.Daniele Mundici - 2014 - Logica Universalis 8 (1):17-24.
    Boolean logic deals with {0, 1}-observables and yes–no events, as many-valued logic does for continuous ones. Since every measurement has an error, continuity ensures that small measurement errors on elementary observables have small effects on compound observables. Continuity is irrelevant for {0, 1}-observables. Functional completeness no longer holds when n-ary connectives are understood as [0, 1]-valued maps defined on [0, 1] n . So one must envisage suitable selection criteria for [0, 1]-connectives. Łukasiewicz implication has a well known characterization as (...)
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  36.  8
    Decidability and godel incompleteness in af c*-algebras.Daniele Mundici - 2005 - Manuscrito 28 (2):547-558.
    In the algebraic treatment of quantum statistical systems, the claim “Nature does not have ideals” is sometimes used to convey the idea that the C*-algebras describing natural systems are simple, i.e., they do not have nontrivial homomorphic images. Using our interpretation of AF C*-algebras as algebras of Lukasiewicz calculus, in a previous paper the claim was shown to be incompatible with the existence of a G¨odel incomplete AF C*-algebra for a quantum physical system existing in nature. In this note we (...)
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  37.  7
    Many-Valued Points And Equality.Costas Drossos & Daniele Mundici - 2000 - Synthese 125 (1):97-101.
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  38.  5
    Foreword DOI:10.5007/1808-1711.2011v15n2p223.Evandro Agazzi, Ítala M. Loffredo D’Ottaviano & Daniele Mundici - 2011 - Principia: An International Journal of Epistemology 15 (2):223-224.
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  39.  4
    Foreword.Daniele Mundici & Itala M. Loffredo D’Ottaviano - 2011 - Studia Logica 97 (1):1-5.
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  40.  3
    Gödel's Incompleteness Theorem and Quantum Thermodynamic Limits.Daniele Mundici - 1997 - In Evandro Agazzi & György Darvas (eds.), Philosophy of Mathematics Today. Kluwer Academic Publishers. pp. 287--298.
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  41. Computational Logic and Proof Theory Third Kurt Gödel Colloquium, Kgc'93 : Brno, Czech Republic, August 1993 : Proceedings. [REVIEW]G. Gottlob, Alexander Leitsch, Daniele Mundici & Kurt Gödel Society - 1993
     
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  42. Computational Logic and Proof Theory 5th Kurt Gödel Colloquium, Kgc '97, Vienna, Austria, August 25-29, 1997 : Proceedings'. [REVIEW]G. Gottlob, Alexander Leitsch, Daniele Mundici & Kurt Gödel Society - 1997
     
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  43. Coherence of the Product Law for Independent Continuous Events.Daniele Mundici - 2018 - In Jacek Malinowski & Walter Carnielli (eds.), Contradictions, from Consistency to Inconsistency. Springer Verlag.
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  44. Decidability And Godel Incompleteness In Af C*-Algebras.Daniele Mundici - 2006 - Manuscrito 29 (2):547-558.
    In the algebraic treatment of quantum statistical systems, the claim “Nature does not have ideals” is sometimes used to convey the idea that the C*-algebras describing natural systems are simple, i.e., they do not have nontrivial homomorphic images. Using our interpretation of AF C*-algebras as algebras of Lukasiewicz calculus, in a previous paper the claim was shown to be incompatible with the existence of a G¨odel incomplete AF C*-algebra for a quantum physical system existing in nature. In this note we (...)
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