52 found
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  1.  53
    Notes on Formal Theories of Truth.Andrea Cantini - 1989 - Zeitshrift für Mathematische Logik Und Grundlagen der Mathematik 35 (1):97--130.
  2.  43
    A Theory of Formal Truth Arithmetically Equivalent to ID.Andrea Cantini - 1990 - Journal of Symbolic Logic 55 (1):244 - 259.
    We present a theory VF of partial truth over Peano arithmetic and we prove that VF and ID 1 have the same arithmetical content. The semantics of VF is inspired by van Fraassen's notion of supervaluation.
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  3.  39
    Notes on Formal Theories of Truth.Andrea Cantini - 1989 - Mathematical Logic Quarterly 35 (2):97-130.
  4.  55
    Logical Frameworks for Truth and Abstraction: An Axiomatic Study.Andrea Cantini - 1996 - Elsevier Science B.V..
    This English translation of the author's original work has been thoroughly revised, expanded and updated. The book covers logical systems known as type-free or self-referential . These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (ii) these (...)
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  5.  2
    A Theory of Formal Truth Arithmetically Equivalent to ID 1.Andrea Cantini - 1990 - Journal of Symbolic Logic 55 (1):244-259.
    We present a theory VF of partial truth over Peano arithmetic and we prove that VF and ID 1 have the same arithmetical content. The semantics of VF is inspired by van Fraassen's notion of supervaluation.
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  6.  49
    The Undecidability of Grisin's Set Theory.Andrea Cantini - 2003 - Studia Logica 74 (3):345 - 368.
    We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.
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  7.  6
    The Undecidability of Grisin's Set Theory.Andrea Cantini - 2003 - Studia Logica 74 (3):345-368.
    We investigate a contractionless naive set theory, due to Gris̆in [11]. We prove that the theory is undecidable.
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  8.  11
    Extending Constructive Operational Set Theory by Impredicative Principles.Andrea Cantini - 2011 - Mathematical Logic Quarterly 57 (3):299-322.
    We study constructive set theories, which deal with operations applying both to sets and operations themselves. Our starting point is a fully explicit, finitely axiomatized system ESTE of constructive sets and operations, which was shown in 10 to be as strong as PA. In this paper we consider extensions with operations, which internally represent description operators, unbounded set quantifiers and local fixed point operators. We investigate the proof theoretic strength of the resulting systems, which turn out to be impredicative . (...)
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  9.  40
    Paradoxes, Self-Reference and Truth in the 20th Century.Andrea Cantini - 2009 - In Dov Gabbay (ed.), The Handbook of the History of Logic. Elsevier. pp. 5--875.
  10.  7
    Thomas Strahm. Autonomous Fixed Point Progressions and Fixed Point Transfinite Recursion. Logic Colloquium '98, Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Prague, Czech Republic, August 9–15, 1998, Edited by Samuel R. Buss, Petr Hájek, and Pavel Pudlák, Lecture Notes in Logic, No. 13, Association for Symbolic Logic, Urbana, and A K Peters, Natick, Mass., 2000, Pp. 449–464. [REVIEW]Andrea Cantini - 2001 - Bulletin of Symbolic Logic 7 (4):535-536.
  11.  6
    Asymmetric Interpretations for Bounded Theories.Andrea Cantini - 1996 - Mathematical Logic Quarterly 42 (1):270-288.
    We apply the method of asymmetric interpretation to the basic fragment of bounded arithmetic, endowed with a weak collection schema, and to a system of “feasible analysis”, introduced by Ferreira and based on weak König's lemma, recursive comprehension and NP-notation induction. As a byproduct, we obtain two conservation results.
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  12.  6
    Solomon Feferman. Toward Useful Type-Free Theories I. The Journal of Symbolic Logic, Vol. 49 , Pp. 75–111. , Pp. 237–287.).Andrea Cantini - 1995 - Journal of Symbolic Logic 60 (1):342-345.
  13.  5
    Solomon Feferman. Reflecting on Incompleteness. The Journal of Symbolic Logic, Vol. 56 , Pp. 1–49.Andrea Cantini - 1995 - Journal of Symbolic Logic 60 (1):345-347.
  14.  25
    Levels of Implication and Type Free Theories of Classifications with Approximation Operator.Andrea Cantini - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):107-141.
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  15.  4
    Wolfram Pohlers. Proof Theory. An Introduction. Lecture Notes in Mathematics, Vol. 1407. Springer-Verlag, Berlin Etc. 1989, Vi + 213 Pp. [REVIEW]Andrea Cantini - 1993 - Journal of Symbolic Logic 58 (1):358-359.
  16.  46
    A Note on Three-Valued Logic and Tarski Theorem on Truth Definitions.Andrea Cantini - 1980 - Studia Logica 39 (4):405 - 414.
    We introduce a notion of semantical closure for theories by formalizing Nepeivoda notion of truth. [10]. Tarski theorem on truth definitions is discussed in the light of Kleene's three valued logic (here treated with a formal reinterpretation of logical constants). Connections with Definability Theory are also established.
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  17.  8
    Levels of Implication and Type Free Theories of Classifications with Approximation Operator.Andrea Cantini - 1992 - Mathematical Logic Quarterly 38 (1):107-141.
    We investigate a theory of Frege structures extended by the Myhill-Flagg hierarchy of implications. We study its relation to a property theory with an approximation operator and we give a proof theoretical analysis of the basic system involved. MSC: 03F35, 03D60.
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  18.  24
    On the Relation Between Choice and Comprehension Principles in Second Order Arithmetic.Andrea Cantini - 1986 - Journal of Symbolic Logic 51 (2):360-373.
    We give a new elementary proof of the comparison theorem relating $\sum^1_{n + 1}-\mathrm{AC}\uparrow$ and $\Pi^1_n -\mathrm{CA}\uparrow$ ; the proof does not use Skolem theories. By the same method we prove: a) $\sum^1_{n + 1}-\mathrm{DC} \uparrow \equiv (\Pi^1_n -CA)_{ , for suitable classes of sentences; b) $\sum^1_{n+1}-DC \uparrow$ proves the consistency of (Π 1 n -CA) ω k, for finite k, and hence is stronger than $\sum^1_{n+1}-AC \uparrow$ . a) and b) answer a question of Feferman and Sieg.
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  19.  6
    Feasible Operations and Applicative Theories Based on Λη.Andrea Cantini - 2000 - Mathematical Logic Quarterly 46 (3):291-312.
    We study a theory PTO of polynomial time computability on the type of binary strings, as embedded in full lambda calculus with total application and extensionality. We prove that the closed terms of type W → W are exactly the polynomial time operations. This answers a conjecture of Strahm [13].
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  20.  29
    Paradoxes and Contemporary Logic.Andrea Cantini - 2008 - Stanford Encyclopedia of Philosophy.
  21. Elementary Constructive Operational Set Theory.Andrea Cantini & Laura Crosilla - 2010 - In Ways of Proof Theory.
    We introduce an operational set theory in the style of [5] and [16]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self{application is permitted. The system we (...)
     
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  22.  18
    Extending the First-Order Theory of Combinators with Self-Referential Truth.Andrea Cantini - 1993 - Journal of Symbolic Logic 58 (2):477-513.
    The aim of this paper is to introduce a formal system STW of self-referential truth, which extends the classical first-order theory of pure combinators with a truth predicate and certain approximation axioms. STW naturally embodies the mechanisms of general predicate application/abstraction on a par with function application/abstraction; in addition, it allows non-trivial constructions, inspired by generalized recursion theory. As a consequence, STW provides a smooth inner model for Myhill's systems with levels of implication.
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  23. Constructive Set Theory with Operations.Andrea Cantini & Laura Crosilla - 2008 - In Logic Colloquium 2004.
    We present an extension of constructive Zermelo{Fraenkel set theory [2]. Constructive sets are endowed with an applicative structure, which allows us to express several set theoretic constructs uniformly and explicitly. From the proof theoretic point of view, the addition is shown to be conservative. In particular, we single out a theory of constructive sets with operations which has the same strength as Peano arithmetic.
     
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  24.  44
    Relating Quine's NF to Feferman's EM.Andrea Cantini - 1999 - Studia Logica 62 (2):141-162.
    We show that, if non-uniform impredicative stratified comprehension is assumed, Feferman's theories of explicit mathematics are consistent with a strong power type axiom. This result answers a problem, raised by Jäger. The proof relies upon an interpretation into Quine's set theory NF with urelements.
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  25.  94
    Hartry Field, Saving Truth From Paradox.Andrea Cantini - 2010 - Erkenntnis 72 (3):417-422.
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  26. Selected Contributed Papers of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995.Andrea Cantini, Etiore Casari & Pierluigi Minari (eds.) - 1999 - Springer-Science+Business Media.
  27.  38
    Uniform Inseparability in Explicit Mathematics.Andrea Cantini & Pierluigi Minari - 1999 - Journal of Symbolic Logic 64 (1):313-326.
    We deal with ontological problems concerning basic systems of explicit mathematics, as formalized in Jäger's language of types and names. We prove a generalized inseparability lemma, which implies a form of Rice's theorem for types and a refutation of the strong power type axiom POW + . Next, we show that POW + can already be refuted on the basis of a weak uniform comprehension without complementation, and we present suitable optimal refinements of the remaining results within the weaker theory.
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  28.  22
    On Weak Theories of Sets and Classes Which Are Based on Strict ∏11-REFLECTION.Andrea Cantini - 1985 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 31 (21-23):321-332.
  29.  23
    Levels of Truth.Andrea Cantini - 1995 - Notre Dame Journal of Formal Logic 36 (2):185-213.
    This paper is concerned with the interaction between formal semantics and the foundations of mathematics. We introduce a formal theory of truth, TLR, which extends the classical first order theory of pure combinators with a primitive truth predicate and a family of truth approximations, indexed by a directed partial ordering. TLR naturally works as a theory of partial classifications, in which type-free comprehension coexists with functional abstraction. TLR provides an inner model for a well known subsystem of second order arithmetic; (...)
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  30.  22
    Sets and Proofs, Invited Papers From Logic Colloquium '97—European Meeting of the Association for Symbolic Logic, Leeds, July 1997. Thomas Strahm. First Steps Into Metapredicativity in Explicit Mathematics. Sets and Proofs, Invited Papers From Logic Colloquium '97—European Meeting of the Association for Symbolic Logic, Leeds, July 1997, Edited by S. Barry Cooper and John K. Truss, London Mathematical Society Lecture Note Series, No. 258, Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1999, Pp. 383–402. [REVIEW]Andrea Cantini - 2002 - Bulletin of Symbolic Logic 8 (4):535-536.
  31.  25
    Two Impredicative Theories of Properties and Sets.Andrea Cantini - 1988 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 34 (5):403-420.
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  32.  15
    On Weak Theories of Sets and Classes Which Are Based on Strict ∏Math Image-REFLECTION.Andrea Cantini - 1985 - Mathematical Logic Quarterly 31 (21-23):321-332.
  33.  23
    A Logic of Abstraction Related to Finite Constructive Number Classes.Andrea Cantini - 1991 - Archive for Mathematical Logic 31 (1):69-83.
  34.  43
    The Axiom of Choice and Combinatory Logic.Andrea Cantini - 2003 - Journal of Symbolic Logic 68 (4):1091-1108.
    We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice.
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  35.  17
    Review: Wolfram Pohlers, Proof Theory. An Introduction. [REVIEW]Andrea Cantini - 1993 - Journal of Symbolic Logic 58 (1):358-359.
  36.  15
    Two Impredicative Theories of Properties and Sets.Andrea Cantini - 1988 - Mathematical Logic Quarterly 34 (5):403-420.
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  37.  8
    Partial Truth.Andrea Cantini - 2003 - In Leon Horsten & Volker Halbach (eds.), Principles of Truth. De Gruyter. pp. 183-202.
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  38.  8
    Remarks on Applicative Theories.Andrea Cantini - 2005 - Annals of Pure and Applied Logic 136 (1-2):91-115.
    We deal with applicative theories which are based on combinatory logic with total application, extensionality and natural numbers. We prove a conservative extension theorem and two consistency results, involving principles of uniformity, reflection, enumeration and choice.
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  39.  4
    Uniform Inseparability in Explicit Mathematics.Andrea Cantini & Pierluigi Minari - 1999 - Journal of Symbolic Logic 64 (1):313-326.
    We deal with ontological problems concerning basic systems of explicit mathematics, as formalized in Jager's language of types and names. We prove a generalized inseparability lemma, which implies a form of Rice's theorem for types and a refutation of the strong power type axiom POW$^+$. Next, we show that POW$^+$ can already be refuted on the basis of a weak uniform comprehension without complementation, and we present suitable optimal refinements of the remaining results within the weaker theory.
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  40.  5
    First Steps Into Metapredicativity in Explicit Mathematics. [REVIEW]Andrea Cantini - 2002 - Bulletin of Symbolic Logic 8 (4):535-535.
  41.  6
    Osservazioni su autoriferimento e verità.Andrea Cantini - 2002 - Annali Del Dipartimento di Filosofia 8:53-76.
    The present essay deals with the fundamental role of self-referential notions in contemporary logic. As a special case study, we survey recent ideas and results in formal semantics.
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  42.  3
    Polynomial Time Operations in Explicit Mathematics.Thomas Strahm & Andrea Cantini - 2002 - Bulletin of Symbolic Logic 8 (4):534-535.
  43.  6
    Review: Solomon Feferman, Reflecting on Incompleteness. [REVIEW]Andrea Cantini - 1995 - Journal of Symbolic Logic 60 (1):345-347.
  44.  6
    Review: Thomas Strahm, S. Barry Cooper, John K. Truss, First Steps Into Metapredicativity in Explicit Mathematics. [REVIEW]Andrea Cantini - 2002 - Bulletin of Symbolic Logic 8 (4):535-536.
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  45.  3
    Review: Solomon Feferman, Toward Useful Type-Free Theories I. [REVIEW]Andrea Cantini - 1995 - Journal of Symbolic Logic 60 (1):342-345.
  46.  3
    Review: Thomas Strahm, Autonomous Fixed Point Progressions and Fixed Point Transfinite Recursion. [REVIEW]Andrea Cantini - 2001 - Bulletin of Symbolic Logic 7 (4):535-536.
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  47. Logic and Foundations of Mathematics Selected Contributed Papers of the Tenth International Congress of Logic, Methodology, and Philosophy of Science, Florence, August 1995.Andrea Cantini, Ettore Casari & Pierluigi Minari - 1999
  48. On Stratified Truth.Andrea Cantini - 2015 - In Kentaro Fujimoto, José Martínez Fernández, Henri Galinon & Theodora Achourioti (eds.), Unifying the Philosophy of Truth. Springer Verlag.
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  49. Su alcuni recenti sviluppi della Logica: temi e problemi della Teoria delle definizioni ricorsive.Andrea Cantini - 1976 - Rivista di Filosofia 6:451.
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  50. Una nota sulla concezione semi-intuizionistica della Matematica.Andrea Cantini - 1978 - Rivista di Filosofia 12:465.
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