55 found
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  1.  68
    Notes on Formal Theories of Truth.Andrea Cantini - 1989 - Zeitshrift für Mathematische Logik Und Grundlagen der Mathematik 35 (1):97--130.
  2.  66
    Notes on Formal Theories of Truth.Andrea Cantini - 1989 - Mathematical Logic Quarterly 35 (2):97-130.
  3.  71
    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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  4.  66
    Logical frameworks for truth and abstraction: an axiomatic study.Andrea Cantini (ed.) - 1996 - New York: 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.  10
    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.  65
    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.
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  7.  58
    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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  8.  8
    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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  9.  80
    Paradoxes and contemporary logic.Andrea Cantini - 2008 - Stanford Encyclopedia of Philosophy.
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  10.  35
    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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  11.  10
    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.  49
    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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  13.  18
    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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  14. On Stratified Truth.Andrea Cantini - 2015 - In T. Achourioti, H. Galinon, J. Martínez Fernández & K. Fujimoto (eds.), Unifying the Philosophy of Truth. Dordrecht: Imprint: Springer.
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  15.  9
    Elementary Constructive Operational Set Theory.Andrea Cantini & Laura Crosilla - 2010 - In Ralf Schindler (ed.), Ways of Proof Theory. De Gruyter. pp. 199-240.
    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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  16.  12
    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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  17.  33
    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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  18.  15
    First Steps into Metapredicativity in Explicit Mathematics.Andrea Cantini - 2002 - Bulletin of Symbolic Logic 8 (4):535-536.
  19.  17
    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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  20.  38
    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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  21.  30
    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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  22.  11
    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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  23.  26
    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.
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  24.  50
    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.  47
    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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  26.  13
    A fixed point theory over stratified truth.Andrea Cantini - 2020 - Mathematical Logic Quarterly 66 (4):380-394.
    We present a theory of stratified truth with a μ‐operator, where terms representing fixed points of stratified monotone operations are available. We prove that is relatively intepretable into Quine's (or subsystems thereof). The motivation is to investigate a strong theory of truth, which is consistent by means of stratification, i.e., by adopting an implicit type theoretic discipline, and yet is compatible with self‐reference (to a certain extent). The present version of is an enhancement of the theory presented in [2].
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  27.  28
    A logic of abstraction related to finite constructive number classes.Andrea Cantini - 1991 - Archive for Mathematical Logic 31 (1):69-83.
  28.  3
    Conservativity of transitive closure over weak constructive operational set theory.Andrea Cantini & Laura Crosilla - 2012 - In Ulrich Berger, Hannes Diener, Peter Schuster & Monika Seisenberger (eds.), Logic, Construction, Computation. De Gruyter. pp. 91-122.
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  29. Hartry field, saving truth from paradox.Andrea Cantini - 2010 - Erkenntnis 72 (3):417-422.
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  30.  3
    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 (eds.) - 1999 - Dordrecht, Netherland: Springer.
    The IOth International Congress of Logic, Methodology and Philosophy of Science, which took place in Florence in August 1995, offered a vivid and comprehensive picture of the present state of research in all directions of Logic and Philosophy of Science. The final program counted 51 invited lectures and around 700 contributed papers, distributed in 15 sections. Following the tradition of previous LMPS-meetings, some authors, whose papers aroused particular interest, were invited to submit their works for publication in a collection of (...)
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  31.  7
    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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  32.  18
    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.
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  33.  15
    Partial Truth.Andrea Cantini - 2003 - In Leon Horsten & Volker Halbach (eds.), Principles of Truth. De Gruyter. pp. 183-202.
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  34.  11
    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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  35. 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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  36. 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.
  37.  12
    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.
  38.  21
    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.
  39.  67
    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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  40.  27
    Two Impredicative Theories of Properties and Sets.Andrea Cantini - 1988 - Mathematical Logic Quarterly 34 (5):403-420.
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  41.  30
    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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  42. 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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  43. Una nota sulla concezione semi-intuizionistica della Matematica.Andrea Cantini - 1978 - Rivista di Filosofia 12:465.
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  44. Una teoria della predicatività secondo Poincaré.Andrea Cantini - 1981 - Rivista di Filosofia 19:32.
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  45.  16
    Conservativity of Transitive Closure over weak operational set theory.Laura Crosilla & Andrea Cantini - 2012 - In Ulrich Berger, Hannes Diener, Peter Schuster & Monika Seisenberger (eds.), Logic, Construction, Computation. De Gruyter.
    Constructive set theory a' la Myhill-Aczel has been extended in (Cantini and Crosilla 2008, Cantini and Crosilla 2010) to incorporate a notion of (partial, non--extensional) operation. Constructive operational set theory is a constructive and predicative analogue of Beeson's Inuitionistic set theory with rules and of Feferman's Operational set theory (Beeson 1988, Feferman 2006, Jaeger 2007, Jaeger 2009, Jaeger 1009b). This paper is concerned with an extension of constructive operational set theory (Cantini and Crosilla 2010) by a uniform operation of Transitive (...)
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  46.  2
    Feasible Operations and Applicative Theories Based on λη.Fernando Ferreira, Thomas Strahm & Andrea Cantini - 2002 - Bulletin of Symbolic Logic 8 (4):534.
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  47.  4
    Polynomial Time Operations in Explicit Mathematics.Thomas Strahm & Andrea Cantini - 2002 - Bulletin of Symbolic Logic 8 (4):534-535.
  48.  13
    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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  49.  12
    Review: Solomon Feferman, Reflecting on Incompleteness. [REVIEW]Andrea Cantini - 1995 - Journal of Symbolic Logic 60 (1):345-347.
  50.  7
    Review: Solomon Feferman, Toward Useful Type-Free Theories I. [REVIEW]Andrea Cantini - 1995 - Journal of Symbolic Logic 60 (1):342-345.
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