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  1. Andrea Cantini (2011). Extending Constructive Operational Set Theory by Impredicative Principles. 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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  2. Andrea Cantini (2010). Hartry Field, Saving Truth From Paradox. Erkenntnis 72 (3):417-422.
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  3. Andrea Cantini (2009). Paradoxes, Self-Reference and Truth in the 20th Century. In Dov Gabbay (ed.), The Handbook of the History of Logic. Elsevier. 5--875.
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  4. Andrea Cantini, Paradoxes and Contemporary Logic. Stanford Encyclopedia of Philosophy.
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  5. Andrea Cantini (2005). Remarks on Applicative Theories. 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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  6. Andrea Cantini & Valentin Goranko (2004). Nicholas Rescher, Paradoxes: Their Roots, Range, and Resolution; Patrick Blackburn, Maarten de Rijke and Yde Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science Vol. 53. Studia Logica 76 (1):135-142.
  7. Andrea Cantini & Valentin Goranko (2004). Nicholas Rescher,; Patrick Blackburn, Maarten de Rijke and Yde Venema, Cambridge Tracts in Theoretical Computer Science Vol. 53. Studia Logica 76 (1):135-142.
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  8. Andrea Cantini (2003). Partial Truth. In Leon Horsten & Volker Halbach (eds.), Principles of Truth. De Gruyter. 183-202.
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  9. Andrea Cantini (2003). The Axiom of Choice and Combinatory Logic. 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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  10. Andrea Cantini (2003). The Undecidability of Grisin's Set Theory. 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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  11. Andrea Cantini (2002). Review: Thomas Strahm, S. Barry Cooper, John K. Truss, First Steps Into Metapredicativity in Explicit Mathematics. [REVIEW] Bulletin of Symbolic Logic 8 (4):535-536.
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  12. Andrea Cantini (2002). First Steps Into Metapredicativity in Explicit Mathematics. [REVIEW] Bulletin of Symbolic Logic 8 (4):535-535.
     
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  13. Andrea Cantini (2002). Osservazioni su autoriferimento e verità. 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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  14. Andrea Cantini (2002). Polytime, Combinatory Logic and Positive Safe Induction. Archive for Mathematical Logic 41 (2):169-189.
    We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.
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  15. Andrea Cantini (2002). Strahm Thomas. 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 Cooper S. Barry and Truss John K., London Mathematical Society Lecture Note Series, No. 258, Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1999, Pp. 383–402. [REVIEW] Bulletin of Symbolic Logic 8 (4):535-536.
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  16. Andrea Cantini (2001). Review: Thomas Strahm, Autonomous Fixed Point Progressions and Fixed Point Transfinite Recursion. [REVIEW] Bulletin of Symbolic Logic 7 (4):535-536.
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  17. Andrea Cantini (1999). Relating Quine's NF to Feferman's EM. 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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  18. Andrea Cantini, Ettore Casari & Pierluigi Minari (1999). Logic and Foundations of Mathematics Selected Contributed Papers of the Tenth International Congress of Logic, Methodology, and Philosophy of Science, Florence, August 1995.
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  19. Andrea Cantini & Pierluigi Minari (1999). Uniform Inseparability in Explicit Mathematics. 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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  20. Andrea Cantini (1996). Asymmetric Interpretations for Bounded Theories. 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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  21. Andrea Cantini (1996). Logical Frameworks for Truth and Abstraction: An Axiomatic Study. 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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  22. Andrea Cantini (1995). Levels of Truth. 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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  23. Andrea Cantini (1995). Review: Solomon Feferman, Reflecting on Incompleteness. [REVIEW] Journal of Symbolic Logic 60 (1):345-347.
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  24. Andrea Cantini (1995). Review: Solomon Feferman, Toward Useful Type-Free Theories I. [REVIEW] Journal of Symbolic Logic 60 (1):342-345.
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  25. Andrea Cantini (1993). Extending the First-Order Theory of Combinators with Self-Referential Truth. 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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  26. Andrea Cantini (1993). Review: Wolfram Pohlers, Proof Theory. An Introduction. [REVIEW] Journal of Symbolic Logic 58 (1):358-359.
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  27. Andrea Cantini (1992). Levels of Implication and Type Free Theories of Classifications with Approximation Operator. 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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  28. Andrea Cantini (1991). A Logic of Abstraction Related to Finite Constructive Number Classes. Archive for Mathematical Logic 31 (1):69-83.
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  29. Andrea Cantini (1990). A Theory of Formal Truth Arithmetically Equivalent to ID. 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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  30. Andrea Cantini (1989). Notes on Formal Theories of Truth. Zeitshrift für Mathematische Logik Und Grundlagen der Mathematik 35 (1):97--130.
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  31. Andrea Cantini (1988). Two Impredicative Theories of Properties and Sets. Mathematical Logic Quarterly 34 (5):403-420.
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  32. Andrea Cantini (1986). On the Relation Between Choice and Comprehension Principles in Second Order Arithmetic. 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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  33. Andrea Cantini (1985). On Weak Theories of Sets and Classes Which Are Based on Strict ∏ 11‐REFLECTION. Mathematical Logic Quarterly 31 (21‐23):321-332.
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  34. Andrea Cantini (1981). Una teoria della predicatività secondo Poincaré. Rivista di Filosofia 19:32.
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  35. Andrea Cantini (1980). A Note on Three-Valued Logic and Tarski Theorem on Truth Definitions. 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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  36. Andrea Cantini (1978). Una nota sulla concezione semi-intuizionistica della Matematica. Rivista di Filosofia 12:465.
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  37. Andrea Cantini (1976). Su alcuni recenti sviluppi della Logica: temi e problemi della Teoria delle definizioni ricorsive. Rivista di Filosofia 6:451.
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