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Fernando Ferreira [58]Fernando J. Ferreira [1]
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  1.  20
    Bounded Functional Interpretation.Fernando Ferreira & Paulo Oliva - 2005 - Annals of Pure and Applied Logic 135 (1):73-112.
    We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of the (...)
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  2. On the Consistency of the Δ11-CA Fragment of Frege's Grundgesetze.Fernando Ferreira & Kai F. Wehmeier - 2002 - Journal of Philosophical Logic 31 (4):301-311.
    It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...)
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  3. Comments on Predicative Logic.Fernando Ferreira - 2006 - Journal of Philosophical Logic 35 (1):1-8.
    We show how to interpret intuitionistic propositional logic into a predicative second-order intuitionistic propositional system having only the conditional and the universal second-order quantifier. We comment on this fact. We argue that it supports the legitimacy of using classical logic in a predicative setting, even though the philosophical cast of predicativism is nonrealistic. We also note that the absence of disjunction and existential quantifications allows one to have a process of normalization of proofs that avoids the use of "commuting conversions.".
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  4. A Feasible Theory for Analysis.Fernando Ferreira - 1994 - Journal of Symbolic Logic 59 (3):1001-1011.
    We construct a weak second-order theory of arithmetic which includes Weak König's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with Σ b 1 -graphs) of this theory are the polynomial time computable functions. It is shown that the first-order strength of this version of WKL is exactly that of the scheme of collection for bounded formulae.
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  5.  28
    Atomic Polymorphism.Fernando Ferreira & Gilda Ferreira - 2013 - Journal of Symbolic Logic 78 (1):260-274.
    It has been known for six years that the restriction of Girard's polymorphic system $\text{\bfseries\upshape F}$ to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each $\beta$-reduction step of the full intuitionistic propositional calculus translates into one or more $\beta\eta$-reduction steps in the restricted Girard system. As a consequence, we obtain (...)
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  6.  16
    The Faithfulness of Fat: A Proof-Theoretic Proof.Fernando Ferreira & Gilda Ferreira - 2015 - Studia Logica 103 (6):1303-1311.
    It is known that there is a sound and faithful translation of the full intuitionistic propositional calculus into the atomic polymorphic system F at, a predicative calculus with only two connectives: the conditional and the second-order universal quantifier. The faithfulness of the embedding was established quite recently via a model-theoretic argument based in Kripke structures. In this paper we present a purely proof-theoretic proof of faithfulness. As an application, we give a purely proof-theoretic proof of the disjunction property of the (...)
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  7.  9
    On the Consistency of the $\Delta_{1}^{1}$-CA Fragment of Frege's "Grundgesetze".Fernando Ferreira & Kai F. Wehmeier - 2002 - Journal of Philosophical Logic 31 (4):301-311.
    It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension schema would already be (...)
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  8. On End‐Extensions of Models of ¬Exp.Fernando Ferreira - 1996 - Mathematical Logic Quarterly 42 (1):1-18.
    Every model of IΔ0 is the tally part of a model of the stringlanguage theory Th-FO . We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IΔ0 + ¬exp is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models.
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  9.  50
    Commuting Conversions Vs. The Standard Conversions of the “Good” Connectives.Fernando Ferreira & Gilda Ferreira - 2009 - Studia Logica 92 (1):63-84.
    Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex (...)
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  10.  23
    Nonstandardness and the Bounded Functional Interpretation.Fernando Ferreira & Jaime Gaspar - 2015 - Annals of Pure and Applied Logic 166 (6):701-712.
  11.  9
    Elementary Proof of Strong Normalization for Atomic F.Fernando Ferreira & Gilda Ferreira - 2016 - Bulletin of the Section of Logic 45 (1).
    We give an elementary proof of the strong normalization of the atomic polymorphic calculus Fat.
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  12.  4
    The FAN Principle and Weak König's Lemma in Herbrandized Second-Order Arithmetic.Fernando Ferreira - 2020 - Annals of Pure and Applied Logic 171 (9):102843.
    We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.
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  13.  9
    Injecting Uniformities Into Peano Arithmetic.Fernando Ferreira - 2009 - Annals of Pure and Applied Logic 157 (2-3):122-129.
    We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finite-type arithmetic. As a consequence, some uniform boundedness principles are interpreted while maintaining unmoved the -sentences of arithmetic. We explain why this interpretation is tailored to yield conservation results.
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  14.  43
    The Co-Ordination Principles: A Problem for Bilateralism.Fernando Ferreira - 2008 - Mind 117 (468):1051-1057.
    In "'Yes" and "No'" (2000), Ian Rumfitt proposed bilateralism--a use-based account of the logical words, according to which the sense of a sentence is determined by the conditions under which it is asserted and denied. One of Rumfitt's key claims is that bilateralism can provide a justification of classical logic. This paper raises a techical problem for Rumfitt's proposal, one that seems to undermine the bilateralist programme.
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  15.  67
    Groundwork for Weak Analysis.António M. Fernandes & Fernando Ferreira - 2002 - Journal of Symbolic Logic 67 (2):557-578.
    This paper develops the very basic notions of analysis in a weak second-order theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we (...)
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  16.  39
    Bounded Modified Realizability.Fernando Ferreira & Ana Nunes - 2006 - Journal of Symbolic Logic 71 (1):329 - 346.
    We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov's principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König's lemma. We discuss some applications, as well as some previous results in the literature.
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  17.  33
    Thomas Strahm. Polynomial Time Operations in Explicit Mathematics. The Journal of Symbolic Logic, Vol. 62 , Pp. 575–594. - Andrea Cantini. Feasible Operations and Applicative Theories Based on Λη. Mathematical Logic Quarterly, Vol. 46 , Pp. 291–312. [REVIEW]Fernando Ferreira - 2002 - Bulletin of Symbolic Logic 8 (4):534-535.
  18.  11
    Interpreting Weak Kőnig's Lemma in Theories of Nonstandard Arithmetic.Bruno Dinis & Fernando Ferreira - 2017 - Mathematical Logic Quarterly 63 (1-2):114-123.
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  19.  13
    Bounded Functional Interpretation and Feasible Analysis.Fernando Ferreira & Paulo Oliva - 2007 - Annals of Pure and Applied Logic 145 (2):115-129.
    In this article we study applications of the bounded functional interpretation to theories of feasible arithmetic and analysis. The main results show that the novel interpretation is sound for considerable generalizations of weak König’s Lemma, even in the presence of very weak induction. Moreover, when this is combined with Cook and Urquhart’s variant of the functional interpretation, one obtains effective versions of conservation results regarding weak König’s Lemma which have been so far only obtained non-constructively.
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  20.  23
    Mitsuru Tada and Makoto Tatsuta. The Function ⌊a/M⌋ in Sharply Bounded Arithmetic. Archive for Mathematical Logic, Vol. 37 No. 1 , Pp. 51–57. [REVIEW]Fernando Ferreira - 2001 - Bulletin of Symbolic Logic 7 (3):391.
  21.  80
    Amending Frege’s Grundgesetze der Arithmetik.Fernando Ferreira - 2005 - Synthese 147 (1):3-19.
    Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this (...)
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  22.  43
    A Most Artistic Package of a Jumble of Ideas.Fernando Ferreira - 2008 - Dialectica 62 (2):205–222.
    In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, discuss the extra principles that (...)
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  23.  7
    A Most Artistic Package of a Jumble of Ideas.Fernando Ferreira - 2008 - Dialectica 62 (2):205-222.
    In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, discuss the extra principles that (...)
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  24.  12
    Interpretability in Robinson's Q.Fernando Ferreira & Gilda Ferreira - 2013 - Bulletin of Symbolic Logic 19 (3):289-317.
    Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is animpassable barrierin the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted inbut also what cannot be so (...)
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  25.  15
    Arithmetic, Proof Theory, and Computational Complexity, Edited by Peter Clote and Krajíček Jan, Oxford Logic Guides, No. 23, Clarendon Press, Oxford University Press, Oxford and New York1993, Xiii + 428 Pp. [REVIEW]Fernando Ferreira - 1995 - Journal of Symbolic Logic 60 (3):1014-1017.
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  26.  8
    Interpretability in Robinson's Q.Fernando Ferreira & Gilda Ferreira - forthcoming - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but (...)
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  27.  15
    A Simple Proof of Parsons' Theorem.Fernando Ferreira - 2005 - Notre Dame Journal of Formal Logic 46 (1):83-91.
    Let be the fragment of elementary Peano arithmetic in which induction is restricted to -formulas. More than three decades ago, Parsons showed that the provably total functions of are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the -consequences of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.
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  28.  38
    The Bounded Functional Interpretation of the Double Negation Shift.Patrícia Engrácia & Fernando Ferreira - 2010 - Journal of Symbolic Logic 75 (2):759-773.
    We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application. we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case.
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  29.  33
    Binary Models Generated by Their Tally Part.Fernando Ferreira - 1994 - Archive for Mathematical Logic 33 (4):283-289.
    We introduce a class of models of the bounded arithmetic theoryPV n . These models, which are generated by their tally part, have a curious feature: they have end-extensions or satisfyB∑ n b only in case they are closed under exponentiation. As an application, we show that if then the polynomial hierarchy does not collapse.
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  30.  7
    What Are the ∀∑1 B-Consequences of T 2 1 and T 2 2?Fernando Ferreira - 1994 - Annals of Pure and Applied Logic 75 (1):79-88.
    We formulate schemes and of the “typical” ∀∑ 1 b -sentences that are provable in T 2 1, respectively T 2 2. As an application, we reprove a recent result of Buss and Krajíček which describes witnesses for the ∀∑ 1 b -sentences provable in T 2 1 in terms of solutions to PLS-problems.
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  31.  5
    What Are the ∀∑1b-Consequences of T21 and T22?Fernando Ferreira - 1995 - Annals of Pure and Applied Logic 75 (1-2):79-88.
    We formulate schemes and of the “typical” ∀∑ 1 b -sentences that are provable in T 2 1 , respectively T 2 2 . As an application, we reprove a recent result of Buss and Krajíček which describes witnesses for the ∀∑ 1 b -sentences provable in T 2 1 in terms of solutions to PLS-problems.
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  32.  56
    A Substitutional Framework for Arithmetical Validity.Fernando Ferreira - 1998 - Grazer Philosophische Studien 56 (1):133-149.
  33.  26
    Extracting Algorithms From Intuitionistic Proofs.Fernando Ferreira & António Marques - 1998 - Mathematical Logic Quarterly 44 (2):143-160.
    This paper presents a new method - which does not rely on the cut-elimination theorem - for characterizing the provably total functions of certain intuitionistic subsystems of arithmetic. The new method hinges on a realizability argument within an infinitary language. We illustrate the method for the intuitionistic counterpart of Buss's theory Smath image, and we briefly sketch it for the other levels of bounded arithmetic and for the theory IΣ1.
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  34.  46
    * Exercícios Eleáticos.Fernando Ferreira - 1997 - Disputatio (2):3-21.
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  35.  20
    Amending Frege’s Grundgesetze der Arithmetik.Fernando Ferreira - 2005 - Synthese 147 (1):3 - 19.
    Frege's "Grundgesetze der Arithmetik" is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege's Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the "Grundgesetze" is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this (...)
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  36.  57
    A Note on Finiteness in the Predicative Foundations of Arithmetic.Fernando Ferreira - 1999 - Journal of Philosophical Logic 28 (2):165-174.
    Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can be derived from a very (...)
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  37.  22
    The Finitistic Consistency of Heck’s Predicative Fregean System.Luís Cruz-Filipe & Fernando Ferreira - 2015 - Notre Dame Journal of Formal Logic 56 (1):61-79.
    Frege’s theory is inconsistent. However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak. We also prove the finitistic consistency of the extension of Heck’s theory to $\Delta^{1}_{1}$-comprehension and of Heck’s ramified predicative second-order system.
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  38. On the Parmenidean Misconception.Fernando Ferreira - 1999 - Logical Analysis and History of Philosophy 2.
    This paper makes two main claims. Firstly, it imputes to Parmenides a misconception rooted in an erroneous theory of the meaning of sentences. In Parmenides' hands, this theory took the extreme form not only of being unable to make sense of falsehoods, but also of being unable to make sense of true negative predications. Secondly, it claims that Plato's double theory of "limited mixing" plus "negation as otherness" - as expounded in the Sophist - is a theory still within the (...)
     
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  39.  17
    What Are the∀∑< Sub> 1< Sup> B-Consequences of< I> T< Sub> 2< Sup> 1< I> And_< I> T< Sub> 2< Sup> 2? [REVIEW]Fernando Ferreira - 1995 - Annals of Pure and Applied Logic 75 (1):79-88.
  40.  24
    On the Notion of Object. A Logical Genealogy.Fernando Ferreira - 2012 - Disputatio 4 (34):609-624.
    Ferreira-Fernando_On-the-notion-of-object.
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  41.  7
    Categoricity and Mathematical Knowledge.Fernando Ferreira - 2017 - Revista Portuguesa de Filosofia 73 (3-4):1423-1436.
    We argue that the basic notions of mathematics can only be properly formulated in an informal way. Mathematical notions transcend formalizations and their study involves the consideration of other mathematical notions. We explain the fundamental role of categoricity theorems in making these studies possible. We arrive at the conclusion that the enterprise of mathematics is not infallible and that it ultimately relies on degrees of evidence.
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  42.  16
    Harrington’s Conservation Theorem Redone.Fernando Ferreira & Gilda Ferreira - 2008 - Archive for Mathematical Logic 47 (2):91-100.
    Leo Harrington showed that the second-order theory of arithmetic WKL 0 is ${\Pi^1_1}$ -conservative over the theory RCA 0. Harrington’s proof is model-theoretic, making use of a forcing argument. A purely proof-theoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cut-elimination argument.
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  43.  13
    A Short Note on Spector's Proof of Consistency of Analysis.Fernando Ferreira - 2012 - In S. Barry Cooper (ed.), How the World Computes. pp. 222--227.
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  44.  15
    Two General Results on Intuitionistic Bounded Theories.Fernando Ferreira - 1999 - Mathematical Logic Quarterly 45 (3):399-407.
    We study, within the framework of intuitionistic logic, two well-known general results of bounded arithmetic. Firstly, Parikh's theorem on the existence of bounding terms for the provably total functions. Secondly, the result which states that adding the scheme of bounded collection to bounded theories does not yield new II2 consequences.
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  45.  7
    A Herbrandized Functional Interpretation of Classical First-Order Logic.Fernando Ferreira & Gilda Ferreira - 2017 - Archive for Mathematical Logic 56 (5-6):523-539.
    We introduce a new typed combinatory calculus with a type constructor that, to each type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, associates the star type σ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*$$\end{document} of the nonempty finite subsets of elements of type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory (...)
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  46.  12
    Computability in Europe 2010.Fernando Ferreira, Martin Hyland, Benedikt Löwe & Elvira Mayordomo - 2012 - Annals of Pure and Applied Logic 163 (6):621-622.
  47.  14
    Review: Mitsuru Tada, Makoto Tatsuta, The Function $Lfloor a/M Rfloor$ in Sharply Bounded Arithmetic. [REVIEW]Fernando Ferreira - 2001 - Bulletin of Symbolic Logic 7 (3):391-391.
  48.  11
    Moscone Center West, San Francisco, CA January 15–16, 2010.Fernando J. Ferreira, John Harrison, François Loeser, Chris Miller, Joseph S. Miller, Slawomir J. Solecki, Stevo Todorcevic & John Steel - 2010 - Bulletin of Symbolic Logic 16 (3).
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  49.  10
    Review: Peter Clote, Jan Krajicek, Arithmetic, Proof Theory, and Computational Complexity. [REVIEW]Fernando Ferreira - 1995 - Journal of Symbolic Logic 60 (3):1014-1017.
  50.  8
    Counting as Integration in Feasible Analysis.Fernando Ferreira & Gilda Ferreira - 2006 - Mathematical Logic Quarterly 52 (3):315-320.
    Suppose that it is possible to integrate real functions over a weak base theory related to polynomial time computability. Does it follow that we can count? The answer seems to be: obviously yes! We try to convince the reader that the severe restrictions on induction in feasible theories preclude a straightforward answer. Nevertheless, a more sophisticated reflection does indeed show that the answer is affirmative.
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