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  1. Review of John Stillwell, Reverse Mathematics: Proofs from the Inside Out. [REVIEW]Benedict Eastaugh - 2020 - Philosophia Mathematica 28 (1):108-116.
    Review of John Stillwell, Reverse Mathematics: Proofs from the Inside Out. Princeton, NJ: Princeton University Press, 2018, pp. 200. ISBN 978-0-69-117717-5 (hbk), 978-0-69-119641-1 (pbk), 978-1-40-088903-7 (e-book).
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  • Otávio Bueno* and Steven French.**Applying Mathematics: Immersion, Inference, Interpretation. [REVIEW]Anthony F. Peressini - 2020 - Philosophia Mathematica 28 (1):116-127.
    Otávio Bueno* * and Steven French.** ** Applying Mathematics: Immersion, Inference, Interpretation. Oxford University Press, 2018. ISBN: 978-0-19-881504-4 978-0-19-185286-2. doi:10.1093/oso/9780198815044. 001.0001. Pp. xvii + 257.
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  • (1 other version)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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  • (1 other version)Sets Completely Creative Via Recursive Permutations.Bruce M. Horowitz - 1978 - Mathematical Logic Quarterly 24 (25‐30):445-452.
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  • Logique mathématique et philosophie des mathématiques.Yvon Gauthier - 1971 - Dialogue 10 (2):243-275.
    Pour le philosophe intéressé aux structures et aux fondements du savoir théorétique, à la constitution d'une « méta-théorétique «, θεωρíα., qui, mieux que les « Wissenschaftslehre » fichtéenne ou husserlienne et par-delà les débris de la métaphysique, veut dans une intention nouvelle faire la synthèse du « théorétique », la logique mathématique se révèle un objet privilégié.
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  • (1 other version)Extensions of the constructive ordinals.Wayne Richter - 1965 - Journal of Symbolic Logic 30 (2):193-211.
    Kleene [5] mentions two ways of extending the constructive ordinals. The first is by relativizing the setOof notations for the constructive ordinals, using fundamental sequences which are partial recursive inO. In this way we obtain the setOOwhich provides notations for the ordinals less than ω1O. Continuing the process, the sequenceO,OO,, … and the corresponding ordinalsare obtained. A second possibility is to define higher number classes in which partial recursive functions are used at limit ordinals to provide an “accessibility” mapping from (...)
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  • The concept of truth in a finite universe.Panu Raatikainen - 2000 - Journal of Philosophical Logic 29 (6):617-633.
    The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.
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  • Formalization, primitive concepts, and purity: Formalization, primitive concepts, and purity.John T. Baldwin - 2013 - Review of Symbolic Logic 6 (1):87-128.
    We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are (...)
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  • The word problem for cancellation semigroups with zero.Yuri Gurevich & Harry R. Lewis - 1984 - Journal of Symbolic Logic 49 (1):184-191.
    By theword problemfor some class of algebraic structures we mean the problem of determining, given a finite setEof equations between words and an additional equationx=y, whetherx=ymust hold in all structures satisfying each member ofE. In 1947 Post [P] showed the word problem for semigroups to be undecidable. This result was strengthened in 1950 by Turing, who showed the word problem to be undecidable forcancellation semigroups,i.e. semigroups satisfying thecancellation propertyNovikov [N] eventually showed the word problem for groups to be undecidable.In 1966 (...)
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  • Quantum theory and consciousness.David L. Wilson - 1993 - Behavioral and Brain Sciences 16 (3):615-616.
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  • (1 other version)Equivalence of some Hierarchies of Primitive Recursive Functions.Keith Harrow - 1979 - Mathematical Logic Quarterly 25 (25‐29):411-418.
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  • Parsimony hierarchies for inductive inference.Andris Ambainis, John Case, Sanjay Jain & Mandayam Suraj - 2004 - Journal of Symbolic Logic 69 (1):287-327.
    Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e., within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computablefunction is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. (...)
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  • Comments on `two undecidable problems of analysis'.Bruno Scarpellini - 2003 - Minds and Machines 13 (1):79-85.
    We first discuss some technical questions which arise in connection with the construction of undecidable propositions in analysis, in particular in connection with the notion of the normal form of a function representing a predicate. Then it is stressed that while a function f(x) may be computable in the sense of recursive function theory, it may nevertheless have undecidable properties in the realm of Fourier analysis. This has an implication for a conjecture of Penrose's which states that classical physics is (...)
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  • (1 other version)Limiting recursion.E. Mark Gold - 1965 - Journal of Symbolic Logic 30 (1):28-48.
    A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in (...)
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  • (1 other version)Elementary Formal Systems for Hyperarithmetical Relations.Melvin Fitting - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (1-6):25-30.
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  • How subtle is Gödel's theorem? More on Roger Penrose.Martin Davis - 1993 - Behavioral and Brain Sciences 16 (3):611-612.
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  • How complicated is the set of stable models of a recursive logic program?W. Marek, A. Nerode & J. Remmel - 1992 - Annals of Pure and Applied Logic 56 (1-3):119-135.
    Gelfond and Lifschitz proposed the notion of a stable model of a logic program. We establish that the set of all stable models in a Herbrand universe of a recursive logic program is, up to recursive renaming, the set of all infinite paths of a recursive, countably branching tree, and conversely. As a consequence, the problem, given a recursive logic program, of determining whether it has at least one stable model, is Σ11-complete. Due to the equivalences established in the authors' (...)
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  • Functional interpretations of feasibly constructive arithmetic.Stephen Cook & Alasdair Urquhart - 1993 - Annals of Pure and Applied Logic 63 (2):103-200.
    A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning IS12, (...)
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  • Finitary inductively presented logics.Solomon Feferman - manuscript
    A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems)met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PRA. The aims of this work are (i)conceptual, (ii)pedagogical and (iii)practical. The system FS0 serves under (i)and (ii)as a theoretical framework for the formalization of metamathematics. The general approach (...)
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  • Wittgenstein and the Logic of Inference.Jan Zwicky - 1982 - Dialogue 21 (4):671-692.
    TheTractatusfirst appeared in 1921, the same year that Post's “Introduction to a General Theory of Elementary Propositions” appeared in theAmerican Journal of Mathematics. As the latter is the first piece clearly to present and exploit the distinction between a deductive system and a truth-functional interpretation of such a system, we may conclude that Wittgenstein's views had been arrived at somewhat before a variety of logical concepts had received the clarification and refinement incipient on the now taken-for-granted distinction between proof and (...)
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  • Two undecidable problems of analysis.Bruno Scarpellini - 2003 - Minds and Machines 13 (1):49-77.
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  • A rudimentary definition of addition.R. W. Ritchie - 1965 - Journal of Symbolic Logic 30 (3):350-354.
    In [S, pp. 77–88], Smullyan introduced the class of rudimentary relations, and showed that they form a basis for the recursively enumerable sets. He also asked [S, p. 81] if the addition and multiplication relations were rudimentary. In this note we answer one of these questions by showing that the addition relation is rudimentary. This result was communicated to Smullyan orally in 1960 and is announced in [S, p. 81, footnote 1]. However, the proof has not yet appeared in print. (...)
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  • Arithmetical Analogues of Productive and Universal Sets.Bruce M. Horowitz - 1982 - Mathematical Logic Quarterly 28 (14-18):203-210.
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  • Dominical categories: recursion theory without elements.Robert A. di Paola & Alex Heller - 1987 - Journal of Symbolic Logic 52 (3):594-635.
    Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic (...)
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  • (1 other version)Equivalence of some Hierarchies of Primitive Recursive Functions.Keith Harrow - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (25-29):411-418.
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  • (1 other version)Effectivizing Inseparability.John Case - 1991 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 37 (7):97-111.
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  • Cognitive mapping and algorithmic complexity: Is there a role for quantum processes in the evolution of human consciousness?Ron Wallace - 1993 - Behavioral and Brain Sciences 16 (3):614-615.
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  • (1 other version)Notes on Formal Theories of Truth.Andrea Cantini - 1989 - Zeitshrift für Mathematische Logik Und Grundlagen der Mathematik 35 (1):97--130.
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  • (1 other version)A note on universal sets.A. H. Lachlan - 1966 - Journal of Symbolic Logic 31 (4):573-574.
    In this note is proved the following:Theorem.Iƒ A × B is universal and one oƒ A, B is r.e. then one of A, B is universal.Letα, τbe 1-argument recursive functions such thatxgoes to, τ) is a map of the natural numbers onto all ordered pairs of natural numbers. A set A of natural numbers is calleduniversalif every r.e. set is reducible to A; A × B is calleduniversalif the set.
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  • Witnessing functions in bounded arithmetic and search problems.Mario Chiari & Jan Krajíček - 1998 - Journal of Symbolic Logic 63 (3):1095-1115.
    We investigate the possibility to characterize (multi) functions that are Σ b i -definable with small i (i = 1, 2, 3) in fragments of bounded arithmetic T 2 in terms of natural search problems defined over polynomial-time structures. We obtain the following results: (1) A reformulation of known characterizations of (multi)functions that are Σ b 1 - and Σ b 2 -definable in the theories S 1 2 and T 1 2 . (2) New characterizations of (multi)functions that are (...)
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  • Injecting inconsistencies into models of pa.Robert M. Solovay - 1989 - Annals of Pure and Applied Logic 44 (1-2):101-132.
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  • (1 other version)Effectivizing Inseparability.John Case - 1991 - Mathematical Logic Quarterly 37 (7):97-111.
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  • (1 other version)Sets Completely Creative Via Recursive Permutations.Bruce M. Horowitz - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (25-30):445-452.
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  • Transfinite induction within Peano arithmetic.Richard Sommer - 1995 - Annals of Pure and Applied Logic 76 (3):231-289.
    The relative strengths of first-order theories axiomatized by transfinite induction, for ordinals less-than 0, and formulas restricted in quantifier complexity, is determined. This is done, in part, by describing the provably recursive functions of such theories. Upper bounds for the provably recursive functions are obtained using model-theoretic techniques. A variety of additional results that come as an application of such techniques are mentioned.
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  • Rudimentary Languages and Second‐Order Logic.Malika More & Frédéric Olive - 1997 - Mathematical Logic Quarterly 43 (3):419-426.
    The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second‐order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing (...)
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  • Are natural languages universal?Robert L. Martin - 1976 - Synthese 32 (3-4):271 - 291.
    We began by distinguishing Tarskian and Fitchean notions of universality in such a way that the claim that no language is universal in the sense of Tarski is compatible with accepting Fitchean universality. Then we examined a proposal involving two truth concepts — one that fit the Fitchean notion and another that followed Tarski's views on truth — finding little advantage in such generosity. We attempted a reformulation of Herzberger's argument for the negative view — the view that no language (...)
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  • (1 other version)Notes on Formal Theories of Truth.Andrea Cantini - 1989 - Mathematical Logic Quarterly 35 (2):97-130.
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  • The categorical and the hypothetical: a critique of some fundamental assumptions of standard semantics.Peter Schroeder-Heister - 2012 - Synthese 187 (3):925-942.
    The hypothetical notion of consequence is normally understood as the transmission of a categorical notion from premisses to conclusion. In model-theoretic semantics this categorical notion is 'truth', in standard proof-theoretic semantics it is 'canonical provability'. Three underlying dogmas, (I) the priority of the categorical over the hypothetical, (II) the transmission view of consequence, and (III) the identification of consequence and correctness of inference are criticized from an alternative view of proof-theoretic semantics. It is argued that consequence is a basic semantical (...)
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  • A Step Towards Absolute Versions of Metamathematical Results.Balthasar Grabmayr - 2024 - Journal of Philosophical Logic 53 (1):247-291.
    There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific (...)
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  • The incompleteness of quantum physics.Euan J. Squires - 1993 - Behavioral and Brain Sciences 16 (3):613-614.
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  • Maximal theories.R. G. Downey - 1987 - Annals of Pure and Applied Logic 33 (C):245-282.
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  • (1 other version)Metarecursive sets.G. Kreisel & Gerald E. Sacks - 1965 - Journal of Symbolic Logic 30 (3):318-338.
    Our ultimate purpose is to give an axiomatic treatment of recursion theory sufficient to develop the priority method. The direct or abstract approach is to keep in mind as clearly as possible the methods actually used in recursion theory, and then to formulate them explicitly. The indirect or experimental approach is to look first for other mathematical theories which seem similar to recursion theory, to formulate the analogies precisely, and then to search for an axiomatic treatment which covers not only (...)
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  • Zwei Unentscheidbare Probleme Der Analysis.Bruno Scarpellini - 1963 - Mathematical Logic Quarterly 9 (18-20):265-289.
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  • Mind the truth: Penrose's new step in the Gödelian argument.Salvatore Guccione - 1993 - Behavioral and Brain Sciences 16 (3):612-613.
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  • Chameleonic languages.Raymond M. Smullyan - 1984 - Synthese 60 (2):201 - 224.
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  • Zur Darstellung koaufzählbarer Prädikate bei Verwendung eines einzigen unbeschränkten Quantors.Michael Deutsch - 1975 - Mathematical Logic Quarterly 21 (1):443-454.
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  • (1 other version)A Basis Theorem for a Class of Two-Way Automata.D. L. Kreider & R. W. Ritchie - 1966 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 12 (1):243-255.
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  • An emperor still without mind.Roger Penrose - 1993 - Behavioral and Brain Sciences 16 (3):616-622.
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  • (1 other version)Elementary Formal Systems for Hyperarithmetical Relations.Melvin Fitting - 1978 - Mathematical Logic Quarterly 24 (1‐6):25-30.
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  • (1 other version)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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