Search results for 'Provability' (try it on Scholar)

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  1.  70
    Holger A. Leuz, Note on Absolute Provability and Cantorian Comprehension.
    We will explicate Cantor’s principle of set existence using the Gödelian intensional notion of absolute provability and John Burgess’ plural logical concept of set formation. From this Cantorian Comprehension principle we will derive a conditional result about the question whether there are any absolutely unprovable mathematical truths. Finally, we will discuss the philosophical significance of the conditional result.
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  2.  28
    Walter Dean (2014). Montague’s Paradox, Informal Provability, and Explicit Modal Logic. Notre Dame Journal of Formal Logic 55 (2):157-196.
    The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory $T\supseteq Q$ over a language containing a predicate $P$ satisfying $P\rightarrow \varphi $ and $T\vdash \varphi \,\therefore\,T\vdash P$ is inconsistent—as a limitative result pertaining to the notions of formal, informal, and constructive provability, in their respective historical contexts. To this end, the paradox is reconstructed in a quantified extension $\mathcal {QLP}$ of Artemov’s logic of proofs. $\mathcal {QLP}$ contains both explicit modalities $t:\varphi (...)
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  3.  4
    Lev Beklemishev & Tommaso Flaminio (2016). Franco Montagna’s Work on Provability Logic and Many-Valued Logic. Studia Logica 104 (1):1-46.
    Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.
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  4.  28
    Paul Égré (2005). The Knower Paradox in the Light of Provability Interpretations of Modal Logic. Journal of Logic, Language and Information 14 (1):13-48.
    This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework (...)
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  5.  6
    Lev D. Beklemishev, David Fernández-Duque & Joost J. Joosten (2014). On Provability Logics with Linearly Ordered Modalities. Studia Logica 102 (3):541-566.
    We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the (...)
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  6.  14
    Mingzhong Cai (2012). Degrees of Relative Provability. Notre Dame Journal of Formal Logic 53 (4):479-489.
    There are many classical connections between the proof-theoretic strength of systems of arithmetic and the provable totality of recursive functions. In this paper we study the provability strength of the totality of recursive functions by investigating the degree structure induced by the relative provability order of recursive algorithms. We prove several results about this proof-theoretic degree structure using recursion-theoretic techniques such as diagonalization and the Recursion Theorem.
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  7.  15
    Vítězslav Švejdar (2003). The Decision Problem of Provability Logic with Only One Atom. Archive for Mathematical Logic 42 (8):763-768.
    The decision problem for provability logic remains PSPACE-complete even if the number of propositional atoms is restricted to one.
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  8.  23
    Thomas F. Icard & Joost J. Joosten (2012). Provability and Interpretability Logics with Restricted Realizations. Notre Dame Journal of Formal Logic 53 (2):133-154.
    The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, the result is the logic of linear frames. However, for the (...)
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  9.  7
    Taishi Kurahashi (2013). On Predicate Provability Logics and Binumerations of Fragments of Peano Arithmetic. Archive for Mathematical Logic 52 (7-8):871-880.
    Solovay proved (Israel J Math 25(3–4):287–304, 1976) that the propositional provability logic of any ∑2-sound recursively enumerable extension of PA is characterized by the propositional modal logic GL. By contrast, Montagna proved in (Notre Dame J Form Log 25(2):179–189, 1984) that predicate provability logics of Peano arithmetic and Bernays–Gödel set theory are different. Moreover, Artemov proved in (Doklady Akademii Nauk SSSR 290(6):1289–1292, 1986) that the predicate provability logic of a theory essentially depends on the choice of a (...)
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  10.  18
    Albert Visser (2008). Closed Fragments of Provability Logics of Constructive Theories. Journal of Symbolic Logic 73 (3):1081-1096.
    In this paper we give a new proof of the characterization of the closed fragment of the provability logic of Heyting's Arithmetic. We also provide a characterization of the closed fragment of the provability logic of Heyting's Arithmetic plus Markov's Principle and Heyting's Arithmetic plus Primitive Recursive Markov's Principle.
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  11.  3
    Paolo Gentilini & P. Gentilini (1992). Provability Logic in the Gentzen Formulation of Arithmetic. Mathematical Logic Quarterly 38 (1):535-550.
    In this paper are studied the properties of the proofs in PRA of provability logic sentences, i.e. of formulas which are Boolean combinations of formulas of the form PIPRA, where h is the Gödel-number of a sentence in PRA. The main result is a Normal Form Theorem on the proof-trees of provability logic sequents, which states that it is possible to split the proof into an arithmetical part, which contains only atomic formulas and has an essentially intuitionistic character, (...)
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  12.  3
    Vedran Čačić & Domagoj Vrgoč (2013). A Note on Bisimulation and Modal Equivalence in Provability Logic and Interpretability Logic. Studia Logica 101 (1):31-44.
    Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see [1]) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that are modally equivalent (...)
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  13.  1
    Larisa Maksimova (2008). Interpolation and Implicit Definability in Extensions of the Provability Logic. Logic and Logical Philosophy 17 (1-2):129-142.
    The provability logic GL was in the field of interest of A.V. Kuznetsov, who had also formulated its intuitionistic analog—the intuitionisticprovability logic—and investigated these two logics and their extensions.In the present paper, different versions of interpolation and of the Bethproperty in normal extensions of the provability logic GL are considered. Itis proved that in a large class of extensions of GL almost all versions of interpolation and of the Beth propertyare equivalent. It follows that in finite slice logics (...)
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  14.  26
    Dov M. Gabbay (2009). Modal Provability Foundations for Argumentation Networks. Studia Logica 93 (2/3):181 - 198.
    Given an argumentation network we associate with it a modal formula representing the 'logical content' of the network. We show a one-to-one correspondence between all possible complete Caminada labellings of the network and all possible models of the formula.
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  15. Sergei N. Artemov (2001). Explicit Provability and Constructive Semantics. Bulletin of Symbolic Logic 7 (1):1-36.
    In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics (...)
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  16.  61
    George Boolos (1993). The Logic of Provability. Cambridge University Press.
    This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (CUP, 1979). Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self referential' (...)
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  17. Jan Krajíček (2004). Dual Weak Pigeonhole Principle, Pseudo-Surjective Functions, and Provability of Circuit Lower Bounds. Journal of Symbolic Logic 69 (1):265 - 286.
    This article is a continuation of our search for tautologies that are hard even for strong propositional proof systems like EF, cf. [14, 15]. The particular tautologies we study, the τ-formulas, are obtained from any ᵊ/poly map g; they express that a string is outside of the range of g. Maps g considered here are particular pseudorandom generators. The ultimate goal is to deduce the hardness of the τ-formulas for at least EF from some general, plausible computational hardness hypothesis. In (...)
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  18.  26
    Rajeev Gore & Revantha Ramanayake (2012). Valentini's Cut-Elimination for Provability Logic Resolved. Review of Symbolic Logic 5 (2):212-238.
    In 1983, Valentini presented a syntactic proof of cut elimination for a sequent calculus GLSV for the provability logic GL where we have added the subscript V for “Valentini”. The sequents in GLSV were built from sets, as opposed to multisets, thus avoiding an explicit contraction rule. From a syntactic point of view, it is more satisfying and formal to explicitly identify the applications of the contraction rule that are ‘hidden’ in these set based proofs of cut elimination. There (...)
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  19. George S. Boolos (2010). The Logic of Provability. Cambridge University Press.
    This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos (...)
     
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  20.  3
    Lev D. Beklemishev (2004). Provability Algebras and Proof-Theoretic Ordinals, I. Annals of Pure and Applied Logic 128 (1-3):103-123.
    We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis and the work which has (...)
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  21.  13
    Lev D. Beklemishev (2010). Kripke Semantics for Provability Logic GLP. Annals of Pure and Applied Logic 161 (6):756-774.
    A well-known polymodal provability logic inlMMLBox due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of inlMMLBox is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for inlMMLBox . First, we isolate a certain (...)
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  22.  7
    Lev Beklemishev & David Gabelaia (2013). Topological Completeness of the Provability Logic GLP. Annals of Pure and Applied Logic 164 (12):1201-1223.
    Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.
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  23. Dick de Jongh & Frank Veltman, Provability Logics for Relative Interpretability.
    In this paper the system IL for relative interpretability described in Visser is studied.1 In IL formulae A|> B are added to the provability logic L. The intended interpretation of a formula A|> B in an theory T is: T + B is relatively interpretable in T + A. The system has been shown to be sound with respect to such arithmetical interpretations. As axioms for IL we take the usual axioms A→ A and → A for the (...) logic L and its rules, modus ponens and necessitation, plus the axioms: → ∧ → ∧ → → A|>A With respect to priority of parentheses |> is treated as →. (shrink)
     
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  24. David Fernández-Duque & Joost J. Joosten (2013). Models of Transfinite Provability Logic. Journal of Symbolic Logic 78 (2):543-561.
    For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary (...)
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  25. Panu Raatikainen (2005). Truth and Provability: A Comment on Redhead. British Journal for the Philosophy of Science 56 (3):611-613.
    Michael Redhead's recent argument aiming to show that humanly certifiable truth outruns provability is critically evaluated. It is argued that the argument is at odds with logical facts and fails.
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  26.  25
    Konstantin N. Ignatiev (1993). On Strong Provability Predicates and the Associated Modal Logics. Journal of Symbolic Logic 58 (1):249-290.
    PA is Peano Arithmetic. Pr(x) is the usual Σ1-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as Pr(·) but is not Σ1. An example: Q is ω-provable if PA + ¬ Q is ω-inconsistent (Boolos [4]). In [5] Dzhaparidze introduced a joint provability logic for iterated ω-provability and obtained its arithmetical completeness. In this paper we prove some further modal properties of Dzhaparidze's logic, e.g., the fixed point (...)
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  27.  6
    Leo Esakia (2009). Around Provability Logic. Annals of Pure and Applied Logic 161 (2):174-184.
    We present some results on algebraic and modal analysis of polynomial distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel–Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system.
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  28.  9
    V. V. Rybakov (1990). Logical Equations and Admissible Rules of Inference with Parameters in Modal Provability Logics. Studia Logica 49 (2):215 - 239.
    This paper concerns modal logics of provability — Gödel-Löb systemGL and Solovay logicS — the smallest and the greatest representation of arithmetical theories in propositional logic respectively. We prove that the decision problem for admissibility of rules (with or without parameters) inGL andS is decidable. Then we get a positive solution to Friedman''s problem forGL andS. We also show that A. V. Kuznetsov''s problem of the existence of finite basis for admissible rules forGL andS has a negative solution. Afterwards (...)
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  29.  28
    Francesca Poggiolesi (2009). A Purely Syntactic and Cut-Free Sequent Calculus for the Modal Logic of Provability. Review of Symbolic Logic 2 (4):593-611.
    In this paper we present a sequent calculus for the modal propositional logic GL (the logic of provability) obtained by means of the tree-hypersequent method, a method in which the metalinguistic strength of hypersequents is improved, so that we can simulate trees shapes. We prove that this sequent calculus is sound and complete with respect to the Hilbert-style system GL, that it is contraction free and cut free and that its logical and modal rules are invertible. No explicit semantic (...)
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  30.  6
    Lev D. Beklemishev (1994). On Bimodal Logics of Provability. Annals of Pure and Applied Logic 68 (2):115-159.
    We investigate the bimodal logics sound and complete under the interpretation of modal operators as the provability predicates in certain natural pairs of arithmetical theories . Carlson characterized the provability logic for essentially reflexive extensions of theories, i.e. for pairs similar to . Here we study pairs of theories such that the gap between and is not so wide. In view of some general results concerning the problem of classification of the bimodal provability logics we are particularly (...)
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  31. Frank Veltman, Provability Logics for Relative Interpretability.
    In this paper the system IL for relative interpretability described in Visser (1988) is studied.1 In IL formulae A|> B (read: A interprets B) are added to the provability logic L. The intended interpretation of a formula A|.
     
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  32.  6
    Albert Visser (1992). An Inside View of Exp; or, the Closed Fragment of the Provability Logic of Iδ0 + Ω1 with a Propositional Constant for $\Operatorname{Exp}$. [REVIEW] Journal of Symbolic Logic 57 (1):131 - 165.
    In this paper I give a characterization of the closed fragment of the provability logic of I ▵0 + EXP with a propositional constant for EXP. In three appendices many details on arithmetization are provided.
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  33.  21
    Sergei Artemov & Giorgie Dzhaparidze (1990). Finite Kripke Models and Predicate Logics of Provability. Journal of Symbolic Logic 55 (3):1090-1098.
    The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that $PA \nvdash fR$ . This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding "the predicate part" as a (...)
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  34.  5
    Rineke Verbrugge & Alessandro Berarducci (1993). On the Provability Logic of Bounded Arithmetic. Annals of Pure and Applied Logic 61 (1-2):75-93.
    Let PLω be the provability logic of IΔ0 + ω1. We prove some containments of the form L ⊆ PLω < Th(C) where L is the provability logic of PA and Th(C) is a suitable class of Kripke frames.
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  35.  8
    Frank Wolter (1998). All Finitely Axiomatizable Subframe Logics Containing the Provability Logic CSM_{0} Are Decidable. Archive for Mathematical Logic 37 (3):167-182.
    In this paper we investigate those extensions of the bimodal provability logic ${\vec CSM}_{0}$ (alias ${\vec PRL}_{1}$ or ${\vec F}^{-})$ which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing ${\vec CSM}_{0}$ are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are (...)
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  36.  2
    Albert Visser (1995). A Course on Bimodal Provability Logic. Annals of Pure and Applied Logic 73 (1):109-142.
    In this paper we study 1. the frame-theory of certain bimodal provability logics involving the reflection principle and we study2. certain specific bimodal logics with a provability predicate for a subtheory of Peano arithmetic axiomatized by a non-standardly finite number of axioms.
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  37.  4
    Samuel R. Buss (1991). The Undecidability of K-Provability. Annals of Pure and Applied Logic 53 (1):75-102.
    Buss, S.R., The undecidability of k-provability, Annals of Pure and Applied Logic 53 75-102. The k-provability problem is, given a first-order formula ø and an integer k, to determine if ø has a proof consisting of k or fewer lines . This paper shows that the k-provability problem for the sequent calculus is undecidable. Indeed, for every r.e. set X there is a formula ø and an integer k such that for all n,ø has a proof of (...)
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  38.  9
    Giorgie Dzhaparidze (1990). Decidable and Enumerable Predicate Logics of Provability. Studia Logica 49 (1):7 - 21.
    Predicate modal formulas are considered as schemata of arithmetical formulas, where is interpreted as the standard formula of provability in a fixed sufficiently rich theory T in the language of arithmetic. QL T(T) and QL T are the sets of schemata of T-provable and true formulas, correspondingly. Solovay's well-known result — construction an arithmetical counterinterpretation by Kripke countermodel — is generalized on the predicate modal language; axiomatizations of the restrictions of QL T(T) and QL T by formulas, which contain (...)
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  39.  29
    Sergei Artëmov & Franco Montagna (1994). On First-Order Theories with Provability Operator. Journal of Symbolic Logic 59 (4):1139-1153.
    In this paper the modal operator "x is provable in Peano Arithmetic" is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.
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  40.  34
    David Fair (1984). Provability and Mathematical Truth. Synthese 61 (3):363 - 385.
    An insight, Central to platonism, That the objects of pure mathematics exist "in some sense" is probably essential to any adequate account of mathematical truth, Mathematical language, And the objectivity of the mathematical enterprise. Yet a platonistic ontology makes how we can come to know anything about mathematical objects and how we use them a dark mystery. In this paper I propose a framework for reconciling a representation-Relative provability theory of mathematical truth with platonism's valid insights. Besides helping to (...)
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  41.  7
    Peter Roeper (2003). Giving an Account of Provability Within a Theory. Philosophia Mathematica 11 (3):332-340.
    This paper offers a justification of the ‘Hilbert-Bernays Derivability Conditions’ by considering what is required of a theory which gives an account of provability in itself.
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  42.  1
    Tatiana Yavorskaya (2001). Logic of Proofs and Provability. Annals of Pure and Applied Logic 113 (1-3):345-372.
    In the paper the joint Logic of Proofs and Provability is presented that incorporates both the modality □ for provability 287–304) and the proof operator tF representing the proof predicate “t is a proof of F” . The obtained system naturally includes both the modal logic of provability GL and Artemov's Logic of Proofs . The presence of the modality □ requires two new operations on proofs that together with operations of allow to realize all the invariant (...)
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  43.  18
    Giorgie Dzhaparidze (1991). Predicate Provability Logic with Non-Modalized Quantifiers. Studia Logica 50 (1):149 - 160.
    Predicate modal formulas with non-modalized quantifiers (call them Q-formulas) are considered as schemata of arithmetical formulas, where is interpreted as the provability predicate of some fixed correct extension T of arithmetic. A method of constructing 1) non-provable in T and 2) false arithmetical examples for Q-formulas by Kripke-like countermodels of certain type is given. Assuming the means of T to be strong enough to solve the (undecidable) problem of derivability in QGL, the Q-fragment of the predicate version of the (...)
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  44.  14
    L. D. Beklemishev (1991). Provability Logics for Natural Turing Progressions of Arithmetical Theories. Studia Logica 50 (1):107 - 128.
    Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced. The corresponding arithmetical completeness theorem is proved.
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  45. Rostislav E. Yavorsky (2001). Provability Logics with Quantifiers on Proofs. Annals of Pure and Applied Logic 113 (1-3):373-387.
    We study here extensions of the Artemov's logic of proofs in the language with quantifiers on proof variables. Since the provability operator □ A could be expressed in this language by the formula u[u]A, the corresponding logic naturally extends the well-known modal provability logic GL. Besides, the presence of quantifiers on proofs allows us to study some properties of provability not covered by the propositional logics.In this paper we study the arithmetical complexity of the provability logic (...)
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  46. William M. Farmer (1991). A Unification-Theoretic Method for Investigating the K-Provability Problem. Annals of Pure and Applied Logic 51 (3):173-214.
    The k-provability for an axiomatic system A is to determine, given an integer k 1 and a formula in the language of A, whether or not there is a proof of in A containing at most k lines. In this paper we develop a unification-theoretic method for investigating the k-provability problem for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata and a finite number of rules of inference. We show that the (...)
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  47.  10
    Shelley L. Trianosky-Stillwell (1983). 'Necessity' and 'Provability' in the Later Wittgenstein. History and Philosophy of Logic 4 (1-2):39-61.
    I present a new interpretation of Wittgenstein's later philosophy of logic and mathematics. This interpretation, like others, emphasizes Wittgenstein's attempt to reconcile platonistic and constructivistic approaches. But, unlike other interpretations, mine explains that attempt in terms of Wittgenstein's position about the relations between our concepts of necessity and provability. If what I say here is correct, then we can rescue Wittgenstein from the charge of naive relativism. For his relativism extends only to provability, and not to necessity.
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  48.  4
    Taishi Kurahashi (2012). Arithmetical Interpretations and Kripke Frames of Predicate Modal Logic of Provability. Review of Symbolic Logic 6 (1):1-18.
    Solovay proved the arithmetical completeness theorem for the system GL of propositional modal logic of provability. Montagna proved that this completeness does not hold for a natural extension QGL of GL to the predicate modal logic. Let Th(QGL) be the set of all theorems of QGL, Fr(QGL) be the set of all formulas valid in all transitive and conversely well-founded Kripke frames, and let PL(T) be the set of all predicate modal formulas provable in Tfor any arithmetical interpretation. Montagna’s (...)
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  49.  12
    Ulrich Nortmann (2001). How to Extend the Dialogical Approach to Provability Logic. Synthese 127 (1-2):95 - 103.
    The core ideas of the dialogicalapproach to modal propositional logic are explainedby means of an elementary example. Subsequently,ways of extending this approach to the system G ofso-called provability logic are checked, therebyraising the question whether the dialogician is inneed of shaping his Nichtverzögerungsregel(non-delay-rule), in order to get it sufficiently precise,in different ways for different modal systems.
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  50.  1
    Konstantin N. Ignatiev (1993). The Provability Logic for Σ1-Interpolability. Annals of Pure and Applied Logic 64 (1):1-25.
    We say that two arithmetical formulas A, B have the Σ1-interpolation property if they have an ‘interpolant’ σ, i.e., a Σ1 formula such that the formulas A→σ and σ→B are provable in Peano Arithmetic PA. The Σ1-interpolability predicate is just a formalization of this property in the language of arithmetic.Using a standard idea of Gödel, we can associate with this predicate its provability logic, which is the set of all formulas that express arithmetically valid principles in the modal language (...)
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