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  1. Self-Referential Theories.Samuel A. Alexander - 2020 - Journal of Symbolic Logic 85 (4):1687-1716.
    We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.
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  2. Another Problem in Possible World Semantics.Yifeng Ding & Wesley H. Holliday - 2020 - In Nicola Olivetti & Rineke Verbrugge (eds.), Advances in Modal Logic, Vol. 13. London: College Publications.
    In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the (...)
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  3. The Logic of Turing Progressions.Eduardo Hermo Reyes & Joost J. Joosten - 2020 - Notre Dame Journal of Formal Logic 61 (1):155-180.
    Turing progressions arise by iteratedly adding consistency statements to a base theory. Different notions of consistency give rise to different Turing progressions. In this paper we present a logic that generates exactly all relations that hold between these different Turing progressions given a particular set of natural consistency notions. Thus, the presented logic is proven to be arithmetically sound and complete for a natural interpretation, named the formalized Turing progressions interpretation.
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  4. A Note on Derivability Conditions.Taishi Kurahashi - 2020 - Journal of Symbolic Logic 85 (3):1224-1253.
    We investigate relationships between versions of derivability conditions for provability predicates. We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by derivability conditions. First, we classify already known versions of the second incompleteness theorem, and exhibit some new sets of conditions which are sufficient for unprovability of Hilbert–Bernays’ consistency statement. Secondly, we improve Buchholz’s schematic proof of provable $\Sigma_1$ -completeness. Then among other things, we show that Hilbert–Bernays’ conditions and Löb’s conditions are (...)
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  5. Non–Well-Founded Derivations in the Gödel-Löb Provability Logic.Daniyar Shamkanov - 2020 - Review of Symbolic Logic 13 (4):776-796.
    We consider Hilbert-style non–well-founded derivations in the Gödel-Löb provability logic GL and establish that GL with the obtained derivability relation is globally complete for algebraic and neighbourhood semantics.
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  6. Complete Additivity and Modal Incompleteness.Wesley H. Holliday & Tadeusz Litak - 2019 - Review of Symbolic Logic 12 (3):487-535.
    In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem, “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question (...)
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  7. The Knower Paradox in the Light of Provability Interpretations of Modal Logic.Paul Égré - 2004 - 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 of first-order (...)
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  8. Johdatus modaalilogiikkaan.Veikko Rantala & Ari Virtanen - 2004 - Gaudeamus.
    The book studies philosophical and mathematical-logical problems of modal notions. Its starting points are possible worlds semantics and Kripke models, and it also concentrates on proof-theoretic methods.
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  9. Can a Turing Machine Know That the Gödel Sentence is True?Storrs McCall - 1999 - Journal of Philosophy 96 (10):525-532.
  10. The Logic of Provability.George S. Boolos - 1993 - 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 does (...)
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  11. On Strong Provability Predicates and the Associated Modal Logics.Konstantin N. Ignatiev - 1993 - 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 property and the Craig (...)
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  12. The Modal Logic of Pure Provability.Samuel R. Buss - 1990 - Notre Dame Journal of Formal Logic 31 (2):225-231.
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  13. Provability Logics for Relative Interpretability.Frank Veltman & Dick De Jongh - 1990 - In Petio Petrov Petkov (ed.), Mathematical Logic. Proceedings of the Heyting '88 Summer School. New York, NY, USA: pp. 31-42.
    In this paper the system IL for relative interpretability is studied.
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  14. On Modal Systems Having Arithmetical Interpretations.Arnon Avron - 1984 - Journal of Symbolic Logic 49 (3):935-942.
  15. On Systems of Modal Logic with Provability Interpretations.George Boolos - 1980 - Theoria 46 (1):7-18.
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  16. Provability, Truth, and Modal Logic.George Boolos - 1980 - Journal of Philosophical Logic 9 (1):1 - 7.
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