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  1. Chrysippus Confronts the Liar: The Case for Stoic Cassationism.Michael Papazian - 2012 - History and Philosophy of Logic 33 (3):197-214.
    The Stoic philosopher Chrysippus wrote extensively on the liar paradox, but unfortunately the extant testimony on his response to the paradox is meager and mainly hostile. Modern scholars, beginning with Alexander Rüstow in the first decade of the twentieth century, have attempted to reconstruct Chrysippus? solution. Rüstow argued that Chrysippus advanced a cassationist solution, that is, one in which sentences such as ?I am speaking falsely? do not express propositions. Two more recent scholars, Walter Cavini and Mario Mignucci, have rejected (...)
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  • Knowledge and belief.Jaakko Hintikka - 1962 - Ithaca, N.Y.,: Cornell University Press.
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  • Saving truth from paradox.Hartry Field - 2008 - New York: Oxford University Press.
    A selective background -- Broadly classical approaches -- Paracompleteness -- More on paracomplete solutions -- Paraconsistent dialetheism.
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  • Solution of a problem of Leon Henkin.M. H. Löb - 1955 - Journal of Symbolic Logic 20 (2):115-118.
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  • (1 other version)A logical analysis of some value concepts.Frederic Fitch - 1963 - Journal of Symbolic Logic 28 (2):135-142.
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  • (1 other version)Paradoxes.R. M. Sainsbury - 1990 - Philosophy 65 (251):106-111.
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  • (1 other version)Three Grades of Modal Involvement.W. V. Quine - 1953 - Proceedings of the XIth International Congress of Philosophy 14:65-81.
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  • The Logic of Provability.George Boolos - 1993 - Cambridge and New York: 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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  • Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability.Richard Montague - 1963 - Journal of Symbolic Logic 40 (4):600-601.
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  • (1 other version)Deontic Logic.Paul McNamara - 2006 - In Dov Gabbay & John Woods (eds.), The Handbook of the History of Logic, vol. 7: Logic and the Modalities in the Twentieth Century. Elsevier Press. pp. 197-288.
    Overview of fundamental work in deontic logic.
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  • The Paradox of the Knower revisited.Walter Dean & Hidenori Kurokawa - 2014 - Annals of Pure and Applied Logic 165 (1):199-224.
    The Paradox of the Knower was originally presented by Kaplan and Montague [26] as a puzzle about the everyday notion of knowledge in the face of self-reference. The paradox shows that any theory extending Robinson arithmetic with a predicate K satisfying the factivity axiom K → A as well as a few other epistemically plausible principles is inconsistent. After surveying the background of the paradox, we will focus on a recent debate about the role of epistemic closure principles in the (...)
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  • Gödel's Incompleteness Theorems.Panu Raatikainen - 2013 - The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (Ed.).
    Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot (...)
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  • Conflicting Rules and Paradox.Colin Johnston - 2014 - Philosophy and Phenomenological Research 88 (2):410-433.
    First paragraph: This paper seeks to understand various paradoxes as cases of conflicting rules. In particular, the ambition is to outline a new perspective on and response to the Liar -- though it will take us a while to get that far. We begin in Section 1 with an account of simple rule confliction. Section 2 then brings this account to bear on a paradox, the Secretary Liberation Paradox, which is readily seen to involve conflicting rules. Finally in Section 3 (...)
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  • A paradox regained.D. Kaplan & R. Montague - 1960 - Notre Dame Journal of Formal Logic 1 (3):79-90.
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  • Provability logic.Rineke Verbrugge - 2008 - Stanford Encyclopedia of Philosophy.
    -/- Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. As a modal logic, provability logic has been studied since the early seventies, and has had important applications in the foundations of mathematics. -/- From a philosophical point of view, provability logic is interesting because (...)
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  • (1 other version)Paradoxes.R. M. Sainsbury - 1991 - Philosophy and Phenomenological Research 51 (2):455-459.
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  • (2 other versions)Philosophy of Logics.Susan Haack - 1978 - Philosophy 56 (217):435-436.
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  • Provability Interpretations of Modal Logic.Robert M. Solovay - 1981 - Journal of Symbolic Logic 46 (3):661-662.
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  • Deontic Tense Logic With Historical Necessity, Frame Constants, and a Solution to the Epistemic Obligation Paradox.Lennart Åqvist - 2014 - Theoria 80 (4):319-349.
    In an earlier paper by the author, Åqvist , I presented an approach to the logic of historical necessity, or inevitability, in the sense of a “two-dimensional” combination of tense and modal logic for worlds, or histories, with the same time order, known as T × W logic. Distinctive features of that approach were, apart from its two-dimensionality, its being based on discrete and finite time, and its use of so-called systematic frame constants in order to enable us to indicate (...)
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  • The Revision Theory of Truth.Anil Gupta & Nuel D. Belnap - 1993 - MIT Press.
    In this rigorous investigation into the logic of truth Anil Gupta and Nuel Belnap explain how the concept of truth works in both ordinary and pathological..
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  • (1 other version)Dialetheism.Francesco Berto, Graham Priest & Zach Weber - 2008 - Stanford Encyclopedia of Philosophy 2018 (2018).
    A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.
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  • (1 other version)Three Grades of Modal Involvement.W. V. Quine - 1976 - In Willard Van Orman Quine (ed.), The ways of paradox, and other essays. Cambridge: Harvard University Press. pp. 158-176.
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  • A Principled Solution to Fitch’s Paradox.Igor Douven - 2005 - Erkenntnis 62 (1):47-69.
    To save antirealism from Fitch's Paradox, Tennant has proposed to restrict the scope of the antirealist principle that all truths are knowable to truths that can be consistently assumed to be known. Although the proposal solves the paradox, it has been accused of doing so in an ad hoc manner. This paper argues that, first, for all Tennant has shown, the accusation is just; second, a restriction of the antirealist principle apparently weaker than Tennat's yields a non-ad hoc solution to (...)
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  • Was Sind und was Sollen Die Zahlen?Richard Dedekind - 1888 - Cambridge University Press.
    This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.
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  • Montague’s Paradox, Informal Provability, and Explicit Modal Logic.Walter Dean - 2014 - 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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  • On bimodal logics of provability.Lev D. Beklemishev - 1994 - 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 interested in such (...)
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  • (1 other version)Self-Reference and Modal Logic.George Boolos & C. Smorynski - 1988 - Journal of Symbolic Logic 53 (1):306.
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  • A mathematical incompleteness in Peano arithmetic.Jeff Paris & Leo Harrington - 1977 - In Jon Barwise (ed.), Handbook of mathematical logic. New York: North-Holland. pp. 90--1133.
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  • Some remarks on the notion of proof.John Myhill - 1960 - Journal of Philosophy 57 (14):461-471.
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  • 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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  • The paradox of the knower.C. Anthony Anderson - 1983 - Journal of Philosophy 80 (6):338-355.
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  • (1 other version)Metamathematics of First-Order Arithmetic.P. Hájek & P. Pudlák - 2000 - Studia Logica 64 (3):429-430.
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  • The paradox of the knower without epistemic closure?Gabriel Uzquiano - 2004 - Mind 113 (449):95-107.
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  • An immaculate conception of modality or how to confuse use and mention.Brian Skyrms - 1978 - Journal of Philosophy 75 (7):368-387.
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  • Three Different Solutions to the Knower Paradox.Francesca Poggiolesi - 2007 - Annali Del Dipartimento di Filosofia 13:147-164.
    In this paper I shall present three solutions to the Knower Paradox which, despite important points in common, differ in several respects. The first solution, proposed by C. A. Anderson [1] is a hierarchical solution, developed in the framework of first-order arithmetic. However I will try toshow that this solution is based on an incorrect argument. The second solution, inspired by a book of R.M. Smullyan [14], is developed in the framework of modal logic and it is based on the (...)
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  • The paradox of the knower without epistemic closure.Charles B. Cross - 2001 - Mind 110 (438):319-333.
    In this essay I present a new version of the Paradox of the Knower and show that this new paradox vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. I conclude by assessing the use of the Knower in arguments against syntactical treatments of knowledge.
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  • (1 other version)Self-Reference and Modal Logic.[author unknown] - 1987 - Studia Logica 46 (4):395-398.
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  • (1 other version)Metamathematics of First-Order Arithmetic.Petr Hajék & Pavel Pudlák - 1994 - Studia Logica 53 (3):465-466.
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  • Glauben, Wissen Und Wahrscheinlichkeit. Systeme Der Epistemischen Logik.Wolfgang Lenzen - 1985 - Erkenntnis 23 (1):97-112.
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  • An Introduction to Gödel's Theorems.Peter Smith - 2009 - Bulletin of Symbolic Logic 15 (2):218-222.
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  • (2 other versions)Philosophy of Logics.Susan Haack - 1980 - Mind 89 (354):303-306.
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  • (2 other versions)Philosophy of Logics.Susan Haack - 1978 - Critica 14 (42):112-119.
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