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Summary A liar paradox is generated by a sentence or proposition (or any truth bearer more generally) that says that it is false.  If we use the name 'S' for the sentence ‘S is false’, then that very sentence says of itself that it is false. We can reason intuitively that if it is true, then what it says is true, namely that it is false. So it is true that it is false, or, more directly, it is false. On the other hand, if it is false, then what it says is false, namely that it is false. So it is false that it is false, or, more directly, it is true. Thus, we derive that it is false from the assumption that it is true, and we derive that it is true from the assumption that it is false. It takes just a couple of steps from here to the claim that S is both true and false.  The above reasoning relies on the following two principles regarding truth (for some class of sentences p of which S is a member): (i) if p is true, then p, and (ii) if p, then p is true.  It also relies on principles of classical (and intuitionistic) logic.  Approaches to the liar paradox usually reject one of the principles of truth, one of the logical principles, or find some defect in S and any other sentence like it.  The liar paradox is closely related to several other paradoxes associated with truth, including Curry's paradox and Yablo's paradox.
Key works The classic work on the liar paradox in analytic philosophy is Tarski 1936. Kripke 1975 solidified the importance of logical rigor in investigations of the liar paradox. Early contextual approaches include Parsons 1974 and Burge 1979Priest 1979 and Chihara 1979 initiated the inconsistency approach. Gupta 1982 and Herzberger 1982 were the first to offer revision theories of truth. 
Introductions Introductory works include the Stanford Encyclopedia of Philosophy entry: Beall & Glanzberg 2010
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  1. Sloman Aaron (1971). Tarski, Frege and the Liar Paradox. Philosophy 46 (176):133-.
    A.1. Some philosophers, including Tarski and Russell, have concluded from a study of various versions of the Liar Paradox ‘that there must be a hierarchy of languages, and that the words “true” and “false”, as applied to statements in any given language, are themselves words belonging to a language of higher order’. In his famous essay on truth Tarski claimed that ‘colloquial’ language is inconsistent as a result of its property of ‘universality’: that is, whatever can be said at all (...)
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  2. Jordi Valor Abad (2010). Saving Truth From Paradox, by Hartry Field. Disputatio.
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  3. Jordi Valor Abad (2008). The Inclosure Scheme and the Solution to the Paradoxes of Self-Reference. Synthese 160 (2):183 - 202.
    All paradoxes of self-reference seem to share some structural features. Russell in 1908 and especially Priest nowadays have advanced structural descriptions that successfully identify necessary conditions for having a paradox of this kind. I examine in this paper Priest’s description of these paradoxes, the Inclosure Scheme (IS), and consider in what sense it may help us understand and solve the problems they pose. However, I also consider the limitations of this kind of structural descriptions and give arguments against Priest’s use (...)
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  4. Jordi Valor Abad & José Martínez Fernández (2009). A Failed "Cassatio": Goldstein on the Liar. Proceedings of the Aristotelian Society 109 (1pt3):327 - 332.
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  5. Carlos Acosta (2012). On the Physiological Generation of Antinomies and Paradoxes. Mind and Matter 10 (1):75 - 114.
    It is proposed that subconscious retro-predictions in conjunction with brain state update cycles are instrumental in the physiological generation of conscious sensations and perceptions, and in all abstract thought. In this paper the hypothesis is supported by conducting a detailed a re-evaluation of the self-referential statements in Set Theory and Formal Logic known as antinomies. This study concludes that the recursive behavior exhibited by abstract enigmas such as "Russell’s Paradox" is analogous to the oscillations typical of bistable perceptual phenomena.
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  6. Zofia Adamowicz, Leszek Aleksander Kołodziejczyk & Jeff Paris (2012). Truth Definitions Without Exponentiation and the Σ₁ Collection Scheme. Journal of Symbolic Logic 77 (2):649-655.
    We prove that: • if there is a model of I∆₀ + ¬ exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then I∆₀ + ¬ exp +¬BΣ₁ is consistent, • there is a model of I∆₀ Ω₁ + ¬ exp with cofinal Σ₁-definable elements, both a Σ₂ and a ∏₂ truth definition for Σ₁ sentences, and for each n > 2, a Σ n truth definition for Σ n sentences. The latter result is obtained by (...)
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  7. Diederik Aerts, Jan Broekaert & Sonja Smets (1999). The Liar-Paradox in a Quantum Mechanical Perspective. Foundations of Science 4 (2):115-132.
    In this paper we concentrate on the nature of the liar paradox asa cognitive entity; a consistently testable configuration of properties. We elaborate further on a quantum mechanical model (Aerts, Broekaert and Smets, 1999) that has been proposed to analyze the dynamics involved, and we focus on the interpretation and concomitant philosophical picture. Some conclusions we draw from our model favor an effective realistic interpretation of cognitive reality.
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  8. Joseph Agassi (1964). Variations on the Liar's Paradox. Studia Logica 15 (1):237 - 238.
    Line 1: The statement on line one is false. Line 2: All statements on line two are false. p and not-p Line 3: All statements on line 3 are true, or all of them are false. p and not-p Line 4: The statement on line 4 is false, or (p and not-p). Line 5: The statement on line 5 is true if and only if (p and not p). Line 6: All statements on line 6 are false. p. Line 7: (...)
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  9. Wayne Aitken & Jeffrey A. Barrett (2004). Computer Implication and the Curry Paradox. Journal of Philosophical Logic 33 (6):631-637.
    There are theoretical limitations to what can be implemented by a computer program. In this paper we are concerned with a limitation on the strength of computer implemented deduction. We use a version of the Curry paradox to arrive at this limitation.
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  10. Seiki Akama (1996). Curry's Paradox in Contractionless Constructive Logic. Journal of Philosophical Logic 25 (2):135 - 150.
    We propose contractionless constructive logic which is obtained from Nelson's constructive logic by deleting contractions. We discuss the consistency of a naive set theory based on the proposed logic in relation to Curry's paradox. The philosophical significance of contractionless constructive logic is also argued in comparison with Fitch's and Prawitz's systems.
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  11. Ken Akiba (2002). A Deflationist Approach to Indeterminacy and Vagueness. Philosophical Studies 107 (1):69 - 86.
    Deflationists cannot make sense ofthe notion of referential indeterminacybecause they deny the existence of substantivereference. One way for them to make sense ofthe objective existence of linguisticindeterminacy is by embracing theworldly (or objectual) view ofindeterminacy, the view that indeterminacyexists not in reference relations but in the(non-linguistic) world itself. On this view,the entire world is divided into precisified worlds, just as it is dividedinto temporal slices and (arguably) alethicpossible worlds. Supervaluationism proves tobe neutral with respect to the debate betweenthe worldly view (...)
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  12. Ken Akiba (2002). Can Deflationism Allow for Hidden Indeterminacy? Pacific Philosophical Quarterly 83 (3):223–234.
  13. Edwin B. Allaire (1975). Truth. Metaphilosophy 6 (3-4):261-276.
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  14. Robert Almeder (1990). Vacuous Truth. Synthese 85 (3):507 - 524.
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  15. William P. Alston (1996). A Realist Conception of Truth. Cornell University Press.
    William P. Alston formulates and defends a realist conception of truth, which he calls alethic realism (from "aletheia", Greek for "truth").
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  16. Ahmed Alwishah & David Sanson (2009). The Early Arabic Liar: The Liar Paradox in the Islamic World From the Mid-Ninth to the Mid-Thirteenth Centuries Ce. Vivarium (1):97-127.
    We describe the earliest occurrences of the Liar Paradox in the Arabic tradition. e early Mutakallimūn claim the Liar Sentence is both true and false; they also associate the Liar with problems concerning plural subjects, which is somewhat puzzling. Abharī (1200-1265) ascribes an unsatisfiable truth condition to the Liar Sentence—as he puts it, its being true is the conjunction of its being true and false—and so concludes that the sentence is not true. Tūsī (1201-1274) argues that self-referential sentences, like the (...)
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  17. Alan Ross Anderson (1970). St. Paul'€™s Epistle to Titus. In Robert L. Martin (ed.), The Paradox of the Liar. Ridgeview.
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  18. Irving H. Anellis (1984). Perun's Revenge: Understanding Theduxovnaja Kul'tura. Studies in East European Thought 27 (1):1-24.
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  19. G. Aldo Antonelli (2000). Virtuous Circles: From Fixed Points to Revision Rules. In Anil Gupta & Andre Chapuis (eds.), Circularity, Definition, and Truth. Indian Council of Philosophical Research. 1--27.
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  20. Gian Aldo Antonelli (1996). Book Review: Keith Simmons. Universality and the Liar: An Essay on Truth and the Diagonal Argument. [REVIEW] Notre Dame Journal of Formal Logic 37 (1):152-159.
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  21. Gian Aldo Antonelli (1994). The Complexity of Revision. Notre Dame Journal of Formal Logic 35 (1):67-72.
    In this paper we show that the Gupta-Belnap systems S# and S* are П12. Since Kremer has independently established that they are П12-hard, this completely settles the problem of their complexity. The above-mentioned upper bound is established through a reduction to countable revision sequences that is inspired by, and makes use of a construction of McGee.
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  22. Gian Aldo Antonelli (1994). Non-Well-Founded Sets Via Revision Rules. Journal of Philosophical Logic 23 (6):633 - 679.
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  23. Ruben G. Apressyan (2009). Revenge Is Mine, I Will Repay. Russian Studies in Philosophy 48 (2):8-27.
    By placing Jesus's commandment not to "resist evil" in its textual and religious context, the author demonstrates that it should not be interpreted in an absolute sense. Christianity does not condemn vengeance as such but makes it a divine prerogative. A critique is presented of Tolstoy's doctrine of nonviolence.
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  24. B. Armour-Garb (2012). No Consistent Way with Paradox. Analysis 72 (1):66-75.
    In ‘A Consistent Way with Paradox’, Laurence Goldstein (2009) clarifies his solution to the liar, which he touts as revenge immune . In addition, he (Ibid.) responds to one of the objections that Armour-Garb and Woodbridge (2006) raise against certain solutions to the open pair and argues that his proffered solution to the liar family of paradoxes undermines what they (Ibid.) call the dialetheic conjecture . In this paper, after critically evaluating Goldstein’s response to A-G&W, I turn to his proposed (...)
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  25. B. Armour-Garb & Jc Beall (2003). Minimalism and the Dialetheic Challenge. Australasian Journal of Philosophy 81 (3):383 – 401.
    Minimalists, following Horwich, claim that all that can be said about truth is comprised by all and only the nonparadoxical instances of (E) p is true iff p. It is, accordingly, standard in the literature on truth and paradox to ask how the minimalist will restrict (E) so as to rule out paradox-inducing sentences (alternatively: propositions). In this paper, we consider a prior question: On what grounds does the minimalist restrict (E) so as to rule out paradox-inducing sentences and, thereby, (...)
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  26. B. Armour-Garb & L. Goldstein (2010). Spandrels of Truth * By JC BEALL. Analysis 70 (3):586-589.
    (No abstract is available for this citation).
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  27. Bradley Armour-Garb (2012). The Monotonicity of 'No' and the No-Proposition View. American Philosophical Quarterly 49 (1):1-14.
    This article reveals a tension between a fairly standard response to "liar sentences," of which -/- (L) Sentence (L) -/- is not true is an instance, and some features of our natural language determiners (e.g., 'every,' 'some,' 'no,' etc.) that have been established by formal linguists. The fairly standard response to liar sentences, which has been voiced by a number of philosophers who work directly on the Liar paradox (e.g., Parsons [1974], Kripke [1975], Burge [1979], Goldstein [1985, 2009], Gaifman [1992, (...)
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  28. Bradley Armour-Garb (2007). Consistent Inconsistency Theories. Inquiry 50 (6):639 – 654.
    In this paper I critically evaluate a number of current "consistent inconsistency theories" and then briefly motivate a rival position. The rival position challenges a consistent inconsistency theory, by sharing many of its basic commitments without suffering the problems that such a theory appears to face.
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  29. Bradley Armour-Garb (2005). Wrestling with (and Without) Dialetheism. Australasian Journal of Philosophy 83 (1):87 – 102.
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  30. Bradley Armour-Garb (2004). Minimalism, the Generalization Problem and the Liar. Synthese 139 (3):491 - 512.
    In defense of the minimalist conception of truth, Paul Horwich(2001) has recently argued that our acceptance of the instances of the schema,`the proposition that p is true if and only if p', suffices to explain our acceptanceof truth generalizations, that is, of general claims formulated using the truth predicate.In this paper, I consider the strategy Horwich develops for explaining our acceptance of truth generalizations. As I show, while perhaps workable on its own, the strategy is in conflictwith his response to (...)
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  31. Bradley Armour-Garb (2001). Deflationism and the Meaningless Strategy. Analysis 61 (4):280–289.
  32. Bradley Armour-Garb & J. C. Beall (2001). Can Deflationists Be Dialetheists? Journal of Philosophical Logic 30 (6):593-608.
    Philosophical work on truth covers two streams of inquiry, one concerning the nature (if any) of truth, the other concerning truth-related paradox, especially the Liar. For the most part these streams have proceeded fairly independently of each other. In his "Deflationary Truth and the Liar" (JPL 28:455-488, 1999) Keith Simmons argues that the two streams bear on one another in an important way; specifically, the Liar poses a greater problem for deflationary conceptions of truth than it does for inflationist conceptions. (...)
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  33. Bradley Armour-Garb & JC Beall (2002). Further Remarks on Truth and Contradiction. Philosophical Quarterly 52 (207):217-225.
    We address an issue recently discussed by Graham Priest: whether the very nature of truth (understood as in correspondence theories) rules out true contradictions, and hence whether a correspondence-theoretic notion of truth rules against dialetheism. We argue that, notwithstanding appearances to the contrary, objections from within the correspondence theory do not stand in the way of dialetheism. We close by highlighting, but not attempting to resolve, two further challenges for dialetheism which arise out of familiar philosophical theorizing about truth.
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  34. Bradley Armour-Garb & James A. Woodbridge (2013). Semantic Defectiveness and the Liar. Philosophical Studies 164 (3):845-863.
    In this paper, we do two things. First, we provide some support for adopting a version of the meaningless strategy with respect to the liar paradox, and, second, we extend that strategy, by providing, albeit tentatively, a solution to that paradox—one that is semantic, rather than logical.
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  35. Bradley Armour-Garb & James A. Woodbridge (2012). Liars, Truthtellers and Naysayers: A Broader View of Semantic Pathology I. Language and Communication 32 (4):293-311.
    Semantic pathology is most widely recognized in the liar paradox, where an apparent inconsistency arises in ‘‘liar sentences’’ and their ilk. But the phenomenon of semantic pathology also manifests a sibling symptom—an apparent indeterminacy—which, while not largely discussed (save for the occasional nod to ‘‘truthteller sentences’’), is just as pervasive as, and exactly parallels, the symptom of inconsistency. Moreover, certain ‘‘dual symptom’’ cases, which we call naysayers, exhibit both inconsistency and indeterminacy and also manifest a higher-order indeterminacy between them. In (...)
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  36. Bradley Armour-Garb & James A. Woodbridge (2010). Truthmakers, Paradox and Plausibility. Analysis 70 (1):11-23.
    In a series of articles, Dan Lopez De Sa and Elia Zardini argue that several theorists have recently employed instances of paradoxical reasoning, while failing to see its problematic nature because it does not immediately (or obviously) yield inconsistency. In contrast, Lopez De Sa and Zardini claim that resultant inconsistency is not a necessary condition for paradoxicality. It is our contention that, even given their broader understanding of paradox, their arguments fail to undermine the instances of reasoning they attack, either (...)
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  37. Bradley Armour-Garb & James A. Woodbridge (2006). Dialetheism, Semantic Pathology, and the Open Pair. Australasian Journal of Philosophy 84 (3):395 – 416.
    Over the past 25 years, Graham Priest has ably presented and defended dialetheism, the view that certain sentences are properly characterized as true with true negations. Our goal here is neither to quibble with the tenability of true, assertable contradictions nor, really, with the arguments for dialetheism. Rather, we wish to address the dialetheist's treatment of cases of semantic pathology and to pose a worry for dialetheism that has not been adequately considered. The problem that we present seems to have (...)
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  38. Bradley Armour-Garb & James A. Woodbridge (2003). Curried Katz with Epimenidean Dilemma. Philosophical Forum 34 (3-4):351-366.
  39. D. M. Armstrong (2004). Truth and Truthmakers. Cambridge University Press.
    Truths are determined not by what we believe, but by the way the world is. Or so realists about truth believe. Philosophers call such theories correspondence theories of truth. Truthmaking theory, which now has many adherents among contemporary philosophers, is the most recent development of a realist theory of truth, and in this book D. M. Armstrong offers the first full-length study of this theory. He examines its applications to different sorts of truth, including contingent truths, modal truths, truths about (...)
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  40. Jamin Asay (forthcoming). Epistemicism and the Liar. Synthese:1-21.
    One well known approach to the soritical paradoxes is epistemicism, the view that propositions involving vague notions have definite truth values, though it is impossible in principle to know what they are. Recently, Paul Horwich has extended this approach to the liar paradox, arguing that the liar proposition has a truth value, though it is impossible to know which one it is. The main virtue of the epistemicist approach is that it need not reject classical logic, and in particular the (...)
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  41. F. G. Asenjo (1966). A Calculus for Antinomies. Notre Dame Journal of Formal Logic 16 (1):103-105.
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  42. Jerome Ashmore (1963). Form and Paradox in Santayana's Truth. Revue Internationale de Philosophie 17 (1=63):18-36.
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  43. E. J. Ashworth (1977). Thomas Bricot (D. 1516) and the Liar Paradox. Journal of the History of Philosophy 15 (3):267-280.
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  44. Richard Kenneth Atkins (2011). This Proposition is Not True: C.S. Peirce and the Liar Paradox. Transactions of the Charles S. Peirce Society 47 (4):421-444.
    Charles Sanders Peirce proposed two different solutions to the Liar Paradox. He proposed the first in 1865 and the second in 1869. However, no one has yet noted in the literature that Peirce rejected his 1869 solution in 1903. Peirce never explicitly proposed a third solution to the Liar Paradox. Nonetheless, I shall argue he developed the resources for a third and novel solution to the Liar Paradox.In what follows, I will first explain the Liar Paradox. Second, I will briefly (...)
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  45. Jody Azzouni (2007). The Inconsistency of Natural Languages: How We Live with It. Inquiry 50 (6):590 – 605.
    I revisit my earlier arguments for the (trivial) inconsistency of natural languages, and take up the objection that no such argument can be established on the basis of surface usage. I respond with the evidential centrality of surface usage: the ways it can and can't be undercut by linguistic science. Then some important ramifications of having an inconsistent natural language are explored: (1) the temptation to engage in illegitimate reductio reasoning, (2) the breakdown of the knowledge idiom (because its facticity (...)
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  46. Jody Azzouni (2003). The Strengthened Liar, the Expressive Strength of Natural Languages, and Regimentation. Philosophical Forum 34 (3-4):329–350.
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  47. Andrew Bacon (forthcoming). Can The Classical Logician Avoid The Revenge Paradoxes? Philosophical Review.
    Most work on the semantic paradoxes within classical logic has centred around what I call `linguistic' accounts of the paradoxes: they attribute to sentences or utterances of sentences some property that is supposed to explain their paradoxical or non-paradoxical status. `No proposition' views are paradigm examples of linguistic theories, although practically all accounts of the paradoxes subscribe to some kind of linguistic theory. This paper shows that linguistic accounts of the paradoxes endorsing classical logic are subject to a particularly acute (...)
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  48. Andrew Bacon (2013). A New Conditional for Naive Truth Theory. Notre Dame Journal of Formal Logic 54 (1):87-104.
    In this paper a logic for reasoning disquotationally about truth is presented and shown to have a standard model. This work improves on Hartry Field's recent results establishing consistency and omega-consistency of truth-theories with strong conditional logics. A novel method utilising the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have heretofore failed to provide.
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  49. Andrew Bacon (2013). Curry's Paradox and Omega Inconsistency. Studia Logica 101 (1):1-9.
    In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...)
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  50. Andrew Bacon (2013). Paradoxes of Logical Equivalence and Identity. Topoi (1):1-10.
    In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.
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