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| '''Dialetheism''' is the view that some [[statement (logic)|statements]] can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose [[negation]] is also true. Such statements are called "true [[contradiction]]s", or ''dialetheia''.
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| Dialetheism is not a [[formal system|system of formal logic]]; instead, it is a thesis about [[truth]], that influences the construction of a formal logic, often based on pre-existing systems. Introducing dialetheism has various [[logical consequence|consequences]], depending on the theory into which it is introduced. For example, in traditional systems of logic (e.g., [[classical logic]] and [[intuitionistic logic]]), every statement becomes true if a contradiction is true; this means that such systems become [[trivialism|trivial]] when dialetheism is included as an axiom. Other logical systems do not [[principle of explosion|explode]] in this manner when contradictions are introduced; such contradiction-tolerant systems are known as [[paraconsistent logic]]s.
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| [[Graham Priest]] defines dialetheism as the view that there are true contradictions.<ref name="Dialetheism, logical consequence and hierarchy">Whittle, Bruno. "Dialetheism, logical consequence and hierarchy." <u>Analysis</u> Vol. 64 Issue 4 (2004): 318-326.</ref> JC Beall is another advocate; his position differs from Priest's in advocating constructive (methodological) [[deflationism]] regarding the truth predicate.<ref name="True and False-As If,">Jc Beall in <u>The Law of Non-Contradiction: New Philosophical Essays</u> (Oxford: Oxford University Press, 2004), pp. 197–219.</ref>
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| ==Motivations==
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| ===Dialetheism resolves certain paradoxes===
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| The [[liar paradox|Liar's paradox]] and [[Russell's paradox]] deal with self-contradictory statements in classical logic and [[naïve set theory]], respectively. Contradictions are problematic in these theories because they cause the theories to [[principle of explosion|explode]]—if a contradiction is true, then every proposition is true. The classical way to solve this problem is to ban contradictory statements, to revise the axioms of the logic so that self-contradictory statements do not appear. Dialetheists, on the other hand, respond to this problem by accepting the contradictions as true. Dialetheism allows for the unrestricted [[Axiom schema of specification#Unrestricted comprehension|axiom of comprehension]] in [[set theory]], claiming that any resulting contradiction is a [[theorem]].<ref name="Transfinite Numbers in Paraconsistent Set Theory"><u>Transfinite Numbers in Paraconsistent Set Theory</u> (Review of Symbolic Logic 3(1), 2010), pp. 71-92..</ref>
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| ===Dialetheism may accurately model human reasoning===
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| Ambiguous situations may cause humans to affirm both a proposition and its negation. For example, if John stands in the doorway to a room, it may seem reasonable both to affirm that John is in the room and to affirm that John is not in the room.
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| Critics argue that this merely reflects an ambiguity in our language rather than a dialetheic quality in our thoughts; if we replace the given statement with one that is less ambiguous (such as "John is halfway in the room" or "John is in the doorway"), the contradiction disappears.
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| However, logic has consistently "ignored the dark processes of thought", Introduction to Logic, Patrick Suppes.
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| Dialetheism, supervening on consciousness, may accurately model human reasoning within All Possible Worlds
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| It may be that Possible World talk (David Lewis) and consciousness talk are required, given that a claim is said to be true and false.
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| ===Apparent dialetheism in other philosophical doctrines===
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| The [[Jain]] philosophical doctrine of [[anekantavada]] — non-one-sidedness — states that<ref>Matilal, Bimal Krishna. (1998), "The character of logic in India" (Albany, State University of New York press), 127-139</ref> all statements are true in some sense and false in another. Some interpret this as saying that dialetheia not only exist but are ubiquitous. Technically, however, a ''logical contradiction'' is a proposition that is true and false in the '''same''' sense; a proposition which is true in one sense and false in another does not constitute a logical contradiction. (For example, although in one sense a man cannot both be a "father" and "celibate", there is no contradiction for a man to be a '''spiritual''' father and also celibate; the sense of the word father is different here.)
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| The [[Buddhist]] logic system named [[Catuṣkoṭi]] similarly implies that a statement and its negation may possibly co-exist.<ref>http://www.iep.utm.edu/nagarjun/#H2</ref> <ref>ed : Ganeri, J. (2002), "The Collected Essays
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| of Bimal Krishna Matilal: Mind, Language and World" (Oxford University Press), 77-79</ref>
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| [[Graham Priest]] argues in ''Beyond the Limits of Thought'' that dialetheia arise at the borders of expressibility, in a number of philosophical contexts other than formal semantics.
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| ==Formal consequences==
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| In some logics, we can show that taking a contradiction <math>p \wedge \neg p</math> as a premise (that is, taking as a premise the truth of both <math>p</math> and <math>\neg p</math>), we can prove any statement <math>q</math>. Indeed, since <math>p</math> is true, the statement <math>p \vee q</math> is true (by generalization). Taking <math>p \vee q</math> together with <math>\neg p</math> is a disjunctive syllogism from which we can conclude <math>q</math>. (This is often called the ''[[principle of explosion]]'', since the truth of a contradiction makes the number of theorems in a system "explode".)
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| Any system in which any formula is provable is trivial and uninformative; this is the motivation for solving the semantic paradoxes. Dialethesists solve this problem by rejecting the principle of explosion, and, along with it, at least one of the more basic principles that lead to it, e.g. [[disjunctive syllogism]] or transitivity of [[Logical consequence|entailment]], or [[disjunction introduction]].
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| ==Advantages==
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| The proponents of dialetheism mainly advocate its ability to avoid problems faced by other more orthodox resolutions as a consequence of their appeals to hierarchies. Graham Priest once wrote "the whole point of the dialetheic solution to the semantic paradoxes is to get rid of the distinction between object language and meta-language".<ref name="Dialetheism, logical consequence and hierarchy"/>
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| There are also dialetheic solutions to the [[sorites paradox]].
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| ==Criticisms==
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| One important criticism of dialetheism is that it fails to capture something crucial about [[negation]] and, consequently, disagreement. Imagine John's utterance of ''P''. Sally's typical way of disagreeing with John is a consequent utterance of ¬''P''. Yet, if we accept dialetheism, Sally's so uttering does not prevent her from also accepting ''P''; after all, ''P'' may be a dialetheia and therefore it and its negation are both true. The absoluteness of disagreement is lost. The dialetheist can respond by saying that disagreement can be displayed by uttering "¬''P'' and, furthermore, ''P'' is not a dialetheia". Again, though, the dialetheist's own theory is his [[Achilles' heel]]: the most obvious codification of "''P'' is not a dialetheia" is ¬(''P'' & ¬''P''). But what if ''this itself'' is a dialetheia as well? One dialetheist response is to offer a distinction between [[Logical assertion|assertion]] and rejection. This distinction might be hashed out in terms of the traditional distinction between [[logical quality|logical qualities]], or as a distinction between two [[illocutionary force|illocutionary]] [[speech acts]]: [[Logical assertion|assertion]] and rejection. Another criticism is that Dialetheism
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| cannot describe logical consequences because of its inability to describe hierarchies.<ref name="Dialetheism, logical consequence and hierarchy"/>
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| ==Works cited==
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| *[[Gottlob Frege|Frege, Gottlob]]. "Negation." ''Logical Investigations''. Trans. P. Geach and R. H Stoothoff. New Haven, Conn.: Yale University Press, 1977. 31–53.
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| *[[Terence Parsons|Parsons, Terence]]. "Assertion, Denial, and the Liar Paradox." ''Journal of Philosophical Logic'' 13 (1984): 137–152.
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| *Parsons, Terence. "True Contradictions." ''Canadian Journal of Philosophy'' 20 (1990): 335–354.
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| *[[Graham Priest|Priest, Graham]]. ''In Contradiction''. Dordrecht: Martinus Nijhoff (1987). (Second Edition, Oxford: Oxford University Press, 2006.)
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| *[[Graham Priest|Priest, Graham]]. "What Is So Bad About Contradictions?" ''Journal of Philosophy'' 95 (1998): 410–426.
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| ==See also==
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| {{Portal|Logic}}
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| *[[Problem of future contingents]]
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| *[[Jorge Luis Borges]]' ''[[The Garden of Forking Paths]]''
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| *[[Many-worlds interpretation]]
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| *[[Gottfried Wilhelm Leibniz|Leibniz]]'s [[compossibility]]
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| *[[Liar paradox]]
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| *[[Gödel's incompleteness theorems]]
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| *[[Doublethink]]
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| *[[Trivialism]]
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| ==References==
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| {{Reflist|1}}
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| ==External links==
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| * Francesco Berto and Graham Priest. [http://plato.stanford.edu/entries/dialetheism/ Dialetheism]. In the [[Stanford Encyclopedia of Philosophy]].
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| *[http://homepages.uconn.edu/~jcb02005/ JC Beall UCONN Homepage]
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| *[http://blogandnot-blog.blogspot.com/ (Blog & ~Blog)]
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| *[http://www.paulkabay.com/ Kabay on dialetheism and trivialism] (includes both published and unpublished works)
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| {{Logic}}
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| [[Category:Theories of deduction]]
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| [[Category:Non-classical logic]]
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| [[Category:Theories of truth]]
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| [[Category:Metaphysical theories]]
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