Dialetheism is the view that some 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 contradictions", or dialetheia.
Dialetheism is not a 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 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 trivial when dialetheism is included as an axiom. Other logical systems do not explode in this manner when contradictions are introduced; such contradiction-tolerant systems are known as paraconsistent logics.
Graham Priest defines dialetheism as the view that there are true contradictions. JC Beall is another advocate; his position differs from Priest's in advocating constructive (methodological) deflationism regarding the truth predicate.
Dialetheism resolves certain paradoxes 
The 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 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 of comprehension in set theory, claiming that any resulting contradiction is a theorem.
Dialetheism may accurately model human reasoning 
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. 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.
Apparent dialetheism in other philosophical doctrines 
The Jain philosophical doctrine of anekantavada — non-one-sidedness — states that 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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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.
Formal consequences 
In some logics, we can show that taking a contradiction as a premise (that is, taking as a premise the truth of both and ), we can prove any statement . Indeed, since is true, the statement is true (by generalization). Taking together with is a disjunctive syllogism from which we can conclude . (This is often called the principle of explosion, since the truth of a contradiction makes the number of theorems in a system "explode".)
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 entailment, or disjunction introduction.
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".
There are also dialetheic solutions to the sorites paradox.
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 assertion and rejection. This distinction might be hashed out in terms of the traditional distinction between logical qualities, or as a distinction between two illocutionary speech acts: assertion and rejection. Another criticism is that Dialetheism cannot describe logical consequences because of its inability to describe hierarchies.
Works cited 
- Frege, Gottlob. "Negation." Logical Investigations. Trans. P. Geach and R. H Stoothoff. New Haven, Conn.: Yale University Press, 1977. 31–53.
- Parsons, Terence. "Assertion, Denial, and the Liar Paradox." Journal of Philosophical Logic 13 (1984): 137–152.
- Parsons, Terence. "True Contradictions." Canadian Journal of Philosophy 20 (1990): 335–354.
- Priest, Graham. In Contradiction. Dordrecht: Martinus Nijhoff (1987). (Second Edition, Oxford: Oxford University Press, 2006.)
- Priest, Graham. "What Is So Bad About Contradictions?" Journal of Philosophy 95 (1998): 410–426.
See also 
- Problem of future contingents
- Jorge Luis Borges' The Garden of Forking Paths
- Many-worlds interpretation
- Leibniz's compossibility
- Liar paradox
- Gödel's incompleteness theorems
- Whittle, Bruno. "Dialetheism, logical consequence and hierarchy." Analysis Vol. 64 Issue 4 (2004): 318-326.
- Jc Beall in The Law of Non-Contradiction: New Philosophical Essays (Oxford: Oxford University Press, 2004), pp. 197–219.
- Transfinite Numbers in Paraconsistent Set Theory (Review of Symbolic Logic 3(1), 2010), pp. 71-92..
- Francesco Berto and Graham Priest. Dialetheism. In the Stanford Encyclopedia of Philosophy.
- JC Beall UCONN Homepage
- (Blog & ~Blog)
- Kabay on dialetheism and trivialism (includes both published and unpublished works)