Dialetheism
From Academic Kids

Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. More specifically, dialetheists believe that some propositions of the form P ∧ ¬P are true.
Like other paraconsistent logics, the motivations for dialetheism might be split into two groups:
 Formal semantic concerns brought about by such puzzles as the Liar's and Russell's paradoxes. These paradoxes show that from the premises of classical logic and set theory one can derive outright contradictions. Historically, this has been an undesirable result of the axioms of classical logic. Dialetheists solve the problem by eliminating it: these contradictions, they say, are simply true.
 It might be argued that our actual thinking is dialetheistic. In other words, it is not completely prima facie implausible that we might affirm both a proposition and its negation. Consider "John is in the room" when John is standing precisely halfway in the room.
It is important to recognize the formal ramifications of accepting a contradiction as true. Using some commonly accepted and intuitively plausible rules of logic, we can easily show that the formula P ∧ ¬P implies everything; taking a contradiction as a premise, we can prove any A. (This is often called the principle of explosion, since the truth of a contradiction makes the number of theorems in your 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 laws of disjunction that lead to it.
Perhaps the most penetrating 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. A first guess at a defense by the dialetheist is to say 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? The dialetheist response is to offer a distinction between two illocutionary speech acts: assertion and rejection (as opposed to the classical view, from Frege, that there is only one such speech actassertionthat is expressed towards a proposition and its negation). This response pushes the burden of argument from the purely logical realm to the theory of speech acts, which is largely pragmatic.
Graham Priest of the University of St. Andrews is dialetheism's most prominent contemporary champion.
Works Cited
Frege, Gottlob. “Negation.” Logical Investigations. Trans. P. Geach and R. H Stoothoff. New Haven, CT: Yale UP, 1977. 3153.
Parsons, Terence. “Assertion, Denial, and the Liar Paradox.” Journal of Philosophical Logic 13 (1984): 137152.
Parsons, Terence. “True Contradictions.” Canadian Journal of Philosophy 20 (1990): 335354.
Priest, Graham. In Contradiction. Dordrecht: Martinus Nijhoff (1987).
Priest, Graham. “What Is So Bad About Contradictions?” Journal of Philosophy 95 (1998): 410426.
External Links
 Graham Priest. Dialetheism (http://plato.stanford.edu/entries/dialetheism/). In the Stanford Encyclopedia of Philosophy, 2004.