Dialetheism

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Dialetheism is the view that there are true contradictions, or dialetheias. More specifically, dialetheists believe that for some sentence or proposition P, both P and its negation (not-P) are true. Dialetheism is not itself a formal logic, but anyone who endorses dialetheism is rationally committed to accepting some version of paraconsistent logic.

The motivations for dialetheism might be split into several groups:

Formal semantic concerns brought about by such puzzles as the Liar's and Russell's paradoxes. From the premises of classical logic and naïve set theory one can derive outright contradictions, a result that is traditionally frowned upon. The classical response to this problem is to revise the axioms of set theory in order to make them consistent. Dialetheists respond to the problem by accepting the contradictions as 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.
The Jain philosophical doctrine of anekantavada — non-one-sidedness is sometimes interpreted as implying that all statements are true in some sense and false in some other sense, and thus that dialetheia are common.
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.

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 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.

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 cashed out in terms of the traditional distinction between logical qualities, or as a distinction between two illocutionary speech acts: assertion and rejection. The second response would push the burden of argument from the purely logical realm to the theory of speech acts, which is largely pragmatic.^{[citation needed]}Graham Priest of the University of Melbourne is dialetheism's most prominent contemporary champion.

[edit] 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).
Priest, Graham. "What Is So Bad About Contradictions?" Journal of Philosophy 95 (1998): 410–426.