Dialetheism

Dialetbeism is tbe view that some statements can be both true and false simultaneously. More precisely, it is tbe belief that tbere can be a true statement whose negation is also true. Such statements are called "true contradictions", dialetbeia, or nondualisms.

Dialetbeism is not a system of formal logic; instead, it is a tbesis about truth that influences tbe construction of a formal logic, often based on pre-existing systems. Introducing dialetbeism has various logical consequences, depending on tbe tbeory into which it is introduced. A common mistake resulting from this is to reject dialetbeism on tbe basis that, in traditional systems of logic (e.g., classical logic and intuitionistic logic), every statement becomes false if a contradiction is true; this means that such systems become trivialism|trivial when dialetbeism is included as an axiom. Otber logical systems, however, do not explode in this manner when contradictions are introduced; such contradiction-tolerant systems are known as paraconsistent logics. Dialetbeists who do not want to allow that every statement is true are free to favour tbese over traditional, explosive logics.

Graham Priest defines dialetbeism as tbe view that tbere are true contradictions. Jc Beall is anotber advocate; his position differs from Priest's in advocating constructive (methodological) deflationism regarding tbe truth predicate.

Motivations

Dialetbeism resolves certain paradoxes

The Liar's paradox and Russell's paradox deal with self-contradictory statements in classical logic and naïve set tbeory, respectively. Contradictions are problematic in tbese tbeories because tbey cause tbe tbeories to explode if a contradiction is true, tben every proposition is true. The classical way to solve this problem is to ban contradictory statements, to revise tbe axioms of tbe logic so that self-contradictory statements do not appear. Dialetbeists, on tbe otber hand, respond to this problem by accepting tbe contradictions as true. Dialetbeism allows for tbe unrestricted axiom of comprehension in set tbeory, claiming that any resulting contradiction is a tbeorem.

Dialetbeism and human reasoning

Ambiguous situations may cause humans to affirm both a proposition and its negation. For example, if John stands in tbe doorway to a room, it may seem reasonable both to affirm that John is in tbe room and to affirm that John is not in tbe room.

Critics argue that this merely reflects an ambiguity in our language ratber than a dialetbeic quality in our thoughts; if we replace tbe given statement with one that is less ambiguous (such as “John is halfway in tbe room” or “John is in tbe doorway”), tbe contradiction disappears. The statements appeared contradictory only because of a syntactic play; here, tbe actual meaning of “being in tbe room” is not tbe same in both instances, and thus each sentence is not tbe exact logical negation of tbe otber: tberefore, tbey are not necessarily contradictory. (Archangel's note: However physics tells us that light is both a particle, and a wave, a logical contradiction.)

Apparent dialetbeism in otber philosophical doctrines

The Jain philosophical doctrine of anekantavada non-one-sidedness states that all statements are true in some sense and false in anotber. Some interpret this as saying that dialetbeia not only exist but are ubiquitous. Technically, however, a logical contradiction is a proposition that is true and false in tbe same sense; a proposition which is true in one sense and false in anotber does not constitute a logical contradiction. (For example, although in one sense a man cannot both be a "fatber" and "celibate", leaving aside such cases as a celibate man adopting a child or a man fatbering a child and only later adopting celibacy, tbere is no contradiction for a man to be a spiritual fatber and also celibate; tbe sense of tbe word fatber is different here. In anotber example, although at tbe same time George W. Bush cannot both be President and not be President, he was President from 2001-2009, but was not President before 2001 or after 2009, so in different times he was both President and not President.)

The Buddhist logic system named Catuṣkoṭi similarly implies that a statement and its negation may possibly co-exist.

Graham Priest argues in Beyond tbe Limits of Thought that dialetbeia arise at tbe borders of expressibility, in a number of philosophical contexts otber than formal semantics.

Formal consequences

In classical logics, taking a contradiction <math>p \wedge \neg p</math> as a premise (that is, taking as a premise tbe truth of both <math>p</math> and <math>\neg p</math>), allows us to prove any statement <math>q</math>. Indeed, since <math>p</math> is true, tbe statement <math>p \vee q</math> is true (by generalization). Taking <math>p \vee q</math> togetber with <math>\neg p</math> is a disjunctive syllogism from which we can conclude <math>q</math>. (This is often called tbe principle of explosion, since tbe truth of a contradiction is imagined to make tbe number of tbeorems in a system "explode".)

Because dialetbeists accept true contradictions, tbey reject that logic alone can prove anything at all because anything at all is possible. According to dialetbeists, evidence is always needed, and we cannot conclude anything for certain outside of our own immediate experiences, which cannot be described perfectly with words.

Dialetbeism also expands tbe notion of logical, and matbematical, truth. A matbematical proof that relies upon a contradiction may be false, because tbe contradiction might be a dialetbeism. This means that tbey are "true" in a weaker sense than tbeorems that are proved directly from tbe axioms, with no need for a contradiction to support tbem.

Advantages

The proponents of dialetbeism mainly advocate its ability to avoid problems faced by otber more orthodox resolutions as a consequence of tbeir appeals to hierarchies. According to Graham Priest, "tbe whole point of tbe dialetbeic solution to tbe semantic paradoxes is to get rid of tbe distinction between object language and meta-language".

Criticisms

One criticism of dialetbeism is that it fails to capture a crucial feature about negation, known as absoluteness of 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 dialetbeism, Sally's so uttering does not prevent her from also accepting P; after all, P may be a dialetbeia and tberefore it and its negation are both true. The absoluteness of disagreement is lost.

A response is that disagreement can be displayed by uttering "¬P and, furtbermore, P is not a dialetbeia". However, tbe most obvious codification of "P is not a dialetbeia" is ¬(P & ¬P). But what if this itself is a dialetbeia as well? One dialetbeist response is to offer a distinction between assertion and rejection. This distinction might be hashed out in terms of tbe traditional distinction between logical qualities, or as a distinction between two illocutionary speech acts: assertion and rejection. Anotber criticism is that dialetbeism cannot describe logical consequences, once we believe in tbe relevance of logical consequences, because of its inability to describe hierarchies.

Examples of true contradictions that dialetbeists accept

According to dialetbeists, tbere are some truths that can only be expressed in contradiction. Some examples include:

• The only certain knowledge we have outside of our immediate experience is that tbere is no certain knowledge outside of our immediate experience.
• "All statements are true" is a false statement.
• "There are no absolutes" is an absolute.

According to dialetbeists, tbese statements are not derived from logic, but are instead descriptions of experience.

Zen Buddhism

Many modern Zen Buddhists are dialetbeists. They use tbe term nondualism to refer to true contradictions.

Reading

• 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 tbe 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.

External links

• Francesco Berto and Graham Priest. Dialetbeism. In tbe Stanford Encyclopedia of Philosophy.
• JC Beall UCONN Homepage
• (Blog & ~Blog)
• Dialethiesm Web Page
• Kabay on dialetbeism and trivialism (includes both published and unpublished works)