Dialetheism
Encyclopedia
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 hypothesis that can be introduced as an axiom within pre-existing systems of formal logic
. 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 logic
s.
Graham Priest
, of the University of Melbourne
and the CUNY Graduate Center, is dialetheism's most prominent contemporary champion. He defines Dialetheism as the view that there are true contradictions. JC Beall of the University of Connecticut is another contemporary advocate. His position differs from Priest's in advocating constructive (methodological) deflationism regarding the truth predicate.
and Russel'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.
It seems that so long as we accept that space is extended (is not a point), "John is in the room" when John is standing in the doorway does not seem to lend support to nor against the idea that our actual thinking is dialetheic.
— 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.)
The Buddhist logic system named Catuṣkoṭi
similarly implies that a statement and its negation may possibly co-exist.
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.
(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
.
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. This tactic however, does not overcome the objection to dialetheism but merely tries to ignore it. Another criticism is that Dialetheism
cannot describe logical consequences because of its inability to describe hierarchies.
Dialetheism is not a system of formal logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
; instead, it is a hypothesis that can be introduced as an axiom within pre-existing systems of formal logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
. 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
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
and intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
), every statement becomes true if a contradiction is true; this means that such systems become trivial
Trivialism
Trivialism is the theory that every proposition is true. A consequence of trivialism is that all statements, including all contradictions of the form "p and not p" , are true.- Further reading :***...
when dialetheism is included as an axiom. Other logical systems do not explode
Principle of explosion
The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
in this manner when contradictions are introduced; such contradiction-tolerant systems are known as paraconsistent logic
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
s.
Graham Priest
Graham Priest
Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
, of the University of Melbourne
University of Melbourne
The University of Melbourne is a public university located in Melbourne, Victoria. Founded in 1853, it is the second oldest university in Australia and the oldest in Victoria...
and the CUNY Graduate Center, is dialetheism's most prominent contemporary champion. He defines Dialetheism as the view that there are true contradictions. JC Beall of the University of Connecticut is another contemporary advocate. His position differs from Priest's in advocating constructive (methodological) deflationism regarding the truth predicate.
Dialetheism resolves certain paradoxes
The Liar's paradoxLiar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
and Russel's paradox deal with self-contradictory statements in classical logic and naïve set theory
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
, respectively. Contradictions are problematic in these theories because they cause the theories to explode
Principle of explosion
The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
—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 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.It seems that so long as we accept that space is extended (is not a point), "John is in the room" when John is standing in the doorway does not seem to lend support to nor against the idea that our actual thinking is dialetheic.
Dialetheism appears in other philosophical doctrines
The Jain philosophical doctrine of anekantavadaAnekantavada
' is one of the most important and fundamental doctrines of Jainism. It refers to the principles of pluralism and multiplicity of viewpoints, the notion that truth and reality are perceived differently from diverse points of view, and that no single point of view is the complete truth.Jains...
— 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.)
The Buddhist logic system named Catuṣkoṭi
Catuṣkoṭi
Catuṣkoṭi is a logical argument of a 'suite of four discrete functions' or 'an indivisible quaternity' that has multiple applications and has been important in the Dharmic traditions of Indian logic and the Buddhadharma logico-epistemological traditions, particularly those of the Madhyamaka...
similarly implies that a statement and its negation may possibly co-exist.
Graham Priest
Graham Priest
Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
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 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
Principle of explosion
The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
, 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
Disjunctive syllogism
A disjunctive syllogism, also known as disjunction-elimination and or-elimination , and historically known as modus tollendo ponens,, is a classically valid, simple argument form:where \vdash represents the logical assertion....
or transitivity of entailment
Entailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
, or disjunction introduction
Disjunction introduction
Disjunction introduction or Addition is a valid, simple argument form in logic:or in logical operator notation: A \vdash A \or B The argument form has one premise, A, and an unrelated proposition, B...
.
Advantages
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".Criticisms
One important criticism of dialetheism is that it fails to capture something crucial about negationNegation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
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
Achilles' heel
An Achilles’ heel is a deadly weakness in spite of overall strength, that can actually or potentially lead to downfall. While the mythological origin refers to a physical vulnerability, metaphorical references to other attributes or qualities that can lead to downfall are common.- Origin :In Greek...
: 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
Logical assertion
A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
and rejection. This distinction might be hashed out in terms of the traditional distinction between logical qualities
Logical quality
In many philosophies of logic statements are categorized into different logical qualities based on how they go about saying what they say. Doctrines of logical quality are an attempt to answer the question: “How many qualitatively different ways are there of saying something?” Aristotle answers,...
, or as a distinction between two illocutionary speech acts: assertion
Logical assertion
A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
and rejection. This tactic however, does not overcome the objection to dialetheism but merely tries to ignore it. Another criticism is that Dialetheism
cannot describe logical consequences because of its inability to describe hierarchies.
Works cited
- Frege, GottlobGottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
. "Negation." Logical Investigations. Trans. P. Geach and R. H Stoothoff. New Haven, Conn.: Yale University Press, 1977. 31–53. - Parsons, TerenceTerence ParsonsTerence Parsons is an American contemporary philosopher of the analytic tradition. Parsons is also a Professor at UCLA in its Department of Philosophy....
. "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, GrahamGraham PriestGraham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
. In Contradiction. Dordrecht: Martinus Nijhoff (1987). (Second Edition, Oxford: Oxford University Press, 2006.) - Priest, GrahamGraham PriestGraham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
. "What Is So Bad About Contradictions?" Journal of Philosophy 95 (1998): 410–426.
See also
- Problem of future contingents
- Jorge Luis BorgesJorge Luis BorgesJorge Francisco Isidoro Luis Borges Acevedo , known as Jorge Luis Borges , was an Argentine writer, essayist, poet and translator born in Buenos Aires. In 1914 his family moved to Switzerland where he attended school, receiving his baccalauréat from the Collège de Genève in 1918. The family...
' The Garden of Forking PathsThe Garden of Forking Paths"The Garden of Forking Paths" is a 1941 short story by Argentine writer and poet Jorge Luis Borges. It is the title story in the collection El jardín de senderos que se bifurcan , which was republished in its entirety in Ficciones in 1944... - Many-worlds interpretationMany-worlds interpretationThe many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction, but denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an...
- Leibniz's compossibilityCompossibilityCompossibility is a philosophical concept from Leibniz. According to Leibniz a complete individual thing is characterized by all its properties, and these determine its relations with other individuals. The existence of one individual may contradict the existence of another...
- Liar paradoxLiar paradoxIn philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
- Gödel's incompleteness theoremsGödel's incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
- DoublethinkDoublethinkDoublethink, a word coined by George Orwell in the novel 1984, describes the act of simultaneously accepting two mutually contradictory beliefs as correct, often in distinct social contexts. It is related to, but distinct from, hypocrisy and neutrality. Its opposite is cognitive dissonance, where...
External links
- Graham Priest. Dialetheism. In the Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
, 2004. - JC Beall UCONN Homepage
- (Blog & ~Blog)
- Kabay on dialetheism and trivialism (includes both published and unpublished works)