![Square of opposition, set diagrams](https://upload.wikimedia.org/wikipedia/commons/5/51/Square_of_opposition%2C_set_diagrams.svg)
In
traditional logic
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, th ...
, a contradiction occurs when a
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
conflicts either with itself or established
fact
A fact is a datum about one or more aspects of a circumstance, which, if accepted as true and proven true, allows a logical conclusion to be reached on a true–false evaluation. Standard reference works are often used to check facts. Scient ...
. It is often used as a tool to detect
disingenuous
Deception or falsehood is an act or statement that misleads, hides the truth, or promotes a belief, concept, or idea that is not true. It is often done for personal gain or advantage. Deception can involve dissimulation, propaganda and sleight o ...
beliefs and
bias
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
. Illustrating a general tendency in applied logic,
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
's
law of noncontradiction
In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...
states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."
In modern
formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
and
type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundat ...
, the term is mainly used instead for a ''single'' proposition, often denoted by the
falsum
The up tack or falsum (⊥, \bot in LaTeX, U+22A5 in Unicode) is a constant symbol used to represent:
* The truth value 'false', or a logical constant denoting a proposition in logic that is always false (often called "falsum" or "absurdum").
* ...
symbol
; a proposition is a contradiction if
false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition). This can be generalized to a collection of propositions, which is then said to "contain" a contradiction.
History
By creation of a
paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
,
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's ''
Euthydemus'' dialogue demonstrates the need for the notion of ''contradiction''. In the ensuing dialogue,
Dionysodorus
Dionysodorus of Caunus ( grc-gre, Διονυσόδωρος ὁ Καύνειος, c. 250 BC – c. 190 BC) was an ancient Greek mathematician.
Life and work
Little is known about the life of Dionysodorus. Pliny the Elder writes about a Dionysodo ...
denies the existence of "contradiction", all the while that
Socrates
Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no te ...
is contradicting him:
Indeed, Dionysodorus agrees that "there is no such thing as false opinion ... there is no such thing as ignorance", and demands of Socrates to "Refute me." Socrates responds "But how can I refute you, if, as you say, to tell a falsehood is impossible?".
In formal logic
In classical logic, particularly in
propositional
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
and
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
, a proposition
is a contradiction
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
. Since for contradictory
it is true that
for all
(because
), one may prove any proposition from a set of axioms which contains contradictions. This is called the "
principle of explosion
In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (, 'from falsehood, anything ollows; or ), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a co ...
", or "ex falso quodlibet" ("from falsity, anything follows").
In a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
logic, a formula is contradictory if and only if it is
unsatisfiable
In mathematical logic, a Well-formed formula, formula is ''satisfiable'' if it is true under some assignment of values to its Variable (mathematics), variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, whil ...
.
Proof by contradiction
For a set of consistent premises
and a proposition
, it is true in
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
that
(i.e.,
proves
) if and only if
(i.e.,
and
leads to a contradiction). Therefore, a
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
that
also proves that
is true under the premises
. The use of this fact forms the basis of a
proof technique called
proof by contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as ...
, which mathematicians use extensively to establish the validity of a wide range of theorems. This applies only in a logic where the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...
is accepted as an axiom.
Using
minimal logic
Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It is an intuitionistic and paraconsistent logic, that rejects both the law of the excluded middle as well as the principle of explosion (' ...
, a logic with similar axioms to classical logic but without ''ex falso quodlibet'' and proof by contradiction, we can investigate the axiomatic strength and properties of various rules that treat contradiction by considering theorems of classical logic that are not theorems of minimal logic. Each of these extensions leads to an
intermediate logic In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate l ...
:
# Double-negation elimination (DNE) is the strongest principle, axiomatised
, and when it is added to minimal logic yields classical logic.
# Ex falso quodlibet (EFQ), axiomatised
, licenses many consequences of negations, but typically does not help inferring propositions that do not involve absurdity from consistent propositions that do. When added to minimal logic, EFQ yields
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
. EFQ is equivalent to ''ex contradictione quodlibet'', axiomatised
, over minimal logic.
#
Peirce's rule (PR) is an axiom
that captures proof by contradiction without explicitly referring to absurdity. Minimal logic + PR + EFQ yields classical logic.
# The Gödel-Dummett (GD) axiom
, whose most simple reading is that there is a linear order on truth values. Minimal logic + GD yields
Gödel-Dummett logic. Peirce's rule entails but is not entailed by GD over minimal logic.
# Law of the excluded middle (LEM), axiomatised
, is the most often cited formulation of the
principle of bivalence
In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called ...
, but in the absence of EFQ it does not yield full classical logic. Minimal logic + LEM + EFQ yields classical logic. PR entails but is not entailed by LEM in minimal logic. If the formula B in Peirce's rule is restricted to absurdity, giving the axiom schema
, the scheme is equivalent to LEM over minimal logic.
# Weak law of the excluded middle (WLEM) is axiomatised
and yields a system where disjunction behaves more like in classical logic than intuitionistic logic, i.e. the
disjunction and existence properties
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005).
Disjunction property
The disjunction property is satisfi ...
don't hold, but where use of non-intuitionistic reasoning is marked by occurrences of double-negation in the conclusion. LEM entails but is not entailed by WLEM in minimal logic. WLEM is equivalent to the instance of
De Morgan's law
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
that distributes negation over conjunction:
.
Symbolic representation
In mathematics, the symbol used to represent a contradiction within a proof varies. Some symbols that may be used to represent a contradiction include ↯, Opq,
, ⊥,
/ , and ※; in any symbolism, a contradiction may be substituted for the truth value "
false", as symbolized, for instance, by "0" (as is common in
boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
). It is not uncommon to see
Q.E.D.
Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...
, or some of its variants, immediately after a contradiction symbol. In fact, this often occurs in a proof by contradiction to indicate that the original assumption was proved false—and hence that its negation must be true.
The notion of contradiction in an axiomatic system and a proof of its consistency
In general, a
consistency proof
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
requires the following two things:
# An
axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
# A demonstration that it is ''not'' the case that both the formula ''p'' and its negation ''~p'' can be derived in the system.
But by whatever method one goes about it, all consistency proofs would ''seem'' to necessitate the primitive notion of ''contradiction.'' Moreover, it ''seems'' as if this notion would simultaneously have to be "outside" the formal system in the definition of tautology.
When
Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Govern ...
, in his 1921 "Introduction to a General Theory of Elementary Propositions", extended his proof of the consistency of the
propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
(i.e. the logic) beyond that of ''
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' (PM), he observed that with respect to a ''generalized'' set of postulates (i.e. axioms), he would no longer be able to automatically invoke the notion of "contradiction"such a notion might not be contained in the postulates:
Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute Proof of Consistency", offered by
Ernest Nagel
Ernest Nagel (November 16, 1901 – September 20, 1985) was an American philosopher of science. Suppes, Patrick (1999)Biographical memoir of Ernest Nagel In '' American National Biograph''y (Vol. 16, pp. 216-218). New York: Oxford University Pr ...
and
James R. Newman in their 1958 ''
Gödel's Proof''. They too observed a problem with respect to the notion of "contradiction" with its usual "truth values" of "truth" and "falsity". They observed that:
Given some "primitive formulas" such as PM's primitives S
1 V S
2 nclusive ORand ~S (negation), one is forced to define the axioms in terms of these primitive notions. In a thorough manner, Post demonstrates in PM, and defines (as do Nagel and Newman, see below) that the property of ''tautologous'' – as yet to be defined – is "inherited": if one begins with a set of tautologous axioms (postulates) and a
deduction system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
that contains
substitution
Substitution may refer to:
Arts and media
*Chord substitution, in music, swapping one chord for a related one within a chord progression
* Substitution (poetry), a variation in poetic scansion
* "Substitution" (song), a 2009 song by Silversun Pi ...
and
modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
, then a ''consistent'' system will yield only tautologous formulas.
On the topic of the definition of ''tautologous'', Nagel and Newman create two
mutually exclusive
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
and
exhaustive classes K
1 and K
2, into which fall (the outcome of) the axioms when their variables (e.g. S
1 and S
2 are assigned from these classes). This also applies to the primitive formulas. For example: "A formula having the form S
1 V S
2 is placed into class K
2, if both S
1 and S
2 are in K
2; otherwise it is placed in K
1", and "A formula having the form ~S is placed in K
2, if S is in K
1; otherwise it is placed in K
1".
Hence Nagel and Newman can now define the notion of ''
tautologous'': "a formula is a tautology if and only if it falls in the class K
1, no matter in which of the two classes its elements are placed". This way, the property of "being tautologous" is described—without reference to a model or an interpretation.
Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulas derived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from either class K
1 or K
2, the deduction violates the inheritance characteristic of tautology (i.e., the derivation must yield an evaluation of a formula that will fall into class K
1). From this, Post was able to derive the following definition of inconsistency—''without the use of the notion of contradiction'':
In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes. An axiomatic system need not include the notion of "contradiction".
Philosophy
Adherents of the
epistemological
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
theory of
coherentism
In philosophical epistemology, there are two types of coherentism: the coherence theory of truth; and the coherence theory of justification (also known as epistemic coherentism).
Coherent truth is divided between an anthropological approach, whic ...
typically claim that as a necessary condition of the justification of a
belief
A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take i ...
, that belief must form a part of a logically non-contradictory
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
of beliefs. Some
dialetheists, including
Graham Priest
Graham Priest (born 1948) is Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at the University of Melbourne, where he was Boyce Gibson Professor of Philosophy and also at the University of St Andr ...
, have argued that coherence may not require consistency.
Pragmatic contradictions
A pragmatic contradiction occurs when the very statement of the argument contradicts the claims it purports. An inconsistency arises, in this case, because the act of utterance, rather than the content of what is said, undermines its conclusion.
Dialectical materialism
In
dialectical materialism
Dialectical materialism is a philosophy of science, history, and nature developed in Europe and based on the writings of Karl Marx and Friedrich Engels. Marxist dialectics, as a materialist philosophy, emphasizes the importance of real-world con ...
: Contradiction—as derived from
Hegelianism
Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German philosopher. He is one of the most important figures in German idealism and one of the founding figures of modern Western philosophy. His influence extends a ...
—usually refers to an opposition inherently existing within one realm, one unified force or object. This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to Marxist theory, such a contradiction can be found, for example, in the fact that:
* (a) enormous wealth and productive powers coexist alongside:
* (b) extreme poverty and misery;
* (c) the existence of (a) being contrary to the existence of (b).
Hegelian and Marxist theory stipulates that the
dialectic
Dialectic ( grc-gre, διαλεκτική, ''dialektikḗ''; related to dialogue; german: Dialektik), also known as the dialectical method, is a discourse between two or more people holding different points of view about a subject but wishing ...
nature of history will lead to the
sublation
() or () is a German word with several seemingly contradictory meanings, including "to lift up", "to abolish", "cancel" or "suspend", or "to sublate". The term has also been defined as "abolish", "preserve", and "transcend". In philosophy, i ...
, or
synthesis
Synthesis or synthesize may refer to:
Science Chemistry and biochemistry
*Chemical synthesis, the execution of chemical reactions to form a more complex molecule from chemical precursors
** Organic synthesis, the chemical synthesis of organ ...
, of its contradictions. Marx therefore postulated that history would logically make
capitalism
Capitalism is an economic system based on the private ownership of the means of production and their operation for Profit (economics), profit. Central characteristics of capitalism include capital accumulation, competitive markets, pric ...
evolve into a
socialist society
The Socialist Society was founded in 1981 by a group of British socialists, including Raymond Williams and Ralph Miliband, who founded it as an organisation devoted to socialist education and research, linking the left of the British Labour Party ...
where the
means of production
The means of production is a term which describes land, labor and capital that can be used to produce products (such as goods or services); however, the term can also refer to anything that is used to produce products. It can also be used as an ...
would equally serve the
exploited and suffering class of society, thus resolving the prior contradiction between (a) and (b).
Mao Zedong's philosophical essay
''On Contradiction'' (1937) furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction.
Outside formal logic
Colloquial usage can label actions or statements as contradicting each other when due (or perceived as due) to
presupposition
In the branch of linguistics known as pragmatics, a presupposition (or PSP) is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions include ...
s which are contradictory in the logical sense.
Proof by contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as ...
is used in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
to construct
proofs
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
.
The
scientific method
The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific m ...
uses contradiction to falsify bad theory.
See also
* , a Monty Python sketch in which one of the two disputants repeatedly uses only contradictions in his argument
*
*
*
*
*
*
Graham's hierarchy of disagreement
*
*
Law of noncontradiction
In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...
*
*
*
*
*
*
Notes and references
Bibliography
*
Józef Maria Bocheński
Józef Maria Bocheński or Innocentius Bochenski (Czuszów, Congress Poland, Russian Empire, 30 August 1902 – 8 February 1995, Fribourg, Switzerland) was a Polish Dominican, logician and philosopher.
Biography
Born on 30 August 1902 in Czu ...
1960 ''Précis of Mathematical Logic'', translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.
* Jean van Heijenoort 1967 ''From Frege to Gödel: A Source Book in Mathematical Logic 1879-1931'', Harvard University Press, Cambridge, MA, (pbk.)
*Ernest Nagel and James R. Newman 1958 ''Gödel's Proof'', New York University Press, Card Catalog Number: 58-5610.
External links
*
*
*
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Propositional calculus
Marxist theory
Mathematical logic
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Immediate inference
Cognitive dissonance