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In
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 prem ...
, 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 same time, e. g. the two propositions "''p is the case''" and "''p is not the case''" are mutually exclusive. Formally this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with 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 noncontradi ...
which states that at least one, "p is the case" or "p is not the case" holds. One reason to have this law is 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 ...
, which states that anything follows from a contradiction. The law is employed in a ''
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
'' proof. To express the fact that the law is tenseless and to avoid
equivocation In logic, equivocation ("calling two different things by the same name") is an informal fallacy resulting from the use of a particular word/expression in multiple senses within an argument. It is a type of ambiguity that stems from a phrase havin ...
, sometimes the law is amended to say "contradictory propositions cannot both be true 'at the same time and in the same sense'". It is one of the so called three laws of thought, along with its complement, the law of excluded middle, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provide
inference rules In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
, such as ''
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 ...
'' or
De Morgan's laws 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 math ...
. The law of non-contradiction and the law of excluded middle create a
dichotomy A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be * jointly exhaustive: everything must belong to one part or the other, and * mutually exclusive: nothing can belong simul ...
in "logical space", wherein the two parts are "mutually exclusive" and "jointly exhaustive". The law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle, an expression of its jointly exhaustive aspect.


Interpretations

One difficulty in applying the law of non-contradiction is ambiguity in the propositions. For instance, if it is not explicitly specified as part of the propositions A and B, then ''A'' may be ''B'' at one time, and not at another. A and B may in some cases be made to sound mutually exclusive linguistically even though ''A'' may be partly ''B'' and partly not ''B'' at the same time. However, it is impossible to predicate of the same thing, at the same time, and in the same sense, the absence and the presence of the same fixed quality.


Heraclitus

According to both Plato and Aristotle,
Heraclitus Heraclitus of Ephesus (; grc-gre, Ἡράκλειτος , "Glory of Hera"; ) was an ancient Greek pre-Socratic philosopher from the city of Ephesus, which was then part of the Persian Empire. Little is known of Heraclitus's life. He wrot ...
was ''said'' to have denied the law of non-contradiction. This is quite likely if, as Plato pointed out, the law of non-contradiction does not hold for changing things in the world. If a philosophy of Becoming is not possible without change, then (the potential of) what is to become must already exist in the present object. In "''We step and do not step into the same rivers; we are and we are not''", both Heraclitus's and Plato's object simultaneously must, in some sense, be both what it now is and have the potential (dynamic) of what it might become. so little remains of
Heraclitus Heraclitus of Ephesus (; grc-gre, Ἡράκλειτος , "Glory of Hera"; ) was an ancient Greek pre-Socratic philosopher from the city of Ephesus, which was then part of the Persian Empire. Little is known of Heraclitus's life. He wrot ...
' aphorisms that not much about his philosophy can be said with certainty. He seems to have held that strife of opposites is universal both within and without, therefore ''both'' opposite existents or qualities must simultaneously exist, although in some instances in different respects. "The ''road up and down are one and the same''" implies either the road leads both ways, or there can be no road at all. This is the logical
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
of the law of non-contradiction. According to Heraclitus, change, and the constant conflict of opposites is the universal
logos ''Logos'' (, ; grc, λόγος, lógos, lit=word, discourse, or reason) is a term used in Western philosophy, psychology and rhetoric and refers to the appeal to reason that relies on logic or reason, inductive and deductive reasoning. Aris ...
of nature.


Protagoras

Personal subjective perceptions or judgments can only be said to be true at the same time in the same respect, in which case, the law of non-contradiction must be applicable to personal judgments. The most famous saying of
Protagoras Protagoras (; el, Πρωταγόρας; )Guthrie, p. 262–263. was a pre-Socratic Greek philosopher and rhetorical theorist. He is numbered as one of the sophists by Plato. In his dialogue '' Protagoras'', Plato credits him with inventing t ...
is: "''Man is the measure of all things: of things which are, that they are, and of things which are not, that they are not''". However, Protagoras was referring to things that are used by or in some way related to humans. This makes a great difference in the meaning of his aphorism. Properties, social entities, ideas, feelings, judgments, etc. originate in the human mind. However, Protagoras has never suggested that man must be the measure of stars or the motion of the stars.


Parmenides

Parmenides Parmenides of Elea (; grc-gre, Παρμενίδης ὁ Ἐλεάτης; ) was a pre-Socratic Greek philosopher from Elea in Magna Graecia. Parmenides was born in the Greek colony of Elea, from a wealthy and illustrious family. His date ...
employed an
ontological In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
version of the law of non-contradiction to prove that being is and to deny the void, change, and motion. He also similarly disproved contrary propositions. In his poem On Nature, he said, The nature of the ‘is’ or what-is in Parmenides is a highly contentious subject. Some have taken it to be whatever exists, some to be whatever is or can be the object of scientific inquiry.


Socrates

In Plato's early dialogues, Socrates uses the elenctic method to investigate the nature or definition of ethical concepts such as justice or virtue. Elenctic refutation depends on a dichotomous thesis, one that may be divided into exactly 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 ...
parts, only one of which may be true. Then Socrates goes on to demonstrate the contrary of the commonly accepted part using the law of non-contradiction. According to Gregory Vlastos, the method has the following steps: # Socrates' interlocutor asserts a thesis, for example, "Courage is endurance of the soul", which Socrates considers false and targets for refutation. # Socrates secures his interlocutor's agreement to further premises, for example, "Courage is a fine thing" and "Ignorant endurance is not a fine thing". # Socrates then argues, and the interlocutor agrees, that these further premises imply the contrary of the original thesis, in this case, it leads to: "courage is not endurance of the soul". # Socrates then claims that he has shown that his interlocutor's thesis is false and that its negation is true.


Plato's synthesis

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 version of the law of non-contradiction states that "''The same thing clearly cannot act or be acted upon in the same part or in relation to the same thing at the same time, in contrary ways''" (The ''
Republic A republic () is a " state in which power rests with the people or their representatives; specifically a state without a monarchy" and also a "government, or system of government, of such a state." Previously, especially in the 17th and 18th ...
'' (436b)). In this, Plato carefully phrases three
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
atic restrictions on ''action'' or reaction: in the same part, in the same relation, at the same time. The effect is to momentarily create a frozen, timeless
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
, somewhat like figures frozen in action on the frieze of the Parthenon. This way, he accomplishes two essential goals for his philosophy. First, he logically separates the Platonic world of constant change from the formally knowable world of momentarily fixed physical objects. Second, he provides the conditions for 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 ...
method to be used in finding definitions, as for example in the ''
Sophist A sophist ( el, σοφιστής, sophistes) was a teacher in ancient Greece in the fifth and fourth centuries BC. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics, and mathematics. They taught ' ...
''. So Plato's law of non-contradiction is the empirically derived necessary starting point for all else he has to say. In contrast, Aristotle reverses Plato's order of derivation. Rather than starting with ''experience'', Aristotle begins ''a priori'' with the law of non-contradiction as the fundamental axiom of an analytic philosophical system. This axiom then necessitates the fixed, realist model. Now, he starts with much stronger logical foundations than Plato's non-contrariety of action in reaction to conflicting demands from the three parts of the soul.


Aristotle's contribution

The traditional source of the law of non-contradiction is
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 ...
's ''
Metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
'' where he gives three different versions. *
Ontological In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
: "It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect." (1005b19-20) *
Psychological Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries bet ...
: "No one can believe that the same thing can (at the same time) be and not be." (1005b23-24) *
Logical 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 ...
(aka the medieval ''Lex Contradictoriarum''): "The most certain of all basic principles is that contradictory
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 ...
s are not true simultaneously." (1011b13-14) Aristotle attempts several proofs of this law. He first argues that every expression has a single meaning (otherwise we could not communicate with one another). This rules out the possibility that by "to be a man", "not to be a man" is meant. But "man" means "two-footed animal" (for example), and so if anything is a man, it is necessary (by virtue of the meaning of "man") that it must be a two-footed animal, and so it is impossible at the same time for it ''not'' to be a two-footed animal. Thus "it is not possible to say truly at the same time that the same thing is and is not a man" (''Metaphysics'' 1006b 35). Another argument is that anyone who believes something cannot believe its contradiction (1008b). :Why does he not just get up first thing and walk into a well or, if he finds one, over a cliff? In fact, he seems rather careful about cliffs and wells.


Avicenna

Avicenna Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islamic ...
's commentary on the ''
Metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
'' illustrates the common view that the law of non-contradiction "and their like are among the things that do not require our elaboration." Avicenna’s words for "the obdurate" are quite facetious: "he must be subjected to the conflagration of fire, since 'fire' and 'not fire' are one. Pain must be inflicted on him through beating, since 'pain' and 'no pain' are one. And he must be denied food and drink, since eating and drinking and the abstention from both are one nd the same"


Indian philosophy

The law of non-contradiction is found in ancient
Indian logic The development of Indian logic dates back to the ''anviksiki'' of Medhatithi Gautama (c. 6th century BCE); the Sanskrit grammar rules of Pāṇini (c. 5th century BCE); the Vaisheshika school's analysis of atomism (c. 6th century BCE to 2nd centu ...
as a meta-rule in the ''
Shrauta Sutras Kalpa ( sa, कल्प) means "proper, fit" and is one of the six disciplines of the Vedānga, or ancillary science connected with the Vedas – the scriptures of Hinduism. This field of study is focused on the procedures and ceremonies asso ...
'', the grammar of
Pāṇini , era = ;;6th–5th century BCE , region = Indian philosophy , main_interests = Grammar, linguistics , notable_works = ' ( Classical Sanskrit) , influenced= , notable_ideas= Descriptive linguistics (Devana ...
, and the ''
Brahma Sutras The ''Brahma Sūtras'' ( sa, ब्रह्मसूत्राणि) is a Sanskrit text, attributed to the sage bādarāyaṇa or sage Vyāsa, estimated to have been completed in its surviving form in approx. 400–450 CE,, Quote: "...we c ...
'' attributed to
Vyasa Krishna Dvaipayana ( sa, कृष्णद्वैपायन, Kṛṣṇadvaipāyana), better known as Vyasa (; sa, व्यासः, Vyāsaḥ, compiler) or Vedavyasa (वेदव्यासः, ''Veda-vyāsaḥ'', "the one who cl ...
. It was later elaborated on by medieval commentators such as
Madhvacharya Madhvacharya (; ; CE 1199-1278 or CE 1238–1317), sometimes anglicised as Madhva Acharya, and also known as Purna Prajna () and Ānanda Tīrtha, was an Indian philosopher, theologian and the chief proponent of the '' Dvaita'' (dualism) sch ...
.


Leibniz and Kant

Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
and
Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and aest ...
both used the law of non contradiction to define the difference between analytic and synthetic propositions. For Leibniz, analytic statements follow from the law of non contradiction, and synthetic ones from the principle of sufficient reason.


Russell

The principle was stated as a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
of
propositional logic 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 ...
by Russell and Whitehead in ''
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. ...
'' as: :: \mathbf. \ \ \vdash. \thicksim(p.\thicksim p)


Dialetheism

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 ...
advocates the view that ''under some conditions'', some statements can be both true and false simultaneously, or may be true and false at different times.
Dialetheism Dialetheism (from Greek 'twice' and 'truth') is the view that there are statements that are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true ...
arises from formal logical
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 ...
es, such as the
Liar's paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth ...
and
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
.


Alleged impossibility of its proof or denial

The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act which is argued to be self-defeating. Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. Logics known as " paraconsistent" are inconsistency-tolerant logics in that there, from P together with ¬P, it doesn't imply that any proposition follows. Nevertheless, not all paraconsistent logics deny the law of non-contradiction and some such logics even prove it. Some, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true. A related objection is that "negation" in paraconsistent logic is not really ''
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
''; it is merely a
subcontrary An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" ...
-forming operator.Béziau (2000)


In popular culture

The '' Fargo'' episode "
The Law of Non-Contradiction "The Law of Non-Contradiction" is the third episode of the third season of the FX anthology series '' Fargo'', and the twenty-third episode of the series overall. It was directed by series executive producer John Cameron, and written by Matt Wol ...
", which takes its name from the law, was noted for its several elements relating to the law of non-contradiction, as the episode's main character faces several paradoxes. For example, she is still the acting
chief of police Chief may refer to: Title or rank Military and law enforcement * Chief master sergeant, the ninth, and highest, enlisted rank in the U.S. Air Force and U.S. Space Force * Chief of police, the head of a police department * Chief of the boa ...
while having been demoted from the position, and tries to investigate a man that both was and was not named Ennis Stussy, and who both was and was not her stepfather. It also features the story of a robot who, after having spent millions of years unable to help humanity, is told that he greatly helped mankind all along by observing history.


See also

*
Contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
*
First principle In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
* Law of identity *
Oxymoron An oxymoron (usual plural oxymorons, more rarely oxymora) is a figure of speech that juxtaposes concepts with opposing meanings within a word or phrase that creates an ostensible self-contradiction. An oxymoron can be used as a rhetorical devi ...
*
Peirce's law In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that inv ...
*
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 ...
* Principle of sufficient reason *
Trivialism Trivialism is the logical theory that all statements (also known as propositions) are true and that all contradictions of the form "p and not p" (e.g. the ball is red and not red) are true. In accordance with this, a trivialist is a person who b ...


References


Bibliography

* * Béziau (2000). * Lewis, David (1982), "Logic for equivocators", reprinted in ''Papers in Philosophical Logic,'' Cambridge University Press (1997), p. 97-110. * . * Slater (1995).


Further reading

*


External links

* S. M. Cohen,
Aristotle on the Principle of Non-Contradiction
, ''Canadian Journal of Philosophy'', Vol. 16, No. 3. * James Danaher (2004),

, ''The Philosopher'', Vol. LXXXXII No. 1. * Paula Gottlieb,
Aristotle on Non-contradiction
(
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
). * Laurence Horn,
Contradiction
(
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
). *
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 ...
and Francesco Berto,
Dialetheism
(
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
). * Graham Priest and Koji Tanaka,
Paraconsistent logic
(
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
). *Orestes J. Gonzalez, ''Actus Essendi'' and the Habit of the First Principle in Thomas Aquinas (New York: Einsiedler Press, 2019)
/span>. * Peter Suber,

, Earlham College. {{Classical logic Classical logic Theorems in propositional logic