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{{short description|Logic theorem}}
{{for|the [[Fargo (TV series)|Fargo]] episode|The Law of Non-Contradiction}}
{{hatnote|This article uses forms of [[Mathematical logic|logical]] notation. For a concise description of the symbols used in this notation, see [[List of logic symbols]].}}

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 same time, e. g. the two propositions "''the house is white''" and "''the house is not white''" are [[Mutual exclusivity|mutually exclusive]]. [[Logical form|Formally]], this is expressed as the [[tautology (logic)|tautology]] ¬(p ∧ ¬p). For example it is tautologous to say "the house is not both white and not white" since this results from putting "the house is white" in that formula, yielding "not (the house is white and not (the house is white))", then rewriting this in natural English. The law is not to be confused with the [[law of excluded middle]] which states that at least one of two propositions like "the house is white" and "the house is not white" holds.

One reason to have this law is the [[principle of explosion]], which states that anything follows from a contradiction. The law is employed in a ''[[reductio ad absurdum]]'' proof.

To express the fact that the law is tenseless and to avoid [[equivocation]], 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 [[Law_of_thought#The_three_traditional_laws|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 [[Rule of inference|inference rules]], such as ''[[modus ponens]]'' or [[De Morgan's laws]].

The law of non-contradiction and the law of excluded middle create a [[dichotomy]] in a so-called logical space, the points in which are all the consistent combinations of propositions. Each combination would contain exactly one member of each pair of contradictory propositions, so the space would have two parts which 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 is an expression of its jointly exhaustive aspect.

== Interpretations ==
One difficulty in applying the law of non-contradiction is ambiguity in the propositions.<ref>{{Cite web|date=2014|title=Logic, Ontological Neutrality, and the Law of Non-Contradiction|url=http://www.columbia.edu/~av72/papers/LNC_2014.pdf|website=Columbeia.edu}}</ref> 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.

=== Indian philosophy ===
The [[Buddhist]] [[Tripitaka]] attributes to [[Mahavira|Nigaṇṭha Nātaputta]], who lived in the 6th century BCE, the implicit formulation of the law of noncontradiction, “‘See how upright, honest and sincere Citta, the householder, is’; and, a little later, he also says: ‘See how Citta, the householder, is not upright, honest or sincere.’ To this, Citta replies: ‘if your former statement is true, your latter statement is false and if your latter statement is true, your former statement is false.’” Early explicit formulations of the law of noncontradiction were [[ontology|ontic]], with later 2nd century Buddhist philosopher [[Nagarjuna]] stating “when something is a single thing, it cannot be both existent and non-existent” similar to Aristotle’s own ontic formulation that “that a thing cannot at the same time be and not be”.<ref>{{Citation |last=Gillon |first=Brendan |title=Logic in Classical Indian Philosophy |date=2011|url=https://plato.stanford.edu/Archives/Spr2012/entries/logic-india/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |access-date=2024-03-03}}.</ref>

=== Heraclitus ===
According to both [[Plato]] and [[Aristotle]],<ref>{{citation|title=Metaphysics (IV,1005b)|last=Aristotle |quote=to suppose that the same thing is and is not, as some imagine that Heraclitus says}}</ref> [[Heraclitus]] was ''said'' to have denied the law of non-contradiction. This is quite likely<ref>{{citation|title=Fragments 36,57,59 (Bywater)|last=Heraclitus}}</ref> if, as Plato [[Law of Contradiction#Plato's synthesis|pointed out]], the law of non-contradiction does not hold for changing things in the world. If a philosophy of [[Becoming (philosophy)|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.<ref>{{citation| last=Cornford| first=F.M.| title=Plato's Theory of Knowledge|page=234}}</ref>

So little remains of Heraclitus' 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 (set theory)|complement]] of the law of non-contradiction. According to Heraclitus, change, and the constant conflict of opposites is the universal [[logos]] 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]] 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".<ref>(80B1 [[Diels-Kranz|DK]]). According to Plato's ''[[Theaetetus (dialogue)|Theaetetus]]'', section 152a. [https://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plat.+Theaet.+152a]</ref> 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]] employed an [[ontological]] 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 (Parmenides)|On Nature]], he said,
{{Quotation|the only routes of inquiry there are for thinking:<br>
the one that [it] is and that [it] cannot not be<br>
is the path of Persuasion (for it attends upon truth)<br>
the other, that [it] is not and that it is right that [it] not be,<br>
this I point out to you is a path wholly inscrutable<br>
for you could not know what is not (for it is not to be accomplished)<br>
nor could you point it out... For the same thing is for thinking and for being}}

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.<ref>Curd, Patricia, "Presocratic Philosophy", ''The Stanford Encyclopedia of Philosophy (Summer 2011 Edition)'', Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/sum2011/entries/presocratics/</ref>

=== Socrates ===
In Plato's early dialogues, Socrates uses the [[Socratic method#Method|elenctic method]] to investigate the nature or definition of ethical concepts such as justice or virtue. Elenctic refutation depends on a [[dichotomy|dichotomous]] thesis, one that may be divided into exactly two [[mutually exclusive]] 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,<ref>Gregory Vlastos, 'The Socratic Elenchus', ''Oxford Studies in Ancient Philosophy I'', Oxford 1983, 27–58.</ref> the method has the following steps:
# Socrates' [[Interlocutor (linguistics)|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]]'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 (dialogue)|Republic]]'' (436b)). In this, Plato carefully phrases three [[axiom]]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 of affairs (philosophy)|state]], somewhat like figures frozen in action on the frieze of the Parthenon.<ref>[http://www.the-philosopher.co.uk/2004/05/the-laws-of-thought-2004.html James Danaher, ''The Laws of Thought''] "The restrictions Plato places on the laws of thought (i.e., "in the same respect," and "at the same time,") are an attempt to isolate the object of thought by removing it from all other time but the present and all respects but one."</ref>

This way, he accomplishes two essential goals for his philosophy. First, he logically separates the Platonic world of constant change<ref>Plato's [[Analogy of the divided line|Divided Line]] describes the four Platonic worlds</ref> from the formally knowable world of momentarily fixed physical objects.<ref>''Cratylus'', starting at [https://www.perseus.tufts.edu/hopper/text?doc=Plat.+Crat.+439e&fromdoc=Perseus%3Atext%3A1999.01.0172 439e]</ref><ref>"A thing which is F at one time, or in one way, or in one relation, or from one point of view, will be all too often not-F, at another time, in another way" ("Metaphysical Paradox" in Gregory Vlastos, ''Platonic Studies'', p.50)</ref> Second, he provides the conditions for the [[dialectic]] method to be used in finding definitions, as for example in the ''[[Sophist (dialogue)|Sophist]]''. So Plato's law of non-contradiction is the empirically derived necessary starting point for all else he has to say.<ref>"Two Principles of Noncontradiction" in Samuel Scolnicov, ''Plato's Parmenides'', pp.12-16</ref>

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.<ref>Similarly, Kant remarked that Newton "''by no means dared to prove this law a priori, and therefore appealed rather to experience''" (''Metaphysical Foundations'', 4:449)</ref> 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]]'s ''[[Metaphysics (Aristotle)|Metaphysics]]'' where he gives three different versions.<ref>{{Harvtxt|Łukasiewicz|1971}} p.487</ref>
*[[Ontological]]: "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]]: "No one can believe that the same thing can (at the same time) be and not be." (1005b23–24)<ref>Whitaker, CWA Aristotle's De Interpretatione: Contradiction and Dialectic page 184</ref>
*[[Logical]] (aka the medieval ''Lex Contradictoriarum''):<ref>{{Cite book |url=https://books.google.com/books?id=Qi8QAQAAIAAJ&q=%22lex+contradictoriarum%22&pg=PA7 |title = Peter of Spain (Petrus Hispanus Portugalensis): Tractatus: Called afterwards Summule logicales. First critical ed. From the manuscripts|isbn = 9789023209751|last1 = Rijk|first1 = Lambertus Marie de|year = 1972| publisher=Van Gorcum }}</ref> "The most certain of all basic principles is that contradictory [[proposition]]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.<ref>1008b, trans. Lawson-Tancred</ref>

=== Avicenna ===
[[Avicenna]]'s [[The Book of Healing#Philosophy|commentary]] on the ''[[Metaphysics (Aristotle)|Metaphysics]]'' 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 [and the same]."<ref>Avicenna, ''[[Metaphysics]]'', I.8 53.13–15 (sect. 12 [p. 43] in ed. Michael Marmura); commenting on Aristotle, ''Topics'' I.11.105a4–5. The editorial addition (brackets) is present in Marmura's translation.</ref>

=== Thomas Aquinas ===
[[Thomas Aquinas]] argued that the principle of non-contradiction is essential to the reasoning of human beings ("One cannot reasonably hold two mutually exclusive beliefs at the same time"). He argued that human reasoning without the principle of non-contradiction is utterly impossible because reason itself can't function with two contradictory ideas. Aquinas argued that this is the same both for moral arguments as well as theological arguments and even machinery (“the parts must work together, the machine can’t work if two parts are incompatible”).<ref>{{cite book | url=https://books.google.com/books?id=Kpa44dKXaiYC&dq=Thomas+Aquinas+nicht-widerspruchsprinzip&pg=PA264 | isbn=978-3-16-148741-5 | title=Contradictio: Theorien und Bewertungen des Widerspruchs in der Theologie des Mittelalters | date=2005 | publisher=Mohr Siebeck }}</ref><ref>{{cite book | chapter-url=https://www.degruyter.com/document/doi/10.1515/9783110837711.753/html?lang=de | doi=10.1515/9783110837711.753 | chapter="Socrates est": Zur Logik singulärer Existenzaussagen nach Thomas von Aquin | title=Sprache und Erkenntnis im Mittelalter, 2. Halbbd | date=1981 | last1=Weidemann | first1=Hermann | isbn=978-3-11-083771-1 }}</ref>

=== Leibniz and Kant ===
[[Gottfried Wilhelm Leibniz|Leibniz]] and [[Immanuel Kant|Kant]] both used the law of non-contradiction to define the difference between [[Analytic–synthetic distinction|analytic and synthetic]] propositions.<ref>{{Citation |last=Wilson |first=Catherine |title=Leibniz's Influence on Kant |date=2018 |url=https://plato.stanford.edu/archives/spr2018/entries/kant-leibniz/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |edition=Spring 2018 |publisher=Metaphysics Research Lab, Stanford University |access-date=2021-02-20}}.</ref> 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]] of [[propositional calculus|propositional logic]] by [[Bertrand Russell|Russell]] and [[Alfred North Whitehead|Whitehead]] in  ''[[Principia Mathematica]]'' as:

:: <math>\mathbf{*3\cdot24}. \ \  \vdash. \thicksim(p.\thicksim p)</math><ref>{{citation |author=[[Alfred North Whitehead]], [[Bertrand Russell]] |title=Principia Mathematica |pages=116–117 |year=1910 |publisher=[[Cambridge]]}}[http://name.umdl.umich.edu/aat3201.0001.001].</ref>

=== Dialetheism ===

[[Graham Priest]] 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]] arises from formal logical [[paradox]]es, such as the [[liar paradox|Liar's paradox]] and [[Russel's paradox|Russell's paradox]], even though it isn't the only solution to them.<ref>{{Citation |last1=Priest |first1=Graham |title=Dialetheism |date=2018 |url=https://plato.stanford.edu/archives/fall2018/entries/dialetheism/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |edition=Fall 2018 |publisher=Metaphysics Research Lab, Stanford University |access-date=2022-07-10 |last2=Berto |first2=Francesco |last3=Weber |first3=Zach}}</ref><ref>{{Cite journal |last1=Hájek |first1=Petr |last2=Paris |first2=Jeff |last3=Shepherdson |first3=John |date=2000 |title=The Liar Paradox and Fuzzy Logic |url=https://www.jstor.org/stable/2586541 |journal=The Journal of Symbolic Logic |volume=65 |issue=1 |pages=339–346 |doi=10.2307/2586541 |jstor=2586541 |s2cid=6865763 |issn=0022-4812}}</ref><ref>{{Cite web |title=Russell's Paradox {{!}} Brilliant Math & Science Wiki |url=https://brilliant.org/wiki/russells-paradox/ |access-date=2023-10-19 |website=brilliant.org |language=en-us}}</ref>

== Alleged impossibility of its proof or denial ==
The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the ground 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 that is argued to be [[Self-refuting idea|self-defeating]].<ref>[http://faculty.washington.edu/smcohen/AristotlePNC.pdf S.M. Cohen, ''Aristotle on the Principle of Non-Contradiction''] "''Aristotle's solution in the Posterior Analytics is to distinguish between episteme (scientific knowledge) and nous (intuitive intellect). First principles, such as PNC, are not objects of scientific knowledge - since they are not demonstrable - but are still known, since they are grasped by nous.''"</ref> Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. 

Logics known as "[[paraconsistent logic|paraconsistent]]" are inconsistency-tolerant logics in that there, from P together with ¬P, it does not imply that any proposition follows. Nevertheless, not all paraconsistent logics deny the law of non-contradiction and some such logics even prove it.<ref>{{cite book |title=The law of non-contradiction : new philosophical essays |date=2004 |publisher=Clarendon Press |location=Oxford |isbn=9780199265176 |doi=10.1093/acprof:oso/9780199265176.003.0024}}</ref><ref>{{cite web |last1=Carnielli |first1=Walter |last2=Rodrigues |first2=Abilio |title=An epistemic approach to paraconsistency: a logic of evidence and truth |url=http://philsci-archive.pitt.edu/14115/1/letj.pdf |publisher=[[University of Pittsburgh|Pittsburg]] |access-date=January 20, 2024}}</ref>

Some, such as [[David Lewis (philosopher)|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.<ref>Lewis (1982)</ref> A related objection is that "negation" in paraconsistent logic is not really ''[[negation]]''; it is merely a [[Square of opposition|subcontrary]]-forming operator.<ref>Slater (1995)</ref>{{full citation needed|date=July 2016}}<ref>Béziau (2000)</ref>{{full citation needed|date=July 2016}} Those who (like the dialetheists) claim that the Law of Non-Contradiction can be violated are in fact using a different definition of negation, and therefore talking about something else other than the Law of Non-Contradiction which is based on a particular definition of negation and therefore cannot be violated.<ref>{{cite journal |last1=Loke |first1=A.T.E. |title=The Law of Non-contradiction and Global Philosophy of Religion |journal=SOPHIA |date=2024 |doi=10.1007/s11841-024-01001-5}}</ref>

== In popular culture ==

The ''[[Fargo (TV series)|Fargo]]'' episode "[[The Law of Non-Contradiction]]", 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]] 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.<ref name=Fargo>{{cite web|url=http://uproxx.com/sepinwall/fargo-the-law-of-non-contradiction-recap-review/|title=Is 'Fargo' Still 'Fargo' If It's In Los Angeles? You Betcha!|work=[[Uproxx]]|date=May 3, 2017|access-date=May 6, 2017}}</ref>

== See also ==
{{div col|colwidth=22em}}
* [[Contradiction]]
* [[First principle]]
* [[Law of identity]]
* [[Oxymoron]]
* [[Peirce's law]]
* [[Principle of bivalence]]
* [[Principle of sufficient reason]]
* [[Trivialism]]
{{div col end}}

== References ==
{{Reflist}}

== Bibliography ==
* {{cite book|title=Aristotle's Metaphysics|author = Aristotle|editor = Lawson-Tancred, H.|year=1998|publisher=Penguin}}
* Béziau, J. Y. (2000). What is paraconsistent logic. Frontiers of paraconsistent logic, 95-111.
* Lewis, David (1982), "Logic for equivocators", reprinted in ''Papers in Philosophical Logic,'' Cambridge University Press (1997), p. 97-110.
* {{Citation|last=Łukasiewicz|first=Jan|year=1971|title=On the Principle of Contradiction in Aristotle|journal=Review of Metaphysics|volume=24|pages=485–509|orig-year=1910 in Polish}}.
* Slater, B. H. (1995). Paraconsistent logics?. Journal of Philosophical logic, 24(4), 451-454.

==Further reading==
*{{cite book|title=Socrates' Second Sailing: On Plato's Republic|last=Benardete|first=Seth|year=1989|publisher=University of Chicago Press}}

== External links ==
* S. M. Cohen, "[http://faculty.washington.edu/smcohen/AristotlePNC.pdf Aristotle on the Principle of Non-Contradiction]", ''Canadian Journal of Philosophy'', Vol. 16, No. 3.
* James Danaher (2004), "[http://www.the-philosopher.co.uk/2004/05/the-laws-of-thought-2004.html The Laws of Thought]", ''The Philosopher'', Vol. LXXXXII No. 1.
* Paula Gottlieb, "[http://plato.stanford.edu/entries/aristotle-noncontradiction/ Aristotle on Non-contradiction]" ([[Stanford Encyclopedia of Philosophy]]).
* [[Laurence Horn]], "[http://plato.stanford.edu/entries/contradiction/ Contradiction]" ([[Stanford Encyclopedia of Philosophy]]).
* [[Graham Priest]] and Francesco Berto, "[http://plato.stanford.edu/entries/dialetheism/ Dialetheism]" ([[Stanford Encyclopedia of Philosophy]]).
* Graham Priest and Koji Tanaka, "[http://plato.stanford.edu/entries/logic-paraconsistent/ Paraconsistent logic]" ([[Stanford Encyclopedia of Philosophy]]).

{{Mathematical logic}}
{{Classical logic}}

[[Category:Classical logic]]
[[Category:Theorems in propositional logic]]