Editing Law of noncontradiction (section)

== Interpretations ==
{{Original research|section|date=May 2025}}
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>