First published Wed Jun 28, 2006; substantive revision Wed Aug 29, 2018
Do I contradict myself?
Very well, then, I contradict myself.
(I am large, I contain multitudes.)
—Walt Whitman, “Song of Myself”
Vorrei e non vorrei.
—Zerlina, “Là ci darem la mano”, Don Giovanni
This entry outlines the role of the law of non-contradiction (LNC) as the foremost among the first (indemonstrable) principles of Aristotelian philosophy and its heirs, and depicts the relation between LNC and LEM (the law of excluded middle) in establishing the nature of contradictory and contrary opposition. §1 presents the classical treatment of LNC as an axiom in Aristotle's “First Philosophy” and reviews the status of contradictory and contrary opposition as schematized on the Square of Opposition. §2 explores in further detail the possible characterizations of LNC and LEM, including the relevance of future contingent statements in which LEM (but not LNC) is sometimes held to fail. §3 addresses the mismatch between the logical status of contradictory negation as a propositional operator and the diverse realizations of contradictory negation within natural language. §4 deals with several challenges to LNC within Western philosophy, including the paradoxes, and the relation between systems with truth-value gaps (violating LEM) and those with truth-value gluts (violating LNC). In §5, the tetralemma of Buddhist logic is discussed within the context of gaps and gluts; it is suggested that apparent violations of LNC in this tradition (and others) can be attributed to either differing viewpoints of evaluation, as foreseen by Aristotle, or to intervening modal and epistemic operators. §6 focuses on the problem of “borderline contradictions”: the range of acceptability judgments for apparently contradictory sentences with vague predicates as surveyed in empirical studies, and the theoretical implications of these studies. Finally, §7 surveys the ways of contradiction and its exploitation in literature and popular culture from Shakespeare to social media.
- 1. LNC as Indemonstrable
- 2. LEM and LNC
- 3. Contradictory Negation in Term and Propositional Logic
- 4. Gaps and Gluts: LNC and Its Discontents
- 5. LNC and the Buddhist Tetralemma
- 6. Vagueness and Borderline Contradictions
- 7. Contradiction in Everyday Life
- Bibliography
- Academic Tools
- Other Internet Resources
- Related Entries
1. LNC as Indemonstrable
The twin foundations of Aristotle's logic are the law of non-contradiction (LNC) (also known as the law of contradiction, LC) and the law of excluded middle (LEM). In Metaphysics Book Γ, LNC—“the most certain of all principles”—is defined as follows:
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, and all other specifications that might be made, let them be added to meet local objections (1005b19–23).
It will be noted that this statement of the LNC is an explicitly modal claim about the incompatibility of opposed properties applying to the same object (with the appropriate provisos). Since Łukasiewicz (1910), this ontological version of the principle has been recognized as distinct from, and for Aristotle arguably prior to, the logical formulation (“The opinion that opposite assertions are not simultaneously true is the firmest of all”—Met. 1011b13–14) and the psychologicalformulation (“It is impossible for anyone to believe that the same thing is and is not, as some consider Heraclitus said”—Met. 1005b23–25) offered elsewhere in Book Γ; we return to Heraclitus below. Wedin (2004a), who argues for the primacy of the ontological version (see also Meyer 2008, Other Internet Resources), formalizes it as ¬◊(∃x)(Fx ∧ ¬Fx). These three formulations of LNC differ in important respects, in particular as to whether the law is explicitly modal in character, whether it applies to propositions or to properties and objects, and whether it requires the invocation of a metalinguistic truth predicate. (See also the entry Aristotle on non-contradiction.)
For Aristotle, the status of LNC as a first, indemonstrable principle is obvious. Those who mulishly demand a proof of LNC clearly “lack education”: since “a demonstration of everything is impossible”, resulting in infinite regress. At least some principles must be taken as primitiveaxiomata rather than derived from other propositions—and what principle more merits this status than LNC? (1006a6–12). In first philosophy, as in mathematics, an axiom is both indemonstrable and indispensable; without LNC, Aristotle argues, “a is F” and “a is not F” are indistinguishable and no argumentation is possible. While Sophists and “even many physicists” may claim that it ispossible for the same thing to be and not to be at the same time and in the same respect, such a position self-destructs “if only our opponent says something”, since as soon as he opens his mouth to make an assertion, any assertion, he must accept LNC. But what if he does not open his mouth? Against such an individual “it is ridiculous to seek an argument” for he is no more than a vegetable (1006a1–15).
The celebrated Arab commentator Avicenna (ibn Sīnā, 980–1037) confronts the LNC skeptic with a more severe outcome than Aristotle's vegetable reduction: “As for the obstinate, he must be plunged into fire, since fire and non-fire are identical. Let him be beaten, since suffering and not suffering are the same. Let him be deprived of food and drink, since eating and drinking are identical to abstaining” (Metaphysics I.8, 53.13–15).
The role of LNC as the basic, indemonstrable “first principle” is affirmed by Leibniz, for whom LNC is taken as interdefinable with the Law of Identity that states that everything is identical to itself: “Nothing should be taken as first principles but experiences and the axiom of identity or (what is the same thing) contradiction, which is primitive, since otherwise there would be no difference between truth and falsehood, and all investigation would cease at once, if to say yes or no were a matter of indifference” (Leibniz 1696/Langley 1916: 13–14). For Leibniz, everybody—even “barbarians”—must tacitly assume LNC as part of innate knowledge implicitly called upon at every moment, thus demonstrating the insufficiency of Locke's empiricism (ibid., 77).[1]
In accounting for the incompatibility of truth and falsity, LNC lies at the heart of Aristotle's theory of opposition, governing both contradictories and contraries. (See traditional square of opposition.) Contradictory opposites (“She is sitting”/“She is not sitting”) are mutually exhaustive as well as mutually inconsistent; one member of the pair must be true and the other false, assuming with Aristotle that singular statements with vacuous subjects are always false. As it was put by the medievals, contradictory opposites divide the true and the false between them; for Aristotle, this is the primary form of opposition.[2] Contrary opposites (“He is happy”/“He is sad”) are mutually inconsistent but not necessarily exhaustive; they may be simultaneously false, though not simultaneously true. LNC applies to both forms of opposition in that neither contradictories nor contraries may belong to the same object at the same time and in the same respect (Metaphysics1011b17–19). What distinguishes the two forms of opposition is a second indemonstrable principle, the law of excluded middle (LEM): “Of any one subject, one thing must be either asserted or denied” (Metaphysics 1011b24). Both laws pertain to contradictories, as in a paired affirmation (“Sis P”) and denial (“S isn't P”): the negation is true whenever the affirmation is false, and the affirmation is true when the negation is false. Thus, a corresponding affirmation and negation cannot both be true, by LNC, but neither can they both be false, by LEM. But while LNC applies both to contradictory and contrary oppositions, LEM holds only for contradictories: “Nothing can exist between two contradictories, but something may exist between contraries” (Metaphysics1055b2): a dog cannot be both black and white, but it may be neither.
As Aristotle explains in the Categories, the opposition between contradictories—“statements opposed to each other as affirmation and negation”—is defined in two ways. First, unlike contrariety, contradiction is restricted to statements or propositions; terms are never related as contradictories. Second, “in this case, and in this case only, it is necessary for the one to be true and the other false” (13b2–3).
Opposition between terms cannot be contradictory in nature, both because only statements (subject-predicate combinations) can be true or false (Categories 13b3–12) and because any two terms may simultaneously fail to apply to a given subject.[3] But two statements may be members of either a contradictory or a contrary opposition. Such statements may be simultaneously false, although (as with contradictories) they may not be simultaneously true. The most striking aspect of the exposition for a modern reader lies in Aristotle's selection of illustrative material. Rather than choosing an uncontroversial example involving mediate contraries, those allowing an unexcluded middle (e.g. “This dog is white”/“This dog is black”; “Socrates is good”/“Socrates is bad”), Aristotle offers a pair of sentences containing immediate contraries, “Socrates is sick”/“Socrates is well”. These propositions may both be false, even though every person is either ill or well: “For if Socrates exists, one will be true and the other false, but if he does not exist, both will be false; for neither ‘Socrates is sick’ nor ‘Socrates is well’ will be true, if Socrates does not exist at all” (13b17–19). But given a corresponding affirmation and negation, one will always be true and the other false; the negation “Socrates is not sick” is true whether the snub-nosed philosopher is healthy or non-existent: “for if he does not exist, ‘he is sick’ is false but ‘he is not sick’ true” (13b26–35).
Members of a canonical pair of contradictories are formally identical except for the negative particle:
An affirmation is a statement affirming something of something, a negation is a statement denying something of something…It is clear that for every affirmation there is an opposite negation, and for every negation there is an opposite affirmation…Let us call an affirmation and a negation which are opposite a contradiction (De Interpretatione 17a25–35).
But this criterion, satisfied simply enough in the case of singular expressions, must be recast in the case of quantified expressions, both those which “signify universally” (“every cat”, “no cat”) and those which do not (“some cat”, “not every cat”).
For such cases, Aristotle shifts from a formal to a semantically based criterion of opposition (17b16–25). Members of an A/O pair (“Every man is white”/“Not every man is white”) or I/E pair (“Some man is white”/“No man is white”) are contradictories because in any state of affairs one member of each pair must be true and the other false. Members of an A/E pair—“Every man is just”/“No man is just”—constitute contraries, since these cannot both be true simultaneously but can both be false. The contradictories of these contraries (“Not every man is just”/“Some man is just”) can be simultaneously true with reference to the same subject (17b23–25). This last opposition of I and O statements, later to be dubbed subcontraries because they appear below the contraries on the traditional square, is a peculiar opposition indeed; Aristotle elsewhere (Prior Analytics 63b21–30) sees I and O as “only verbally opposed”, given the consistency of the Istatement, e.g. “Some Greeks are bald”, with the corresponding O statement, “Some Greeks aren't bald” (or “Not all Greeks are bald”, which doesn't necessarily amount to the same thing, given existential import; see traditional square of opposition).
The same relations obtain for modal propositions, for propositions involving binary connectives like “and” and “or”, for quantificational adverbs, and for a range of other operators that can be mapped onto the square in analogous ways utilizing the same notions of contradictory and contrary opposition and with unilateral entailment definable on duals (see Horn 1989). Thus for example we have the modal square below, based on De Interpretatione 21b10ff. and Prior Analytics 32a18–28, where the box and diamond symbols denote necessity and possibility, respectively. As with universal affirmatives and universal negatives, necessity and impossibility constitute contraries: “A priest must marry” and “A priest can’t marry” can both be (and, on the Episcopalian reading, are) false but cannot both be true. “A priest can marry” and “A priest can (if he wants) not marry” are subcontraries; these can be simultaneously true but not simultaneously false. And necessity, as in “A priest must marry”, unilaterally entails its possibility dual counterpart, “A priest may marry”.
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(1) Modal Square
