In classical logic, the law of non-contradiction (LNC) (or the principle of non-contradiction (PNC), or the principle of contradiction) is the second of the so-called three classic laws of thought. It states that contradictory statements cannot both at the same time be true, e.g. the two propositions "A is B" and "A is not B" are mutually exclusive.
In the symbolism of propositional logic, the law can expressed as "¬ (P ∧ ¬P)".
The law of noncontradiction, along with its complement, the law of excluded middle (the third of the three classic laws of thought), are correlates of the law of identity (the first of the three laws). Because the law of identity partitions its logical Universe into exactly two parts: a "logical object" and everything else, it creates a dichotomy wherein the two parts are "mutually exclusive" and "jointly exhaustive". The law of noncontradiction 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.
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One difficulty in applying the law of noncontradiction is ambiguity in the propositions. For instance, if time 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.
The law of noncontradiction is found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini,[1] and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya.[2]
According to both Plato and Aristotle, Heraclitus was said to have denied the law of noncontradiction. This is quite likely if, as Plato pointed out, the law of noncontradiction 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", an object simultaneously must be both what it now is and what it will become.
Unfortunately, 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 of the law of noncontradiction. According to Heraclitus, change, and the constant conflict of opposites is the universal logos of nature.
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 noncontradiction 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".[3] 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, judgements, 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, employed an ontological version of the law of noncontradiction 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 only routes of inquiry there are for thinking:
the one that [it] is and that [it] cannot not be
is the path of Persuasion (for it attends upon truth)
the other, that [it] is not and that it is right that [it] not be,
this I point out to you is a path wholly inscrutable
for you could not know what is not (for it is not to be accomplished)
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[4].
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 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 noncontradiction. According to Gregory Vlastos,[5] the method has the following steps:
Plato's version of the law of noncontradiction 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 (436b)). In this, Plato carefully phrases three axiomatic restrictions on action or reaction: 1) in the same part, 2) in the same relation, 3) at the same time. The effect is to momentarily create a frozen, timeless state, somewhat like figures frozen in action on the frieze of the Parthenon[6].
This way, he accomplishes two essential goals for his philosophy. First, he logically separates the Platonic world of constant change[7] from the formally knowable world of fixed physical objects[8]. Second, he provides the conditions for the dialectic method to be used in finding definitions, as for example in the Sophist. So Plato's law of noncontradiction 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 noncontradiction as the fundamental axiom of an analytic philosophical system[9]. 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.
The traditional source of the law of noncontradiction is Aristotle's Metaphysics where he gives three different versions.[10]
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).
Avicenna gives a similar argument:
Leibniz and Kant adopted a different statement, by which the law assumes an essentially different meaning. Their formula is A is not not-A; in other words it is impossible to predicate of a thing a quality which is its contradictory. Unlike Aristotle's law this law deals with the necessary relation between subject and predicate in a single judgment. For example, in Gottlob Ernst Schulze's Aenesidemus, it is asserted, "… nothing supposed capable of being thought may contain contradictory characteristics." Whereas Aristotle states that one or other of two contradictory propositions must be false, the Kantian law states that a particular kind of proposition is in itself necessarily false. On the other hand there is a real connection between the two laws. The denial of the statement A is not-A presupposes some knowledge of what A is, i.e. the statement A is A. In other words a judgment about A is implied.
Kant's analytical judgments of propositions depend on presupposed concepts which are the same for all people. His statement, regarded as a logical principle purely and apart from material facts, does not therefore amount to more than that of Aristotle, which deals simply with the significance of negation.
Traditionally, in Aristotle's classical logical calculus, in evaluating any proposition there are only two possible truth values, "true" and "false." An obvious extension to classical two-valued logic is a many-valued logic for more than two possible values. In logic, a many- or multi-valued logic is a propositional calculus in which there are more than two values. Those most popular in the literature are three-valued (e.g., Łukasiewicz's and Kleene's), which accept the values "true", "false", and "unknown", finite-valued with more than three values, and the infinite-valued (e.g. fuzzy logic and probability logic) logics.
Recently, Graham Priest pointed out that under some conditions, some statements can be both true and false simultaneously, or may be true and false at different times. Applied universally, without specified conditions or axiomatic restrictions, this dialetheism will cause every statement, to explode, to become true. Dialetheism arises from formal logical paradoxes, such as the Liar's paradox and Russell's paradox.
As is true of all axioms of logic, 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 would essentially be self-defeating[14]. Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. Collectively, these logics are known as "paraconsistent" or "inconsistency-tolerant" logics. Graham Priest advances the strongest thesis of this sort, which he calls "dialetheism".
In several axiomatic derivations of logic[15], this is effectively resolved by showing that (P ∨ ¬P) and its negation are constants, and simply defining TRUE as (P ∨ ¬P) and FALSE as ¬(P ∨ ¬P), without taking a position as to the principle of bivalence or the law of excluded middle.
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.[16] A related objection is that "negation" in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator.[17]