Law of noncontradiction
Encyclopedia
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.
as a meta-rule in the Shrauta Sutras
, the grammar of Pāṇini, and the Brahma Sutras
attributed to Vyasa
. It was later elaborated on by medieval commentators such as Madhvacharya
.
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.
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". 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.
, 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 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.
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, the method has the following steps:
'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.
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 fixed physical objects. 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. 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.
's Metaphysics
where he gives three different versions.
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:
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.
'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.
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.
. 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, 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. A related objection is that "negation" in paraconsistent logic is not really negation
; it is merely a subcontrary
-forming operator.
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
, 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
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
(the third of the three classic laws of thought), are correlates of the law of identity
Law of identity
In logic, the law of identity is the first of the so-called three classic laws of thought. It states that an object is the same as itself: A → A ; While this can also be listed as A ≡ A this is redundant Any reflexive relation upholds 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
Concept and object
In the philosophy of language, the distinction between concept and object is attributable to the German philosopher Gottlob Frege.According to Frege, any sentence that expresses a singular thought consists of an expression that signifies an Object together with a predicate In the philosophy of...
" and everything else, it creates a dichotomy
Dichotomy
A dichotomy is any splitting of a whole into exactly two non-overlapping parts, meaning it is a procedure in which a whole is divided into two parts...
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.
Interpretations
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.Eastern philosophy
The law of noncontradiction is found in ancient Indian logicIndian logic
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...
as a meta-rule in the Shrauta Sutras
Kalpa (Vedanga)
Kalpa is one of the six disciplines of Vedanga, treating ritual.Tradition does not single out any special work in this branch of the Vedanga; but sacrificial practice gave rise to a large number of systematic sutras for the several classes of priests...
, the grammar of Pāṇini, and the Brahma Sutras
Brahma Sutras
The Brahma sūtras , also known as Vedānta Sūtras , are one of the three canonical texts of the Vedānta school of Hindu philosophy. A thorough study of Vedānta requires a close examination of these three texts, known in Sanskrit as the Prasthanatrayi, or the three starting points...
attributed to Vyasa
Vyasa
Vyasa is a central and revered figure in most Hindu traditions. He is also sometimes called Veda Vyasa , or Krishna Dvaipayana...
. It was later elaborated on by medieval commentators such as Madhvacharya
Madhvacharya
Madhvācārya was the chief proponent of Tattvavāda "Philosophy of Reality", popularly known as the Dvaita school of Hindu philosophy. It is one of the three most influential Vedānta philosophies. Madhvācārya was one of the important philosophers during the Bhakti movement. He was a pioneer in...
.
Heraclitus
According to both Plato and Aristotle, HeraclitusHeraclitus
Heraclitus of Ephesus was a pre-Socratic Greek philosopher, a native of the Greek city Ephesus, Ionia, on the coast of Asia Minor. He was of distinguished parentage. Little is known about his early life and education, but he regarded himself as self-taught and a pioneer of wisdom...
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
Becoming (philosophy)
The concept of becoming was born in eastern ancient Greece by the philosopher Heraclitus of Hephesus, who in the Sixth century BC, said that nothing in this world is constant except change or 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
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of the law of noncontradiction. According to Heraclitus, change, and the constant conflict of opposites is the universal logos
Logos
' is an important term in philosophy, psychology, rhetoric and religion. Originally a word meaning "a ground", "a plea", "an opinion", "an expectation", "word," "speech," "account," "reason," it became a technical term in philosophy, beginning with Heraclitus ' is an important term in...
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 noncontradiction must be applicable to personal judgments.The most famous saying of Protagoras
Protagoras
Protagoras was a pre-Socratic Greek philosopher and is numbered as one of the sophists by Plato. In his dialogue Protagoras, Plato credits him with having invented the role of the professional sophist or teacher of virtue...
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, 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
ParmenidesParmenides
Parmenides of Elea was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy. The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form. In this poem, 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
On Nature
On Nature was a philosophical poem which details Anaximander's theories about the evolution of the Earth, plants, animals and humankind. Anaximander described his theory that humans and other animals descended from fish once the world's oceans began to dry up. Also he described a theory of...
, 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 dichotomousDichotomy
A dichotomy is any splitting of a whole into exactly two non-overlapping parts, meaning it is a procedure in which a whole is divided into two parts...
thesis, one that may be divided into exactly two mutually exclusive
Mutually exclusive
In layman's terms, two events are mutually exclusive if they cannot occur at the same time. An example is tossing a coin once, which can result in either heads or tails, but not both....
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, the method has the following steps:
- Socrates' interlocutorInterlocutorInterlocutor may refer to:* Interlocutor , the master of ceremonies of a minstrel show* Interlocutor , someone who informally explains the views of a government and also can relay messages back to a government...
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
PlatoPlato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
'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
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
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
State of affairs (philosophy)
In philosophy, a state of affairs, or a situation, is a way the actual world must be in order to make some given proposition about the actual world true; in other words, a state of affairs is a truth-maker, whereas a proposition is a truth-bearer...
, 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 fixed physical objects. Second, he provides the conditions for the dialectic
Dialectic
Dialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...
method to be used in finding definitions, as for example in the Sophist
Sophist (dialogue)
The Sophist is a Platonic dialogue from the philosopher's late period, most likely written in 360 BCE. Having criticized his Theory of Forms in the Parmenides, Plato presents a new conception of the forms in the Sophist, more mundane and down-to-earth than its predecessor...
. 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. 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 noncontradiction is AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's Metaphysics
Metaphysics (Aristotle)
Metaphysics is one of the principal works of Aristotle and the first major work of the branch of philosophy with the same name. The principal subject is "being qua being", or being understood as being. It examines what can be asserted about anything that exists just because of its existence and...
where he gives three different versions.
- 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)
- logical: "The most certain of all basic principles is that contradictory propositionPropositionIn logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
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
Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā , commonly known as Ibn Sīnā or by his Latinized name Avicenna, was a Persian polymath, who wrote almost 450 treatises on a wide range of subjects, of which around 240 have survived...
gives a similar argument:
- Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.
Leibniz and Kant
Leibniz and KantKANT
KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...
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
Gottlob Ernst Schulze
Gottlob Ernst Schulze was born in Heldrungen . Schulze was a professor at Wittenberg, Helmstedt, and Göttingen...
's Aenesidemus
Aenesidemus (book)
Aenesidemus was a German book published anonymously by Professor Gottlob Ernst Schulze of Helmstedt in 1792. It attempted to refute the principles that Karl Leonhard Reinhold established in support of Immanuel Kant's Critique of Pure Reason. The title's reference is to Aenesidemus, who was an...
, 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.
Modern logics
Traditionally, in AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's classical logical calculus
Term logic
In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...
, in evaluating any proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
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
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, a many- or multi-valued logic is a propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
in which there are more than two values. Those most popular in the literature are three-valued (e.g., Łukasiewicz's and Kleene's
Stephen Cole Kleene
Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science...
), which accept the values "true", "false", and "unknown", finite-valued with more than three values, and the infinite-valued (e.g. fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
and probability logic
Probabilistic logic
The aim of a probabilistic logic is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. The result is a richer and more expressive formalism with a broad range of possible application areas...
) logics.
Dialetheism
Recently, Graham PriestGraham Priest
Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
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
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
restrictions, this dialetheism
Dialetheism
Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...
will cause every statement, to explode
Principle of explosion
The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
, to become true. Dialetheism arises from formal logical paradoxes
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
, such as the Liar's paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
and Russell's paradox.
Alleged impossibility of its proof or denial
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-defeatingSelf-refuting idea
Self-refuting ideas are ideas or statements whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are accused by their detractors of being self-refuting, and such accusations are therefore almost always controversial, with defenders claiming that...
. Since the early 20th century, certain logicians have proposed logics that deny the validity of the law. Collectively, these logics are known as "paraconsistent
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
" or "inconsistency-tolerant" logics. Graham Priest
Graham Priest
Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
advances the strongest thesis of this sort, which he calls "dialetheism
Dialetheism
Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...
".
In several axiomatic derivations of logic, 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
Principle of bivalence
In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
or the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
.
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
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
; it is merely a subcontrary
Square of opposition
In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...
-forming operator.
See also
- ContradictionContradictionIn classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
- First principle
- Identity (philosophy)Identity (philosophy)In philosophy, identity, from , is the relation each thing bears just to itself. According to Leibniz's law two things sharing every attribute are not only similar, but are the same thing. The concept of sameness has given rise to the general concept of identity, as in personal identity and...
- Law of excluded middleLaw of excluded middleIn logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
- Law of identityLaw of identityIn logic, the law of identity is the first of the so-called three classic laws of thought. It states that an object is the same as itself: A → A ; While this can also be listed as A ≡ A this is redundant Any reflexive relation upholds the law of identity...
- Laws of thought
- Peirce's lawPeirce's lawIn 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 involves only one sort of connective, namely...
- Principle of bivalencePrinciple of bivalenceIn logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
- Principle of explosionPrinciple of explosionThe principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
- Reductio ad absurdumReductio ad absurdumIn logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
- Three classic laws of thought
- OxymoronOxymoronAn oxymoron is a figure of speech that combines contradictory terms...
External links
- S.M. Cohen, "Aristotle on the Principle of Non-Contradiction", Canadian Journal of Philosophy, Vol. 16, No. 3
- James Danaher, "The Laws of Thought", The Philosopher, Vol. LXXXXII No. 1
- Laurence Horn, "Contradiction" (Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
) - Graham PriestGraham PriestGraham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
and Francesco Berto, "Dialetheism" (Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
) - Graham Priest and Koji Tanaka, "Paraconsistent logic" (Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
) - Peter Suber, "Non-Contradiction and Excluded Middle", Earlham College