A contradiction is a logical incompatibility between two or more statements or propositions. It occurs when those statements or propositions, taken together, yield a falsehood. By extension, outside of logic, contradictions are also said to occur between actions for which the motives are held to be contradictory, or between beliefs or principles when their content is contradictory.

Pointing to this principle in applied logic, Aristotle’s law of noncontradiction states that “One cannot say of something that it is and that it is not in the same respect and at the same time.”

The simplest or classic form of contradiction is the assertion both of some statement or proposition and its negation. So, for example, the statement-pair: "All fire engines are red," and "It is not true that all fire engines are red" is contradictory. This means that one of those statements must be false; they cannot both be true at the same time and in the same manner.

The logical form of a simple contradiction is "Statement + negation of that statement." Stated in symbolic form, this would be:

'p and not p', or 'p•~p'

Where 'p' is some statement or proposition (it can be either a simple statement or proposition, or a complex one), '•' is the symbol for conjunction (or "and"), and '~' is the symbol for negation.

The problem with any statement-set or proposition-set of the form 'p•~p'is that it is always false. This occurs because if 'p' is true, then 'not p' is false, and if 'p' is false, then 'not p' is true. But a conjunction, in order to be true, must have both of its conjuncts ( i.e. both of the statements that make up the conjunction) true. So, whether or not 'p' is true, 'p•~p' is always false.

The problem about that is that, in logic, a false statement implies anything. To put it in logical form,

'P implies Q' is always true, regardless of the truth or falsity of 'Q', when 'P' is false.

Thus, if 'P' is a contradiction (i.e. if its inner form is 'p•~p'), then it is always false, and thus it implies anything. This is often expressed as "A contradiction implies anything and everything." So a contradiction is useless for purposes of logic or evidence because it is always false, and thus anything at all follows logically from it.

In actual speech or practice, people rarely assert some proposition or statement 'P' and its negation 'not P' in such an obvious or bald form. More often, they assert some claim or statement or proposition 'P' and then go on to make other claims or statements or propositions, not realizing that those additional claims or statements or propositions have 'not P' as a logical (or other) consequence. Thus they do not realize that the totality of their claims or assertions or statements is contradictory—i.e. embodies a contradiction—because they fail to notice that one of their statements or propositions has a logical consequence that is the negation of one of their other claims or assertions or statements. But the logical result is there and is the same: the fact that the totality of their claims or assertions or statements embodies a contradiction means that this totality implies anything and everything, thus it is useless for evidentiary or logical purposes. (It may have the rhetorical effect of getting their hearers to assent or agree to something, but this is assenting or agreeing to a falsehood.)

Also known as "reductio ad absurdum"

In logic and mathematics, a proposition or statement $\varphi$ is a tautology if it is always true.

For example, the statement

'either p or not p'
symbolized as 'p v ~p'
where 'p' is any statement, ' v ' is the symbol for 'or' or disjunction, and '~' is the symbol for 'not' or negation,

is always true.

This occurs because a disjunction is true just in case one of the disjuncts (one of the statements on either side of the 'or' or ' v ') is true. But if 'p' is true, then this makes 'p v ~p' true. But if 'p' is false, then '~p' is true, and this also makes 'p v ~p' true. Thus 'p v ~p' is a tautology because it is always true.

Moreover, any proposition or statement $\varphi$ is a contradiction if it is always false.

But if something is a tautology (i.e. always true), then its negation is always false, and if something is a contradiction (i.e. always false) then its negation is always true (i.e. it is a tautology). This can be summed up as: The negation of a contradiction is a tautology, and the negation of a tautology is a contradiction. "Contradiction" and "always false" mean the same thing, logically speaking, as do "tautology" and "always true."

Thus, if $\varphi$ is a contradiction, then the negation of $\varphi$, or ~$\varphi$ is true. So proving that something is a contradiction constitutes a proof that its negation is true, because the negation of a contradiction—i.e. the negation of something that is false—is always true.

This method is known as proof by contradiction (or reductio ad absurdum), and is used extensively in logic and mathematics. The method consists of assuming the truth of some statement or proposition, showing that this assumption leads to a contradiction, and thus concluding that the negation of the original assumption is true.

## Contradiction outside of formal logic

### In colloquial speech

Colloquially, actions or statements, or both, are said to contradict each other when they are or are perceived as being in opposition to one another, due to presuppositions or necessary conditions or hidden background statements or beliefs which are contradictory in the logical sense.

### Marxism

In dialectical materialism contradiction, derived by Karl Marx from Hegelianism, usually refers to an opposition of social forces. Most prominently, according to Marx, capitalism entails a social system that has contradictions because the social classes have conflicting collective goals. These contradictions are based in the social structure of society and inherently lead to class conflict, crisis, and eventually revolution, the existing order’s overthrow and the formerly oppressed classes’ ascension to political power.

### Liberalism

The idea of a contradiction as a conflict based in a social structure is not unique to Marxist thought. For liberal thinkers, the problem of public goods may be interpreted as a contradiction in that there is a conflict between what is good for society, e. g., the production of a public good, and what is good for individual free riders who refuse to pay the costs of the public good. This is another interpretation of the Hegelian contradiction.

### Environmental Ethics

In environmental ethics or environmental concerns, the tragedy of the commons (see external link below), also known as the paradox of the commons, first presented by Garrett Hardin, is based on the inherent contradiction between private rationality and collective rationality; this paradox shows that what is rational behavior for an individual often becomes irrational and tragic if everyone were to do it.

### Chinese thought: Han Feizi

In Han Feizi, Han Feizi illustrated a contradiction by a man who was selling both: a shield that no spear can penetrate, and a spear that can pierce any shield. A member of the audience asked the man, "What is going to happen if you used the spear to pierce the shield?"

Coherentism is an epistemological theory in which a belief is justified based at least in part on being part of a logically non-contradictory system of beliefs. The coherence theory of truth takes the fact that a body of propositions or statements does not contain any contradiction as showing that they are true.

The coherence theory of truth stands in opposition to the semantic theory of truth that holds that a statement is true just in case the state of affairs that it asserts is true. Thus the semantic theory of truth asserts that the statement "Snow is white" is true just in case snow is white. (See the article 'Alfred Tarski" for an account of the semantic theory of truth, first put forward by Tarski.)

It often occurs in philosophy that the presence of the argument contradicts with the claims of the argument. An example of this is Heraclitus’s proposition that knowledge is impossible, or, arguably, Nietzsche’s statement that one should not obey others.

## References

Almost all elementary logic textbooks contain discussions of contradiction, tautology, and the relationship between them. They also discuss the fact that 'P implies Q' is always true when 'P' is false.

Two examples of such logic texts are:

Some other useful works:

• Hardin, Garrett. 1968. "The Tragedy of the Commons." Science, 162: 1243-1248.
• Quine, Willard Van Orman. 1961. The Ways of Paradox. Cambridge, MA: Harvard University Press.
• Reese, William L. 1996. "Contradiction, Principle of." pp. 139-140 in Dictionary of Philosophy and Religion. Highlands NJ: Humanities Press. ISBN 0391038656