Contradiction

A contradiction is a logical incompatibility between two or more statements or propositions. It occurs when the 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 presumed to be contradictory. Illustrating a general tendency 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 Logic of Contradictions

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.

The logical form of a 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, '•' is the symbol for conjunction, 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. 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, for logical purposes, is that a false statement implies anything. To put it in logical form, 'P implies Q' is always true 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. 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.

Contradiction outside of formal logic

In colloquial speech

^{This short section requires expansion.}

Colloquially, actions or statements, or both, are said to contradict each other when they are, or are perceived as being, due to presuppositions which are contradictory in the logical sense.

In Dialectics

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.^{[citation needed]}

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.^[1]

Proof by contradiction

Also known as "reductio ad absurdum"

For a proposition $\varphi$ it is true that $\varphi \vdash \top$ , i. e. that $\varphi$ is a tautology, i. e. that it is always true, if and only if $\neg \varphi \vdash \bot$ , i. e. if the negation of $\varphi$ is a contradiction. Therefore, a proof that $\neg \varphi \vdash \bot$ also proves that $\varphi$ is true. The use of this fact constitutes the technique of the proof by contradiction, which is used extensively in mathematics.

Contradictions and philosophy

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.

Meta-contradiction

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.

Notes

↑ Deising, Paul. Hegel’s Dialectical Political Economy. ISBN 0813391318.

External link

Stanford Encyclopedia of Philosophy, Contradiction.

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[deising-1] Deising, Paul. Hegel’s Dialectical Political Economy. ISBN 0813391318.

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