In 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 inversions of each other. 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.”
By extension, outside of classical logic, one can speak of contradictions between actions when one presumes that their motives contradict each other.
Contradiction in formal logic
In classical logic
, particularly in propositional
and first-order logic
, a proposition
is a contradiction if and only if
. Since for contradictory
it is true that
), one may prove any proposition from a set of axioms which contains contradictions.
Contradictions and philosophy
Adherents of the epistemological
theory of coherentism
typically claim that as a necessary condition of the justification of a belief
, that belief must form a part of a logically non-contradictory (consistent) system
of beliefs. Some dialetheists
, including Graham Priest
, have argued that coherence may not require consistency.
A pragmatic contradiction occurs when the very statement of the argument contradicts the claims it purports. An inconsistency arises, in this case, because the act of utterance, rather than the content of what is said, undermines its conclusion.
For examples, Heraclitus
’s proposition that knowledge is impossible; or, arguably, Nietzsche
’s statement that one should not obey others, or Moore's paradox
. These are self-refuting statements
and performative contradictions
Contradiction outside formal logic
can label actions or statements (or both) as contradicting each other when due (or perceived as due) to presuppositions
which are contradictory in the logical sense.
In dialectical materialism, contradiction, as 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 stem from the social structure of society and inherently lead to class conflict, economic crisis, and eventually revolution, the existing order’s overthrow and the formerly oppressed classes’ ascension to political power.
Mao Zedong's philosophical essay furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction.
Proof by contradiction is used in mathematics to construct proofs.