Contradictions

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, usually opposite 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.

By creation of a paradox, Plato's Euthydemus dialogue demonstrates the need for the notion of contradiction. In the ensuing dialogue Dionysodorus denies the existence of "contradiction", all the while that Socrates is contradicting him:

Indeed, Dionysodorus agrees that "there is no such thing as false opinion ... there is no such thing as ignorance" and demands of Socrates to "Refute me." Socrates responds "But how can I refute you, if, as you say, to tell a falsehood is impossible?".

In classical logic, particularly in propositional and first-order logic, a proposition ${\displaystyle \varphi }$ is a contradiction if and only if ${\displaystyle \varphi \vdash \bot }$. Since for contradictory ${\displaystyle \varphi }$ it is true that ${\displaystyle \vdash \varphi \rightarrow \psi }$ for all ${\displaystyle \psi }$ (because ${\displaystyle \bot \rightarrow \psi }$), one may prove any proposition from a set of axioms which contains contradictions. This is called the "principle of explosion" or "ex falso quodlibet" ("from falsity, whatever you like").

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