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Since for contradictory it is true that for all (because ), 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, anything follows").
In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion [a] [b] is the law according to which any statement can be proven from a contradiction. [1] [2] [3] That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion. [4] [5]
(This set is sometimes called "the Russell set".) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:
A theory is consistent if it is not possible to prove a contradiction from the axioms of the theory. A theory is complete if, for every formula in its signature, either that formula or its negation is a logical consequence of the axioms of the theory.
So Plato's law of non-contradiction is the empirically derived necessary starting point for all else he has to say. [13] In contrast, Aristotle reverses Plato's order of derivation. Rather than starting with experience, Aristotle begins a priori with the law of non-contradiction as the fundamental axiom of an analytic philosophical system. [14]
Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884. In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
Contradictory impulses in Nazi views on masculinity made it impossible to police when much-vaunted male camaraderie crossed the line into unacceptable homoerotic intimacy, precisely because there ...
An axiomatic system is said to be consistent if it lacks contradiction.That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explo