Self-refuting idea

Self-refuting ideas or self-defeating ideas are ideas or statements whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are called self-refuting by their detractors, and such accusations are therefore almost always controversial, with defenders stating that the idea is being misunderstood or that the argument is invalid. For these reasons, none of the ideas below are unambiguously or incontrovertibly self-refuting. These ideas are often used as axioms, which are definitions taken to be true (tautological assumptions), and cannot be used to test themselves, for doing so would lead to only two consequences: consistency (circular reasoning) or exception (self-contradiction). It is important to know that the conclusion of an argument that is self-refuting is not necessarily false, since it could be supported by another, more valid, argument.

The Epimenides paradox is a statement of the form "this statement is false". Such statements troubled philosophers, especially when there was a serious attempt to formalize the foundations of logic. Bertrand Russell developed his "Theory of Types" to formalize a set of rules that would prevent such statements (more formally Russell's paradox) being made in symbolic logic. This work has led to the modern formulation of axiomatic set theory. While Russell's formalization did not contain such paradoxes, Kurt Gödel showed that it must contain independent statements. Any logical system that is rich enough to contain elementary arithmetic contains at least one proposition whose interpretation is this proposition is unprovable (from within the logical system concerned), and hence no such system can be both complete and consistent.

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