A self-refuting idea or self-defeating idea is an idea or statement 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).
Variations
Directly self-denying statements Directly self-denying statements are characterised by being necessarily (or inherently) false. 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.
Indirectly self-denying statements
One form of an indirect self-denying statement is the "Stolen Concept": the act of using a concept while ignoring, contradicting or denying the validity of the concepts on which it logically and/or genetically depends. The idea of the "Stolen Concept" is generally attributed to be first noted by Ayn Rand and then later supported by followers of Objectivism. Much like Objectivism the idea of the "Stolen Concept" does not have mainstream acceptance in academia. An example of the stolen concept fallacy is anarchist Pierre-Joseph Proudhon's statement, "All property is theft".
While discussing the hierarchical nature of knowledge, Nathaniel Branden states, "Theft" is a concept that logically and genetically depends on the antecedent concept of "rightfully owned property"—and refers to the act of taking that property without the owner's consent. If no property is rightfully owned, that is, if nothing is property, there can be no such concept as "theft." Thus, the statement "All property is theft" has an internal contradiction: to use the concept "theft" while denying the validity of the concept of "property," is to use "theft" as a concept to which one has no logical right—that is, as a stolen concept. Others have said the statement is fallacious only on a superficial reading of Proudhon, devoid of context. Proudhon used the term "property" with reference to claimed ownership in land, factories, etc. He believed such claims were illegitimate, and thus a form of theft from the commons. Proudhon explicitly states that the phrase "property is theft" is analogous to the phrase "slavery is murder". According to Proudhon, the slave, though biologically alive, is clearly in a sense "murdered". The "theft" in his terminology does not refer to ownership any more than the "murder" refers directly to physiological death, but rather both are meant as terms to represent a denial of specific rights.
In logic Self-refutation plays an important role in some inconsistency tolerant logics (e.g. paraconsistent logics and direct logic) that lack proof by contradiction. For example, the negation of a proposition can be proved by showing that the proposition implies its own negation. Likewise, it can be inferred that a proposition cannot be proved by (1) showing that a proof would imply the negation of the proposition or by (2) showing a proof would imply that the negation of the proposition can be proved.
Examples
Brain in a vat Brain in a vat is a thought experiment in philosophy which is premised upon the skeptical hypothesis that one could actually be a brain in a vat receiving electrical input identical to that which would be coming from the nervous system. Similar premises are found in Descartes's evil demon and dream argument. Philosopher Hilary Putnam argues that some versions of the thought experiment would be inconsistent due to semantic externalism. For a brain in a vat that had only ever experienced the simulated world, the statement "I'm not a brain in a vat" is true. The only possible brains and vats it could be referring to are simulated, and it is true that it is not a simulated brain in a simulated vat. By the same argument, saying "I'm a brain in a vat" would be false.
Determinism It has been argued by advocates of libertarian free will that to call determinism a rational statement is doubly self-defeating.
To count as rational, a belief must be freely chosen, which according to the determinist is impossible Any kind of debate seems to be posited on the idea that the parties involved are trying to change each other's minds.
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