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In formal [[logic]], the philosophical definition takes on a technical precision. Within a logical system such as [[proposition]]al [[calculus]] or [[intuitionistic logic]], a [[mathematical formula|formula]] φ is a tautology if it is its own [[proof]] (written symbolically as <math>\phi \vdash \phi</math>). This is equivalent to saying that φ is true under all truth assignments, or, by [[Gödel's completeness theorem]], that every collection of formulas containing φ is [[sound]].
In formal [[logic]], the philosophical definition takes on a technical precision. Within a logical system such as [[proposition]]al [[calculus]] or [[intuitionistic logic]], a [[mathematical formula|formula]] φ is a tautology if it is its own [[proof]] (written symbolically as <math>\phi \vdash \phi</math>). This is equivalent to saying that φ is true under all truth assignments, or, by [[Gödel's completeness theorem]], that every collection of formulas containing φ is [[sound]].
An example of a rhetorical tautology is presented in Christian YouTube producer [[shockofgod]]'s "challenge to atheists", in which he requests that atheist's give "proof and evidence" that atheism is "accurate and correct"; two prime examples of rhetorical tautologies. 'Proof' and 'evidence' are semantically different only to a negligible degree; whilst 'accurate' and 'correct' are in fact exactly the same.
== References ==
== References ==