Difference between revisions of "Tautology"
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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]]. | ||
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| + | 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 == | ||
Revision as of 23:06, January 6, 2011
The concept of tautology has two definitions, one philosophical, one rhetorical.
Rhetorical tautology [1]is defined as "needless repetition of an idea, statement, or word". An example of a rhetorical tautology would be, for instance, a "godless atheist", "secular science" or a "three-sided triangle" (as opposed to "all triangles have three sides", which is a definition).
A philosophical tautology is [2]"a compound propositional form all of whose instances are true", such as “A or not A.” For example, "This candidate will win or will not win.” Another example is "If it rains, it will rain."
In formal logic, the philosophical definition takes on a technical precision. Within a logical system such as propositional calculus or intuitionistic logic, a formula φ is a tautology if it is its own proof (written symbolically as ). 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.