Difference between revisions of "Tautology"
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Another example is "If it [[rain]]s, it will rain." | Another example is "If it [[rain]]s, it will rain." | ||
| + | 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]]. | ||
== References == | == References == | ||
Revision as of 03:29, March 31, 2008
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.
