Difference between revisions of "Tautology"
From Encyclopedia of Mathematics
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| − | A formula of the language of propositional calculus taking the [[ | + | A formula of the language of [[propositional calculus]] taking the [[truth value]] "true" independently of the truth values "true" or "false" taken by its propositional variables. Examples: $A\supset A$, $A\lor\neg A$, $(A\supset B)\supset(\neg B\supset\neg A)$. |
| − | In general one can check whether a given propositional formula is a tautology by simply examining the finite set of all combinations of values of its propositional variables. | + | In general one can check whether a given propositional formula is a tautology by simply examining its [[truth table]]: the finite set of all combinations of values of its propositional variables. It is usual to give a presentation of propositional calculus which is both ''[[Sound rule|sound]]'': every theorem deducible in the system is a tautology; and ''[[Completeness (in logic)|complete]]'': every tautology is a theorem. |
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Yu.I. Manin, "A course in mathematical logic" , Springer (1977) pp. 31, 54 (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> Yu.I. Manin, "A course in mathematical logic" , Springer (1977) pp. 31, 54 (Translated from Russian)</TD></TR> | ||
| + | <TR><TD valign="top">[b1]</TD> <TD valign="top"> Peter J. Cameron, "Sets, Logic and Categories" Springer (2012) ISBN 1447105893</TD></TR> | ||
| + | </table> | ||
Revision as of 12:30, 20 November 2016
A formula of the language of propositional calculus taking the truth value "true" independently of the truth values "true" or "false" taken by its propositional variables. Examples: $A\supset A$, $A\lor\neg A$, $(A\supset B)\supset(\neg B\supset\neg A)$.
In general one can check whether a given propositional formula is a tautology by simply examining its truth table: the finite set of all combinations of values of its propositional variables. It is usual to give a presentation of propositional calculus which is both sound: every theorem deducible in the system is a tautology; and complete: every tautology is a theorem.
Comments
References
| [a1] | Yu.I. Manin, "A course in mathematical logic" , Springer (1977) pp. 31, 54 (Translated from Russian) |
| [b1] | Peter J. Cameron, "Sets, Logic and Categories" Springer (2012) ISBN 1447105893 |
How to Cite This Entry:
Tautology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tautology&oldid=32584
Tautology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tautology&oldid=32584
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article