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Difference between revisions of "Tautology"

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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: $A\supset A$, $A\lor\neg A$, $(A\supset B)\supset(\neg B\supset\neg A)$.
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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: $A\supset A$, $A\lor\neg A$, $(A\supset B)\supset(\neg B\supset\neg A)$.
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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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====Comments====
 
  
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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.
  
 
====References====
 
====References====
 
<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">[a1]</TD> <TD valign="top">  Yu.I. Manin,  "A course in mathematical logic" , Springer  (1977)  pp. 31, 54  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  Peter J. Cameron, "Sets, Logic and Categories" Springer (2012) ISBN 1447105893</TD></TR>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  Peter J. Cameron, "Sets, Logic and Categories" Springer (2012) {{ISBN|1447105893}}</TD></TR>
 
</table>
 
</table>

Latest revision as of 14:26, 12 November 2023

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.

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=54410
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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