contradictory statementContradictoryStatement2006-12-09 12:15:562008-03-13 10:18:41Definitionfalse% this is the default PlanetMath preamble. as your knowledge
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A contradictory statement is a statement (or form) which is false due to its logical form rather than because of the meaning of the terms employed.
In propositional logic, a {\em contradictory statement}, a.k.a. {\em contradiction}, is a statement which is false regardless of the truth values of the substatements which form it.\, According to G. Peano, one may generally denote a contradiction with the symbol $\curlywedge$.
For a simple example, the statement\, $P\!\wedge\!\lnot P$\, is a contradiction for any statement $P$.
The negation $\lnot Q$ of every contradiction $Q$ is a tautology, and vice versa:
$$\lnot\curlywedge = \curlyvee, \;\;\; \lnot\curlyvee = \curlywedge$$
To test a given statement or form to see if it is a contradiction, one may construct its truth table.\, If it turns out that every value of the last column is ``F'', then the statement is a contradiction.
Cf. the entry ``\PMlinkname{contradiction}{Contradiction}''.