contradictory statement

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 contradictory statement, a.k.a. 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 $⋏$.

For a simple example, the statement  $P\wedge \mathrm{\neg }P$  is a contradiction for any statement $P$.

The negation $\mathrm{\neg }Q$ of every contradiction $Q$ is a tautology, and vice versa:

$$\mathrm{\neg }⋏=⋎,\mathrm{\neg }⋎=⋏$$

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 “contradiction (http://planetmath.org/Contradiction)”.