The Tautology and Contradiction commands test whether the given Boolean expression b is a tautology or a contradiction.

The Tautology(b) calling sequence returns true if b is a tautology (true for every valuation of its variables) and false otherwise. Similarly, Contradiction(b) returns true if b is a contradiction (false for every valuation of its variables) and false otherwise.

If the parameter p is supplied and the test returns false, then a valuation is assigned to p which demonstrates a negative result. Otherwise, p is assigned NULL. Note that the test may be significantly faster if p is not given.

The problem of determining whether every possible assignment of truth values satisfies a given Boolean formula is the dual of the Boolean satisfiability problem and is known to be co-NP-complete, meaning that no polynomial-time algorithm is presently known.

$\mathrm{with}\left(\mathrm{Logic}\right)\:$

$\mathrm{Tautology}\left(\left(a\&andb\&or\mathrm{`\&not`}\left(a\right)\right)\&or\mathrm{`\&not`}\left(b\right)\right)$

$\mathrm{true}$

$\mathrm{Tautology}\left(\left(a\&iffb\right)\&orb\,\'p\'\right)$

$\mathrm{false}$

$p$

$\left\{a=\mathrm{true}\,b=\mathrm{false}\right\}$

$\mathrm{Tautology}\left(a\&or\mathrm{`\&not`}\left(a\right)\,\'p\'\right)$

$\mathrm{true}$

$p$

$\mathrm{Contradiction}\left(\left(a\&or\mathrm{`\&not`}\left(a\&andb\right)\right)\&norb\right)$

$\mathrm{true}$

$\mathrm{Contradiction}\left(\left(a\&iffb\right)\&orb\,\'p\'\right)$

$\mathrm{false}$

$p$

$\left\{a=\mathrm{false}\,b=\mathrm{true}\right\}$

$\mathrm{Contradiction}\left(\mathrm{`\&not`}\left(a\right)\&anda\,\'p\'\right)$

$\mathrm{true}$

$p$