Consistent Glossary
Definition
A set of formulae is inconsistent if there exists a derivation of a contradiction from it. The set is consistent otherwise.
Comments
A non-trivial fact, amounting to the soundness and completeness of the natural deduction system, is that a set is consistent if it is satisfiable.
A set of sentences in a natural language may similarly be said to be consistent or inconsistent, according to whether or not it logically implies a contradiction.
Examples
-
The set consisting of the formulae
p OR q, p IMP q, NOTp
is consistent. -
The set consisting of the formulae
p OR q, q IMP p, NOTp
is inconsistent, because it implies both p and NOTp: it already contains NOTp, and p OR q, q IMP p ⊢ p.