THE LOGIC NOTES

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

  1. The set consisting of the formulae
              p OR q,   p IMP q,   NOTp
    is consistent.
  2. 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.

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