Proof procedure

In logic, and in particular proof theory, a proof procedure is a method of proving statements. A statement p that is provable from a non-empty set &Gamma; of statements in a theory K is called a deduction of p from &Gamma; in K. If &Gamma; is empty then p is either a theorem of K (i.e., $$p \in K$$) or an axiom of K if K is axiomatic. We express that p is deducible (or provable or derivable or demonstrable) from &Gamma; in K in symbols as


 * $$ \Gamma \vdash_K p$$.

There are several types of proof procedures. The most popular are natural deduction, sequent calculi (i.e., Gentzen type systems), Hilbert type axiomatic systems, and semantic tableaux or trees.

We call a proof procedure M for a theory K sound if each and every member of the set of deducible formulas is valid (with respect to the semantics of K). We say that M is complete if the set of deducible formulas is a superset of the set of valid formulas of K. (Generally this last statement regarding completeness states that M is complete if the set of deducible formulas is exactly the set of valid formulas of K. But proof procedures that prove, e.g., every formula, are trivially complete even if K is consistent. They are, however, unsound.)