General validity

The property of a logical formula of being true for any interpretation of the non-logical symbols that it contains, that is, the predicate and propositional variables. Logical formulas with this property are called generally valid, identically true or tautologies. Every generally-valid formula expresses a logical law. Instead of the words "A is generally valid" one often writes "A". The most important forms of logical formulas are propositional and predicate formulas. In the classical sense of logical operations (cf. Logical operation), the general validity of a propositional formula is verified by the construction of a truth table: A formula is generally valid if and only if it takes the value T ("true") for any truth values of the propositional variables. General validity of a predicate formula means truth in any model (in logic). The set of generally-valid predicate formulas is undecidable, that is, there is no algorithm allowing one to decide whether an arbitrary predicate formula is generally valid or not. The Gödel completeness theorem implies that all generally-valid predicate formulas, and only those, are derivable in classical predicate calculus.

References

[1]	S.C. Kleene, "Mathematical logic", Wiley (1967)

Comments

The term "universally valid" is often used instead of "generally valid".