The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity.
Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true.
Semantic completeness is the converse of soundness for formal systems.
That is: A formal system S is refutation-complete if it is able to derive false from every unsatisfiable set of formulas.
Intuitively, strong completeness means that, given a formula set
, while refutation completeness means that, given a formula set
Examples of refutation-complete systems include: SLD resolution on Horn clauses, superposition on equational clausal first-order logic, Robinson's resolution on clause sets.
Gödel's incompleteness theorem shows that any computable system that is sufficiently powerful, such as Peano arithmetic, cannot be both consistent and syntactically complete.
In this sense, a formal system is syntactically complete if and only if no unprovable sentence can be added to it without introducing an inconsistency.
Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example, the propositional logic statement consisting of a single propositional variable A is not a theorem, and neither is its negation).