Consistency, validity, soundness, and completeness

Among the important properties that logical systems can have:

• Consistency, which means that no theorem of the system contradicts another.^[14]
• Validity, which means that the system’s rules of proof never allow a false inference from true premises. A logical system has the property of soundness when the logical system has the property of validity and uses only premises that prove true (or, in the case of axioms, are true by definition).^[14]
• Completeness, of a logical system, which means that if a formula is true, it can be proven (if it is true, it is a theorem of the system).
• Soundness, the term soundness has multiple separate meanings, which creates a bit of confusion throughout the literature. Most commonly, soundness refers to logical systems, which means that if some formula can be proven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is the converse of completeness. A distinct, peripheral use of soundness refers to arguments, which means that the premises of a valid argument are true in the actual world.

Some logical systems do not have all four properties. As an example, Kurt Gödel‘s incompleteness theorems show that sufficiently complex formal systems of arithmetic cannot be consistent and complete;^[9] however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.^[15]