Editing Theorem (section)

=== Syntax and semantics ===
{{Main|Syntax (logic)|Formal semantics (logic)}}

The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a ''true proposition,'' which introduces [[semantics]]. Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. [[belief]], [[Theory of justification|justification]] or other [[Modal logic|modalities]]). The [[soundness]] of a formal system depends on whether or not all of its theorems are also [[Validity (logic)|validities]]. A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are [[tautology (logic)|tautologies]]). A formal system is considered [[completeness (logic)|semantically complete]] when all of its theorems are also tautologies.