 One of the goals of a logical system is to arrive at a rigorous characterization of the notion of validity. In a logical system, which normally consists of a language, a proof-theory and a semantics, validity can be defined proof-theoretically or semantically. Proof-theoretically, validity is defined in terms of formal proofs. An argument is proof-theoretically valid (\(\vdash\)) if there is a proof from some or all of its premises to its conclusion. Semantically or model-theoretically, validity is normally defined in terms of truth-preservation. An argument is model-theoretically valid (\(\models\)) if there is no interpretation (in the semantics) in which its premises are all true and its conclusion false. In classical logic, the proof-theoretical and the model-theoretical perspectives have been proven to characterize exactly the same notion of validity (soundness and completeness theorems): an argument is proof-theoretically valid iff it is model-theoretically valid.