Propositional calculus
Proof systems

Proof systems in propositional logic can be broadly classified into semantic proof systems and syntactic proof systems, according to the kind of logical consequence that they rely on: semantic proof systems rely on semantic consequence (φ ⊨ ψ {\displaystyle \varphi \models \psi }), whereas syntactic proof systems rely on syntactic consequence (φ ⊢ ψ {\displaystyle \varphi \vdash \psi }). Semantic consequence deals with the truth values of propositions in all possible interpretations, whereas syntactic consequence concerns the derivation of conclusions from premises based on rules and axioms within a formal system. This section gives a very brief overview of the kinds of proof systems, with anchors to the relevant sections of this article on each one, as well as to the separate Wikipedia articles on each one.
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