2.5.14: Summary

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Proof systems provide purely syntactic methods for characterizing consequence and compatibility between sentences. Natural deduction is one such proof system. A derivation in it consists of a tree formulas. The topmost formulas in a derivation are assumptions. All other formulas, for the derivation to be correct, must be correctly justified by one of a number of inference rules. These come in pairs; an introduction and an elimination rule for each connective and quantifier. For instance, if a formula \(A\) is justified by a \(\Elim{\lif}\) rule, the preceding formulas (the premises) must be \(B \lif A\) and \(B\) (for some \(B\)). Some inference rules also allow assumptions to be discharged. For instance, if \(A \lif B\) is inferred from \(B\) using \(\Intro{\lif}\), any occurrences of \(A\) as assumptions in the derivation leading to the premise \(B\) may be discharged, and is given a label that is also recorded at the inference.

If there is a derivation with end formula \(A\) and all assumptions are discharged, we say \(A\) is a theorem and write \(\Proves A\). If all undischarged assumptions are in some set \(\Gamma\), we say \(A\) is derivable from \(\Gamma\) and write \(\Gamma \Proves A\). If \(\Gamma \Proves \lfalse\) we say \(\Gamma\) is inconsistent, otherwise consistent. These notions are interrelated, e.g., \(\Gamma \Proves A\) iff \(\Gamma \cup \{\lnot A\}\) is inconsistent. They are also related to the corresponding semantic notions, e.g., if \(\Gamma \Proves A\) then \(\Gamma \Entails A\). This property of proof systems—what can be derived from \(\Gamma\) is guaranteed to be entailed by \(\Gamma\)—is called soundness. The soundness theorem is proved by induction on the length of derivations, showing that each individual inference preserves entailment of its conclusion from open assumptions provided its premises are entailed by their undischarged assumptions.