2.4.15: Summary
Proof systems provide purely syntactic methods for characterizing consequence and compatibility between sentences. The sequent calculus is one such proof system. A derivation in it consists of a tree of sequents (a sequent \(\Gamma \Sequent \Delta\) consists of two sequences of formulas separated by \(\Sequent\) ). The topmost sequents in a derivation are initial sequents of the form \(A \Sequent A\) . All other sequents, for the derivation to be correct, must be correctly justified by one of a number of inference rules . These come in pairs; a rule for operating on the left and on the right side of a sequent for each connective and quantifier. For instance, if a sequent \(\Gamma \Sequent \Delta, A \lif B\) is justified by the \(\RightR{\lif}\) rule, the preceding sequent (the premise ) must be \(A, \Gamma \Sequent \Delta, B\) . Some rules also allow the order or number of sentences in a sequent to be manipulated, e.g., the \(\RightR{\Exchange}\) rule allows two formulas on the right side of a sequent to be switched.
If there is a derivation of the sequent \(\quad \Sequent A\) , we say \(A\) is a theorem and write \(\Proves A\) . If there is a derivation of \(\Gamma_0 \Sequent A\) where every \(B\) in \(\Gamma_0\) is in \(\Gamma\) , we say \(A\) is derivable from \(\Gamma\) and write \(\Gamma \Proves A\) . If there is a derivation of \(\Gamma_0 \Sequent \quad\) where every \(B\) in \(\Gamma_0\) is in \(\Gamma\) , 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 validity of the conclusion sequent provided the premise sequents are valid.