2.4.5: Derivations
We’ve said what an initial sequent looks like, and we’ve given the rules of inference. Derivations in the sequent calculus are inductively generated from these: each derivation either is an initial sequent on its own, or consists of one or two derivations followed by an inference.
An \(\Log{LK}\) -derivation of a sequent \(S\) is a tree of sequents satisfying the following conditions:
- The topmost sequents of the tree are initial sequents.
- The bottommost sequent of the tree is \(S\).
- Every sequent in the tree except \(S\) is a premise of a correct application of an inference rule whose conclusion stands directly below that sequent in the tree.
We then say that \(S\) is the end-sequent of the derivation and that \(S\) is derivable in \(\Log{LK}\) (or \(\Log{LK}\) -derivable).
Every initial sequent, e.g., \(C \Sequent C\) is a derivation. We can obtain a new derivation from this by applying, say, the \(\LeftR{\Weakening}\) rule,
The rule, however, is meant to be general: we can replace the \(A\) in the rule with any sentence, e.g., also with \(D\) . If the premise matches our initial sequent \(C \Sequent C\) , that means that both \(\Gamma\) and \(\Delta\) are just \(C\) , and the conclusion would then be \(D, C \Sequent C\) . So, the following is a derivation:
We can now apply another rule, say \(\LeftR{\Exchange}\) , which allows us to switch two sentences on the left. So, the following is also a correct derivation:
In this application of the rule, which was given as
both \(\Gamma\) and \(\Pi\) were empty, \(\Delta\) is \(C\) , and the roles of \(A\) and \(B\) are played by \(D\) and \(C\) , respectively. In much the same way, we also see that
is a derivation. Now we can take these two derivations, and combine them using \(\RightR{\land}\) . That rule was
In our case, the premises must match the last sequents of the derivations ending in the premises. That means that \(\Gamma\) is \(C, D\) , \(\Delta\) is empty, \(A\) is \(C\) and \(B\) is \(D\) . So the conclusion, if the inference should be correct, is \(C, D \Sequent C \land D\) .
Of course, we can also reverse the premises, then \(A\) would be \(D\) and \(B\) would be \(C\) .