2.4.1: Rules and Derivations

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For the following, let \(\Gamma, \Delta, \Pi, \Lambda\) represent finite sequences of sentences.

Definition \(\PageIndex{1}\): Sequent

A sequent is an expression of the form \[\Gamma \Sequent \Delta\nonumber\] where \(\Gamma\) and \(\Delta\) are finite (possibly empty) sequences of sentences of the language \(\Lang L\). \(\Gamma\) is called the antecedent, while \(\Delta\) is the succedent.

The intuitive idea behind a sequent is: if all of the sentences in the antecedent hold, then at least one of the sentences in the succedent holds. That is, if \(\Gamma = \tuple{A_1, \dots, A_m}\) and \(\Delta = \tuple{B_1, \dots, B_n}\), then \(\Gamma \Sequent \Delta\) holds iff \[(A_1 \land \cdots \land A_m) \lif (B_1 \lor \cdots \lor B_n)\nonumber\] holds. There are two special cases: where \(\Gamma\) is empty and when \(\Delta\) is empty. When \(\Gamma\) is empty, i.e., \(m = 0\), \(\quad \Sequent \Delta\) holds iff \(B_1 \lor \dots \lor B_n\) holds. When \(\Delta\) is empty, i.e., \(n = 0\), \(\Gamma \Sequent \quad\) holds iff \(\lnot(A_1 \land \dots \land A_m)\) does. We say a sequent is valid iff the corresponding sentence is valid.

If \(\Gamma\) is a sequence of sentences, we write \(\Gamma, A\) for the result of appending \(A\) to the right end of \(\Gamma\) (and \(A, \Gamma\) for the result of appending \(A\) to the left end of \(\Gamma\)). If \(\Delta\) is a sequence of sentences also, then \(\Gamma, \Delta\) is the concatenation of the two sequences.

Definition \(\PageIndex{2}\): Initial Sequent

An initial sequent is a sequent of one of the following forms:

1. \(A \Sequent A\)
2. \(\lfalse \Sequent \quad\)

for any sentence \(A\) in the language.

Derivations in the sequent calculus are certain trees of sequents, where the topmost sequents are initial sequents, and if a sequent stands below one or two other sequents, it must follow correctly by a rule of inference. The rules for \(\Log{LK}\) are divided into two main types: logical rules and structural rules. The logical rules are named for the main operator of the sentence containing \(A\) and/or \(B\) in the lower sequent. Each one comes in two versions, one for inferring a sequent with the sentence containing the logical operator on the left, and one with the sentence on the right.