A Systematic Approach to Canonicity in the Classical Sequent Calculus

The sequent calculus is often criticized for requiring proofs to contain large amounts of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps—such as instantiating a block of quantifiers—by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical firstorder logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means of abstracting away the details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that make the foci as parallel as possible are canonical. Moreover, such proofs are isomorphic to expansion proofs—a well known, minimalistic, and parallel generalization of Herbrand disjunctions—for classical firstorder logic. This technique is a systematic way to recover the desired essence of any sequent proof without abandoning the sequent calculus. 1998 ACM Subject Classification F.4.1 Mathematical Logic: Proof theory