Turning Cycles into Spirals

In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noting that very large numbers can be actually constructed through very short proofs, A more refined analysis of these short proofs reveals the presence of cyclic paths in their logical graphs, Indeed, in [6] it is shown that cycles need to exist for the proofs to be short. Here, we present a new sequent calculus for classical logic which is close to linear logic in spirit, enjoys cut-elimination, is acyclic and its proofs are just &~errtar~~ larger than proofs in LK. The proofs in the new calculus can bc obtained by a srn~ll perturhntim of proofs in LK and they represent a geometrical alternative for studying structural properties of LK-proofs. They satisfy the constructive disjunction property and most important. simpler geometrical properties of their logical graphs. The geometrical counterpart to a cycle in LK is represented in the new setting by a spiwl which is passing through sets of formulas logically grouped together by the nesting of their quantifiers. @ 1999 Elsevier Science B.V. All rights reserved. <4M.S clmv~ficution: 03F03; 03F05; 03F07; 03F20 &~orrls: Cut elimination; Gentzen sequent calculus: Cycles in proofs; Structure of proofs: Complexity of proofs