Pomset logic: the other approach to non commutativity in logic

Abstract: Thirty years ago, I introduced a non-commutative variant of classical linear logic, called "pomset logic", issued from a particular categorical interpretation of linear logic known as coherence spaces. In addition to the usual commutative multiplicative connectives of linear logic, pomset logic includes a non-commutative connective, "$\triangleleft$" called "before", associative and self-dual: $(A\triangleleft B)^\perp=A^\perp \triangleleft B^\perp$. The conclusion of a pomset logic proof is a Partially Ordered MultiSET of formulas. Pomset logic enjoys a proof net calculus with cut-elimination, denotational semantics, and faithfully embeds sequent calculus. The study of pomset logic has reopened with recent results on handsome proof nets, on its sequent calculus, or on its following calculi like deep inference by Guglielmi and Strassburger. Therefore, it is high time we published a thorough presentation of pomset logic, including published and unpublished material, old and new results. Pomset logic (1993) is a non-commutative variant of linear logic (1987) as for Lambek calculus (1958!) and it can also be used as a grammatical formalism. Those two calculi are quite different, but we hope that the algebraic presentation we give here, with formulas as algebraic terms and with a semantic notion of proof (net) correctness, better matches Lambek's view of what a logic should be.