Linear logic

Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction. Linear logic lends itself to many different presentations, explanations, and intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and weakening are carefully controlled. Operationally, this means that logical deduction is no longer merely about an ever-expanding collection of persistent "truths", but also a way of manipulating resources that cannot always be duplicated or thrown away at will. In terms of simple denotational models, linear logic may be seen as refining the interpretation of intuitionistic logic by replacing cartesian (closed) categories by symmetric monoidal (closed) categories, or the interpretation of classical logic by replacing Boolean algebras by C*-algebras. The language of classical linear logic (CLL) is defined inductively by the BNF notation Here p and p⊥ range over logical atoms. For reasons to be explained below, the connectives ⊗, ⅋, 1, and ⊥ are called multiplicatives, the connectives &, ⊕, ⊤, and 0 are called additives, and the connectives ! and ? are called exponentials. We can further employ the following terminology: Binary connectives ⊗, ⊕, & and ⅋ are associative and commutative; 1 is the unit for ⊗, 0 is the unit for ⊕, ⊥ is the unit for ⅋ and ⊤ is the unit for &. Every proposition A in CLL has a dual A⊥, defined as follows: Observe that (-)⊥ is an involution, i.e., A⊥⊥ = A for all propositions.