Editing Linear logic (section)

{{Short description|System of resource-aware logic}}
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'''Linear logic''' is a [[substructural logic]] proposed by French [[logician]] [[Jean-Yves Girard]] as a refinement of [[classical logic|classical]] and [[intuitionistic logic]], joining the [[Duality (mathematics)#Duality in logic and set theory|dualities]] of the former with many of the [[Constructivism (mathematics)|constructive]] properties of the latter.{{sfn|Girard|1987}} 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]]),{{sfn|Baez|Stay|2008}} as well as [[linguistics]],{{sfn|De Paiva|Van Genabith|Ritter|1999}} particularly because of its emphasis on resource-boundedness, duality, and interaction.

Linear logic lends itself to many different presentations, explanations, and intuitions.
[[Proof theory|Proof-theoretically]], it derives from an analysis of classical [[sequent calculus]] in which uses of (the [[structural rule]]s) [[Idempotency of entailment|contraction]] and [[Monotonicity of entailment|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 semantics|denotational models]], linear logic may be seen as refining the interpretation of intuitionistic logic by replacing [[cartesian closed categories|cartesian (closed) categories]] by [[symmetric monoidal categories|symmetric monoidal (closed) categories]], or the interpretation of classical logic by replacing [[Boolean algebras]] by [[C*-algebras]].{{Citation needed|date=October 2015}}