Concept
Linear logic
Overview
Linear logic is a substructural logic proposed by French logician 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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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(2) If two magnitudes are equal to a third they are equal to one another; this statement is, as Hegel has shown, a conclusion, upon the correctness of which all logic depends, and which is demonstrated therefore outside of pure mathematics. The remaining axioms with regard to equality and inequality are merely logical extensions of this conclusion. Such barren statements are not enticing either in mathematics or…
Friedrich Engels, Landmarks of Scientific Socialism
States what Linear logic is.
The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of…
Alfred North Whitehead, Science and the modern world
States what Linear logic is.
Hence, in the conclusion of a syllogism we restrict a predicate to a certain object, after having thought it in the major in its whole extent under a certain condition. This complete quantity of the extent in relation to such a condition is called universality (universalitas). To this corresponds totality (universitas) of conditions in the synthesis of intuitions.
Immanuel Kant, The Critique of Pure Reason
States what Linear logic is.
Philosophers who discussed Linear logic
Related ideas
- First-order logic
- Intuitionistic logic
- Logical conjunction
- Logical connective
- Proof theory
- Relevance logic
- Constructivism (philosophy of mathematics)
- Linguistics
- Quantum mechanics
- Sequent calculus
- Morphism
- Logical consequence
- Atomic formula
- Logic
- Classical logic
- Higher-order logic
- Deductive reasoning
- De Morgan's laws
- Logical disjunction
- Normal modal logic
- Recursive definition
- Jean-Yves Girard
- Structural rule
- Cut-elimination theorem
Opposing perspectives
Place in the tradition
Part of Outline of philosophy.
Narrower topics: