Logic where truths are resources that can be used up

Science · Published 20 October 2026 · Updated 21 October 2026 Swipe more like this →

In short: In classical logic, if you know a fact, you can use it forever without it disappearing. Linear logic changes the rules. It treats logical propositions like physical resources: once you use them, they are gone. This shift from permanent truths to consumable resources has profound implications for quantum physics and computer science.

Proposed by French logician Jean-Yves Girard, linear logic is a substructural logic that refines classical and intuitionistic systems. Its core innovation lies in the careful control of 'structural rules'—specifically contraction and weakening. In standard logic, you can duplicate a premise (contraction) or ignore one (weakening) without consequence. Linear logic forbids this by default, meaning logical deduction becomes a process of manipulating resources that cannot be arbitrarily duplicated or discarded. This resource-bounded approach creates a unique distinction between different types of conjunctions and disjunctions. For example, multiplicative conjunction (⊗) splits a context between premises, while additive conjunction (&) carries the entire context into both. This nuance allows linear logic to model systems where the state of the world changes upon interaction. Because of these properties, it is frequently cited as the natural logic for quantum information theory, where information cannot be cloned. While linear logic is distinct, it is not isolated. It can recover classical and intuitionistic logic through the use of 'exponentials' (! and ?). These operators act as controlled valves, allowing specific formulas to be used as many times as needed, mimicking the behavior of traditional logic. This ability to embed other systems, combined with its 'cut-elimination' property, makes it a powerful tool for proof search and developing resource-aware programming languages.

topics [ logic quantum-physics computer-science

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