Substructural Logic

Substructural logics are non-classical logics obtained by modifying the structural rules of classical logic — the rules that govern how premises can be used and combined in proofs, rather than the rules that govern particular logical connectives. The most famous structural rules are weakening (the ability to add irrelevant premises to a proof) and contraction (the ability to use a premise multiple times). By restricting or eliminating these rules, substructural logics model reasoning in contexts where resources, relevance, or linearity matter.

The three principal families are:

Linear logic (Girard, 1987): forbids both weakening and contraction. Every premise must be used exactly once. This makes linear logic a logic of resources: proving A ⊗ B consumes one copy of A and one copy of B. It has direct applications in computer science — particularly in programming language semantics, concurrency theory, and compiler optimization — where the tracking of resource consumption is essential.

Relevant logic (Anderson & Belnap): forbids weakening but permits contraction. Every premise in a valid proof must be actually used in deriving the conclusion. This blocks the classical paradoxes of material implication (e.g., 'if the moon is made of cheese, then 2+2=4') by demanding a genuine connection between antecedent and consequent.

Affine logic (a variant of linear logic): forbids contraction but permits weakening. Premises may be ignored but cannot be duplicated. This models reasoning about resources that can be wasted but not cloned — a useful abstraction for certain physical and computational processes.

The philosophical significance of substructural logics is that they make explicit what classical logic leaves implicit: the assumption that information is freely available, infinitely copyable, and irrelevantly accumulable. These assumptions hold for abstract mathematical truths but fail for physical resources, computational processes, and empirical evidence. Substructural logics are therefore the natural logical frameworks for quantum mechanics (where no-cloning theorems forbid contraction), for economic reasoning (where resources are scarce), and for type systems that track ownership and borrowing.

Classical logic treats truth as a static, inexhaustible commodity. Substructural logics treat inference as a process that consumes and transforms resources. The question is not which logic is 'correct' — it is which structural assumptions are appropriate for the domain of reasoning.