Affine logic¶

Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7899
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Substructural Logic, Mathematical Logic, Proof Theory → Mathematics

Core Idea¶

Affine logic is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. It can also be characterized as linear logic with weakening. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. Jean-Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterizes linear logic in terms.

How would you explain it like I'm…

Use-Each-Ticket-Once Logic

Imagine each fact you know is a ticket you can use to prove things. In affine logic, you can use each ticket at most once, and you're allowed to leave some tickets unused. You can't make copies of a ticket to use it twice.

No Copying Assumptions

Logic is about rules for building proofs from assumptions. In ordinary logic, you can use an assumption as many times as you like. Affine logic changes that: each assumption can be used at most once, because the rule that lets you copy an assumption, called contraction, is removed. But you're still allowed to ignore an assumption you don't need, which is called weakening. That makes it close to linear logic, where each assumption must be used exactly once.

Contraction-Free Logic with Weakening

Affine logic is a substructural logic, which means it drops some of the structural rules that ordinary logic takes for granted about how assumptions can be handled. Specifically, it rejects contraction, the rule that lets you use an assumption more than once by duplicating it. It keeps weakening, the rule that lets you add or ignore unused assumptions. That makes it linear logic plus weakening: linear logic rejects both contraction and weakening, so every assumption is used exactly once, while in affine logic each is used at most once. Jean-Yves Girard named it, alluding to affine transformations. Dropping contraction matters: Grishin observed in 1974 that Russell's paradox can't be derived in a set theory without contraction, even with unrestricted comprehension.

Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction while retaining weakening, so hypotheses may be discarded but not duplicated; in resource terms, each assumption is used at most once. Equivalently, it is linear logic with weakening added. Jean-Yves Girard introduced the name within the geometry of interaction semantics of linear logic, which characterizes linear logic via linear algebra, alluding to affine transformations of vector spaces. Grishin used this logic in 1974 after observing that Russell's paradox cannot be derived in a set theory without contraction, even with unrestricted comprehension. The logic also formed the basis of 'Direct logic', a decidable sub-theory of predicate logic developed by Ketonen and collaborators. The defining test is proof-theoretic: absence of contraction, not merely a resource-flavored interpretation.

Scope of Application¶

Documented setting. Grishin used this logic in 1974, after observing that Russell's paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom.
Documented setting. Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.
Documented setting. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule.
Documented setting. Jean-Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterizes linear logic in terms of linear algebra; here he alludes to affine transformations.
Documented setting. Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989).

Clarity¶

A clear use of Affine logic names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. The strongest recognition evidence in the frozen account is: Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.

Manages Complexity¶

Affine logic compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989).—and the practical consequence—grishin used this logic in 1974, after observing that Russell's paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom.

Abstract Reasoning¶

Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
State the relation. Use the source-grounded identity: Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.
Check operation and conditions. Affine logic can be embedded into linear logic by rewriting the affine arrow A \rightarrow B as the linear arrow A \multimap B \otimes \top .
Demand recognition evidence. Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.
Test variation.

Knowledge Transfer¶

Within the home domain. Knowledge about Affine logic transfers literally when a new case preserves the same carrier type, relation, and recognition test. Grishin used this logic in 1974, after observing that Russell's paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom. Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. Beyond the home domain. No canonical parent is asserted for Affine logic.

Relationships to Other Abstractions¶

Current abstraction Affine logic Domain-specific

Parents (1) — more general patterns this builds on

Affine logic is a kind of Formal System Prime

Affine logic is a formal deductive system distinguished by rejecting contraction; it is not a specialization of Omega-logic.

Hierarchy paths (2) — routes to 2 parentless roots

Affine logic → Formal System → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space¶

Affine logic sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08