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 mathematics_logic_statistics 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 of linear algebra; here he alludes to affine transformations on 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 an unbounded comprehension axiom. Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989).

For Affine logic, the abstraction is narrower than the article's general subject matter: a positive case must preserve Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

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.

Structural Signature¶

Sig role-phrases:

Defining carrier — The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule.
Constitutive relation — Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989).
Operating condition — 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 .
Recognition evidence — Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.
Admissible variation — 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 on vector spaces.
Characteristic 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.
Failure boundary — Whereas full linear logic (i.e. propositional linear logic with multiplicatives, additives, and exponentials) is undecidable, full affine logic is decidable.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.
Not an over-broad reading. Whereas full linear logic (i.e. propositional linear logic with multiplicatives, additives, and exponentials) is undecidable, full affine logic is decidable.
Not an over-broad reading. Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.
Not an over-broad reading. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule.
Not automatically Complex Affine Space. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Affine logic applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

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 on vector spaces.
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).
Documented setting. 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 .

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

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. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Whereas full linear logic (i.e. propositional linear logic with multiplicatives, additives, and exponentials) is undecidable, full affine logic is decidable. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Affine logic compresses multiple mathematics_logic_statistics 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. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics 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. Change an implementation or setting while preserving 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 on vector spaces.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

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. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction; recognition evidence → Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction

Applied / In Practice¶

The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → the applied context; invariant → Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction; boundary → the case exits the class when whereas full linear logic (i.e. propositional linear logic with multiplicatives, additives, and exponentials) is undecidable, full affine logic is decidable

Structural Tensions¶

T1 — Stable identity versus admissible variation. Whereas full linear logic (i.e. propositional linear logic with multiplicatives, additives, and exponentials) is undecidable, full affine logic is decidable. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. 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 on vector spaces. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Affine logic literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Affine logic distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Affine logic is structural-leaning. Its structural side is the repeatable organization summarized by Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: 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 . Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989). It further constrains recognition and variation through: 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 . Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Affine logic literal. Its documented scope includes the condition that 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. Another bounded application condition is that Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—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 on vector spaces.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Formal System.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Affine logic. The reviewed identity is: Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

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

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction?
Complex Affine Space. A torsor for a complex vector space: points admit complex-vector differences, translations, and affine combinations, but no point is distinguished as the origin. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Linear separability. The property that two labeled point sets lie on opposite sides of at least one affine hyperplane. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Vector logic. An algebraic representation of logical truth values and connectives as vectors and matrices acting on a finite-dimensional space. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Affine logic remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Affine_logic (revision 1344586076).
Preserved source candidate: http://www.seas.upenn.edu/~sweirich/types/archive/1997-98/msg00134.html

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.