Affine logic is a substructural logic

Substructural logic

In logic, a substructural logic is a logic lacking one of the usual structural rules , such as weakening, contraction or associativity...

 whose proof theory rejects the structural rule

Structural rule

In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgements or sequents directly. Structural rules often mimic intended meta-theoretic properties of the logic...

 of contraction

Idempotency of entailment

Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one...

. It can also be characterized as linear logic

Linear logic

Linear logic is a substructural logic proposed by 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...

 with weakening.

The name "affine logic" is associated with linear logic

Linear logic

Linear logic is a substructural logic proposed by 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...

, to which is differs by allowing the weakening rule. Jean-Yves Girard

Jean-Yves Girard

Jean-Yves Girard is a French logician working in proof theory. His contributions include a proof of strong normalization in a system of second-order logic called system F; the invention of linear logic; the geometry of interaction; and ludics...

 introduced the name as part of the geometry of interaction

Geometry of interaction

The geometry of interaction was introduced by Jean-Yves Girard shortly after his work on Linear Logic. In linear logic, proofs can be seen as some kind of networks instead of the flat tree structure of sequent calculus. To distinguish the real proof nets from all the possible networks, Girard...

 semantics of linear logic, which characterises linear logic in terms of linear algebra; here he alludes to affine transformation

Affine transformation

In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

s on vector spaces.

The logic predated linear logic. V. N. Grishin used this logic in 1974, after observing that Russell's paradox

Russell's paradox

In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...

 cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom. Likewise, the logic formed the basis of a decidable subtheory of predicate logic

Predicate logic

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...

, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989).

Affine logic can be embedded into linear logic by rewriting the affine arrow as the linear arrow .

Whereas full linear logic (ie. propositional linear logic with multiplicatives, additives and exponentials) is undecidable, full affine logic is decidable.

Affine logic forms the foundation of ludics

Ludics

In proof theory, ludics is an analysis of the principles governing inference rules of mathematical logic. Key features of ludics are its notion of compound connectives using a technique known as focusing or focalisation , and its use of locations or loci over a base instead of propositions.More...

.