nLab affine logic





The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



In formal logic, affine logic refers to substructural logics which omit the contraction rule but retain the weakening rule. Conversely, affine logic is linear logic adjoined with the weakening rule.

In terms of categorical semantics, affine logic is modeled by (symmetric) monoidal categories whose tensor unit II is terminal, also known as semicartesian monoidal categories, an example of which is the category of affine spaces, whence the name.

(In contrast, the category of linear spaces, ie. vector spaces, serves as categorical semantics for the multiplicative fragment of linear logic [eg. Murfet 2014], where also the weakening rule is dropped.)

Categorical semantics

One might imagine that a more general notion of categorical semantics would be given by monoidal categories equipped with a natural (in AA) family AIA\to I of morphisms implementing weakening for each object. However, such an interpretation is in general badly behaved, unless one additionally requires these natural transformations to be monoidal, but it can be shown that this additional requirement already forces the tensor unit to be terminal (specifically, this follows from the component at II being the identity).

An example of the badly behaved case – where the transformation is not monoidal, and the tensorial unit is not terminal – is given by the category Rel of relations, with cartesian product as tensor product (i.e., with RelRel as cartesian bicategory). Here a natural family of relations AIA\to I is given by picking empty relations everywhere. In the corresponding interpretation of affine logic, any weakening yields an empty relation, which contradicts intuitive principles like for example that “adding a dummy variable to a proof and then substituting a closed term” should not change the semantics.



The substructural part of many forms of bunched logic are affine instead of linear, sometimes inadvertently, ultimately due to former technical problems with formulating dependent linear types (see review in Riley 2022 §1.7.2).


See also:

Last revised on September 4, 2023 at 10:12:41. See the history of this page for a list of all contributions to it.