nLab bunched logic

Bunched logic


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


Bunched logic


In logic, and more specifically in sequent calculus/natural deduction, a bunched logic is a logic in which the formulas in the context are not just a list or set but have some additional, usually tree-like, structure.

This can be indicated syntactically by the use of two or more punctuation symbols, such as comma and semicolon, along with parentheses for grouping. Thus for instance a sequent with bunches might be written like

A,B,(C;(D,E);F),(G;H)K A,\; B,\; \big( C;\; (D,\,E); \; F \big),\; (G;\,H) \;\;\;\;\vdash\;\;\;\; K

The contexts put together with both commas and semicolons are called bunches. The general phrase bunched logic is not entirely standard, although the word “bunches” has been used with more than one logic of this form (the original is “bunched implication (BI)”, below).

Each type of punctuation also comes with a nullary version. The punctuation symbols like comma and semicolon are sometimes called structural connectives, since they are part of the judgmental structure (like structural rules) but are closely related to the logical connectives such as conjunction and disjunction.

Usually the reason for using a bunched logic is that the different structural connectives obey different structural rules. For instance, the semicolon might allow the contraction rule and/or the weakening rule, while the comma does not.


Bunched implications

In bunched implications logic (BI), the semicolon allows contraction and weakening while the comma does not. This allows defining both an additive conjunction and a multiplicative conjunction, internalizing the semicolon and the comma respectively, such that both distribute over the additive disjunction and moreover both come with a corresponding implication: the additive implication? and the multiplicative implication.

Relevance logic

Some forms of relevance logic can be presented with a bunched sequent calculus that is similar to BI, but in which the comma also has contraction (though not weakening) and there is no additive implication. See for instance (Mints).

Classical bunched implication

A logic of classical bunched implication is like BI but with arbitrary bunches on the right as well as on the left. On the right, the semicolon represents the additive disjunction and the comma represents a multiplicative disjunction, and there are both an additive and a multiplicative negation that move formulas back and forth. Both negations are “classical” with respect to their corresponding connectives, e.g. we have AA\sim\sim A \multimap A (where \sim is the multiplicative negation and \multimap the multiplicative implication) and also ¬¬AA\neg\neg A \to A (where ¬\neg is the additive negation and \to the additive implication). See Pym 2002 and this discussion.

Dependent versions

Bunched logics are also used to combine linear type theories with dependent type theories. See some of the references at dependent linear type theory.

Categorical semantics

Bunched logics naturally have semantics in categories with more than one monoidal structure, so that a bunch such as (A,(B;C),((D,E);F))(A,(B;C),((D,E);F)) can be interpreted as A(BC)((DE)F)A \otimes (B\boxtimes C) \otimes ((D\otimes E)\boxtimes F). Frequently (e.g. if one kind of bunch admits contraction and weakening) one of the two monoidal structures is a cartesian one. A typical and motivating example of a model for BI is the category of presheaves [C op,Set][C^{op},Set] over a monoidal category CC, which comes equipped both with the ordinary ccc structure on presheaves as well as the closed monoidal structure given by Day convolution.


  • Peter O'Hearn, David J. Pym, The Logic of Bunched Implications, The Bulletin of Symbolic Logic 5 2 (1999) 215-244 [pdf, doi:10.2307/421090]

  • G. E. Mints. Cut-elimination theorem for relevant logics, Zap. Nauchn. Sem. LOMI, 1972, Volume 32, Pages 90–97. ( An English translation appears in the Journal of Soviet Mathematics 6 (1976) pp.422-8. (doi)

  • Bodil Biering, On the logic of bunched implications - and its relation to separation logic, masters thesis

  • David Pym, The Semantics and Proof Theory of the Logic of Bunched Implications, Applied Logic Series 26, Springer (2002) [doi:10.1007/978-94-017-0091-7, GoogleBooks]

  • Brotherston and Calcagno, Classical BI: Its Semantics and Proof Theory, arxiv

A bunched logic for dependent linear types, reflecting contexts built not just from Cartesian products but also from tensor products:

Last revised on March 24, 2023 at 17:09:35. See the history of this page for a list of all contributions to it.