nLab
multiplicative conjunction

Contents

Context

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) 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

semantics

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

In linear logic/linear type theory, one of the two linear versions of logical conjunction is called the multiplicative conjunction and usually denoted “\otimes”. The analogous connective in relevance logic is sometimes called the intensional conjunction or the fusion or cotenability, and sometimes denoted “\circ”.

The categorical semantics of the multiplicative conjunction is as the tensor product with respect to a (symmetric) monoidal category structure on the collection of types.

There are various fragments of linear logic that contain the multiplicative conjunction. For instance, multiplicative linear logic (MLL) contains \otimes along with its unit 1\mathbf{1} and also the multiplicative disjunction \parr and its unit \bot and the linear negation () (-)^\bot. While multiplicative intuitionistic linear logic (MILL) contains \otimes and 1\mathbf{1} along with the linear implication \multimap.

basic symbols used in logic

A\phantom{A}symbolA\phantom{A}A\phantom{A}meaningA\phantom{A}
A\phantom{A}\inA\phantom{A}element relation
A\phantom{A}:\,:A\phantom{A}typing relation
A\phantom{A}==A\phantom{A}equality
A\phantom{A}\vdashA\phantom{A}A\phantom{A}entailment / sequentA\phantom{A}
A\phantom{A}\topA\phantom{A}A\phantom{A}true / topA\phantom{A}
A\phantom{A}\botA\phantom{A}A\phantom{A}false / bottomA\phantom{A}
A\phantom{A}\RightarrowA\phantom{A}implication
A\phantom{A}\LeftrightarrowA\phantom{A}logical equivalence
A\phantom{A}¬\notA\phantom{A}negation
A\phantom{A}\neqA\phantom{A}negation of equality / apartnessA\phantom{A}
A\phantom{A}\notinA\phantom{A}negation of element relation A\phantom{A}
A\phantom{A}¬¬\not \notA\phantom{A}negation of negationA\phantom{A}
A\phantom{A}\existsA\phantom{A}existential quantificationA\phantom{A}
A\phantom{A}\forallA\phantom{A}universal quantificationA\phantom{A}
A\phantom{A}\wedgeA\phantom{A}logical conjunction
A\phantom{A}\veeA\phantom{A}logical disjunction
A\phantom{A}\otimesA\phantom{A}A\phantom{A}multiplicative conjunctionA\phantom{A}
A\phantom{A}\oplusA\phantom{A}A\phantom{A}multiplicative disjunctionA\phantom{A}

Last revised on November 12, 2019 at 09:19:00. See the history of this page for a list of all contributions to it.