nLab multiplicative conjunction

Contents

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

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

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

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

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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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 $\mathbf{1}$ and also the multiplicative disjunction $\parr$ and its unit $\bot$ and the linear negation $(-)^\bot$ . While multiplicative intuitionistic linear logic (MILL) contains $\otimes$ and $\mathbf{1}$ along with the linear implication $\multimap$ .

Related concepts

$\phantom{-}$ symbol $\phantom{-}$	$\phantom{-}$ in logic $\phantom{-}$
$\phantom{A}$ $\in$	$\phantom{A}$ element relation
$\phantom{A}$ $\,:$	$\phantom{A}$ typing relation
$\phantom{A}$ $=$	$\phantom{A}$ equality
$\phantom{A}$ $\vdash$ $\phantom{A}$	$\phantom{A}$ entailment / sequent $\phantom{A}$
$\phantom{A}$ $\top$ $\phantom{A}$	$\phantom{A}$ true / top $\phantom{A}$
$\phantom{A}$ $\bot$ $\phantom{A}$	$\phantom{A}$ false / bottom $\phantom{A}$
$\phantom{A}$ $\Rightarrow$	$\phantom{A}$ implication
$\phantom{A}$ $\Leftrightarrow$	$\phantom{A}$ logical equivalence
$\phantom{A}$ $\not$	$\phantom{A}$ negation
$\phantom{A}$ $\neq$	$\phantom{A}$ negation of equality / apartness $\phantom{A}$
$\phantom{A}$ $\notin$	$\phantom{A}$ negation of element relation $\phantom{A}$
$\phantom{A}$ $\not \not$	$\phantom{A}$ negation of negation $\phantom{A}$
$\phantom{A}$ $\exists$	$\phantom{A}$ existential quantification $\phantom{A}$
$\phantom{A}$ $\forall$	$\phantom{A}$ universal quantification $\phantom{A}$
$\phantom{A}$ $\wedge$	$\phantom{A}$ logical conjunction
$\phantom{A}$ $\vee$	$\phantom{A}$ logical disjunction

symbol	in type theory (propositions as types)
$\phantom{A}$ $\to$	$\phantom{A}$ function type (implication)
$\phantom{A}$ $\times$	$\phantom{A}$ product type (conjunction)
$\phantom{A}$ $+$	$\phantom{A}$ sum type (disjunction)
$\phantom{A}$ $0$	$\phantom{A}$ empty type (false)
$\phantom{A}$ $1$	$\phantom{A}$ unit type (true)
$\phantom{A}$ $=$	$\phantom{A}$ identity type (equality)
$\phantom{A}$ $\simeq$	$\phantom{A}$ equivalence of types (logical equivalence)
$\phantom{A}$ $\sum$	$\phantom{A}$ dependent sum type (existential quantifier)
$\phantom{A}$ $\prod$	$\phantom{A}$ dependent product type (universal quantifier)

symbol	in linear logic
$\phantom{A}$ $\multimap$ $\phantom{A}$	$\phantom{A}$ linear implication $\phantom{A}$
$\phantom{A}$ $\otimes$ $\phantom{A}$	$\phantom{A}$ multiplicative conjunction $\phantom{A}$
$\phantom{A}$ $\oplus$ $\phantom{A}$	$\phantom{A}$ additive disjunction $\phantom{A}$
$\phantom{A}$ $\&$ $\phantom{A}$	$\phantom{A}$ additive conjunction $\phantom{A}$
$\phantom{A}$ $\invamp$ $\phantom{A}$	$\phantom{A}$ multiplicative disjunction $\phantom{A}$
$\phantom{A}$ $\;!$ $\phantom{A}$	$\phantom{A}$ exponential conjunction $\phantom{A}$
$\phantom{A}$ $\;?$ $\phantom{A}$	$\phantom{A}$ exponential disjunction $\phantom{A}$

References

Eg.

Daniel Mihályi, Valerie Novitzká, p. 3, 8 of: What about Linear Logic in Computer Science?, Acta Polytechnica Hungarica 10 4 (2013) 147-160 [pdf, pdf]

For more see the references at linear logic.

Last revised on March 29, 2023 at 10:24:16. See the history of this page for a list of all contributions to it.