nLab
de Morgan duality

Context

Duality

(0,1)(0,1)-Category theory

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 rulecompositionsubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
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 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

homotopy levels

semantics

De Morgan duality

Idea

In logic, de Morgan duality is a duality between intuitionistic logic and dual-intuitionistic paraconsistent logic. In classical logic and linear logic, it is a self-duality mediated by negation. Although it goes back to Aristotle (at least), its discovery is generally attributed to Augustus de Morgan.

The dualities

More explicitly, de Morgan duality is the duality between logical operators as shown in the table below:

Intuitionistic operatorDual-intuitionistic operator
\top (truth)\bot (falsehood)
\wedge (conjunction)\vee (disjunction)
\Rightarrow (conditional)\setminus (without?)
\Leftrightarrow (biconditional)++ (exclusive disjunction)
¬\neg (pp \Rightarrow \bot)- (p\top \setminus p)
\forall (universal quantification)\exists (existential quantification)
\Box (necessity)\lozenge (possibility)

The first two operators in each column exist in both intuitionistic and dual-intuitionistic propositional logic and the last two in each column exist in both forms of predicate logic and modal logic (respectively), but they are still dual as shown. All of these exist in classical logic (although some of the paraconsistent operators are not widely used), and the two forms of negation (¬\neg and -) are the same there.

In linear logic, this extends to a duality between conjunctive and disjunctive operators:

Conjunctive operatorDisjunctive operator
\top00
11\bot
&\&\oplus
\otimes\parr
^\bot^\bot
\bigwedge\bigvee
!!??

As with classical negation, linear negation is self-dual. (For the categorical semantics of this see at dualizing object and at Wirthmüller context – Comparison of push-forwards.)

The first two rows of the intuitionistic/dual-intuitionistic/classical duality generalise to arbitrary lattices, including subobject lattices in coherent categories, and from there to the duality between limits and colimits in category theory:

LimitColimit
topbottom
meetjoin
intersectionunion
terminal objectinitial object

So in a way, all duality in category theory is a generalisation of de Morgan duality.

The de Morgan laws

The de Morgan laws are the statements, valid in various forms of logic, that de Morgan duality is mediated by negation. For example, using the second line of the first table, we have

¬(pq)¬p¬q, ¬(pq)¬p¬q. \array { \neg(p \wedge q) \equiv \neg{p} \vee \neg{q} ,\\ \neg(p \vee q) \equiv \neg{p} \wedge \neg{q} . }

Traditionally, the term is reserved for this line.

In the foundations of constructive mathematics, de Morgan's Law usually means the statement

¬(pq)¬p¬q, \neg(p \wedge q) \vdash \neg{p} \vee \neg{q} ,

since every other aspect of the first two lines is already constructively valid, the claim that negation mediates the de Morgan self-duality of negation already has a name (the double negation law, equivalent to the principle of excluded middle), and no other line involves only operators that appear in intuitionstic propositional calculus.

Revised on January 25, 2014 14:00:41 by Urs Schreiber (89.204.137.78)