nLab stable proposition

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
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 setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
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

semantics

Modalities, Closure and Reflection

Contents

Definition

In (modal) logic, a proposition pp is stable under a modality \diamond if ppp \equiv \diamond{p}. If \diamond is monadic?, then pp is stable iff pqp \equiv \diamond{q} for some qq.

(In terms of modal type theory and propositions as types this means that pp is a modal type.)

Examples

In intuitionistic logic, the default is the double negation modality ¬¬\neg\neg. Since p¬¬pp \Rightarrow \neg\neg{p} regardless, pp is stable iff ¬¬pp\neg\neg{p} \Rightarrow p. Being stable is weaker than being decidable; however, if every proposition is stable, then every proposition is decidable and the logic becomes classical. (This is because pp is decidable iff p¬pp \vee \neg{p} is stable.) Double negation is monadic, so by the previous paragraph, pp is stable iff p¬¬qp \equiv \neg\neg{q} for some qq; in fact, pp is stable iff p¬qp \equiv \neg{q} for some qq. (I guess that this has to do with negation forming a monadic adjunction with itself, or something like that.) In the topological semantics? of intuitionistic logic (where propositions correspond to open sets), the stable propositions correspond to the regular open sets.

In the internal language of a category of sheaves Sh(X)Sh(X), a proposition pp is ¬¬\neg\neg-stable iff it is enough for pp to hold on a dense open subset of XX to be able to conclude that pp holds on the whole of XX. For instance, the proposition f=0f = 0 about a section ff of the sheaf of real functions on XX is ¬¬\neg\neg-stable.

Last revised on January 27, 2014 at 09:37:49. See the history of this page for a list of all contributions to it.