nLab K modal logic



(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

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





The flavor of modal logic called KK is propositional logic equipped with a single modality usually written “\Box” subject to the rules that for all propositions p,q:Propp, q \colon Prop we have

  • K:(pq)(pq)\Box K \colon \Box(p \to q) \to (\Box p \to \Box q) (K modal logic)

Often one adds to this the following further axioms

KK is the basic epistemic logic.



  • (Taut) All (instances of ) propositional tautologies.

  • For each i=1,,mi = 1,\ldots, m, the axiom, (K iK_i):

(K iϕK i(ϕψ))K iψ.(K_i\phi \wedge K_i(\phi \to \psi))\to K_i\psi.

Derivation rules

  • (MP)
ϕϕψψ\frac{\phi \quad \phi\to \psi}{\psi} \quad

(i.e. modus ponens);

  • (Generalisation)
ϕK iϕ.\frac{\phi}{K_i\phi}.

The second deduction rule corresponds to the idea that if a statement has been proved, then it is known to all ‘agents’.


This logic is the smallest normal modal logic.


The semantics of K (m)K_{(m)} is just the Kripke semantics of this context, so a frame, 𝔉\mathfrak{F} is just a set, WW of possible worlds with mm relations R iR_i. A model, 𝔐=(𝔉,V)\mathfrak{M} = (\mathfrak{F},V), is a frame in that sense together with a valuation, V:Prop𝒫(W)V \colon Prop \to \mathcal{P}(W), and the satisfaction relation is as described in geometric models for modal logics with just the difference implied by the fact that that page correspond to the use of i=M i\Diamond_i = M_i whilst this uses K iK_i. This means that

  • 𝔐,wK iϕ\mathfrak{M},w \models K_i \phi if and only if, for all vWv \in W such that R iwv R_i w v, 𝔐,vϕ\mathfrak{M},v \models \phi.
type of modal logicrelation in its Kripke frames
K modal logicany relation
K4 modal logic?transitive relation
T modal logicreflexive relation
B modal logic?symmetric relation
S4 modal logicreflexive & transitive relation
S5 modal logicequivalence relation

(following BdRV (2001) Table 4.1; cf. Fagin, Halpern, Moses & Vardi (1995) Thm. 3.1.5)


The notation “K” for a modal operator meant to express epistemic modality originates with:

Last revised on August 1, 2023 at 10:46:51. See the history of this page for a list of all contributions to it.