nLab S4 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 (classical) modal logic called S4S4 is (classical) 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

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

  2. T:pp\Box T \colon \Box p \to p (T modal logic)

  3. 4:pp\Box 4 \colon \Box p \to \Box \Box p. (S4 modal logic)

  4. (in addition in S5 modal logic one adds: pp\lozenge p \to \Box \lozenge p).

Traditionally the canonical interpretation of the Box operator is that p\Box p is the statement that “pp is necessarily true.” (The de Morgan dual to the necessity modality is the “possibility” modality.) Then the interpretation of 4\Box 4 is that “If pp is necessarily true then it is necessarily necessarily true.” S4 modal logic appears in many temporal logics.

If instead of a single Box operator one considers nn \in \mathbb{N} box operators i\Box_i of this form, the resulting modal logic is denoted S4(n)S4(n). Here ip\Box_i p is sometimes interpreted as “the iith agent knows pp.”

In intuitionistic (or constructive) Modal Logic, one does not expect the possibility modality to be dual to the necessity modality, necessarily. Alex Simpson has written in his thesis the most used constructive modal logic system, but other systems exist.


Relation to Kripke frames

The models for T modal logic corresponded to Kripke frames where each relation R iR_i is reflexive.
For S4S4 modal logic they are furthermore transitive.

Theorem (Soundness of S4 (m)S4_{(m)})

S4 (m)ϕ𝒮4(m)ϕ.S4_{(m)}\vdash \phi \Rightarrow \mathcal{S}4(m)\models \phi.


(We show this for m=1m = 1.)

Suppose that 𝔐=((W,R),V)\mathfrak{M}= ((W,R),V) where RR is a reflexive transitive relation on WW. We have to check that 𝔐(4)\mathfrak{M}\models (4).

Suppose 𝔐,wKp\mathfrak{M},w\models K p, then, for every tt with RwtR w t, we have 𝔐,tp\mathfrak{M},t\models p. Now suppose we seek to check that 𝔐,wKKp\mathfrak{M},w\models K K p so we have tt with RwtR w t and want 𝔐,tKp\mathfrak{M},t\models K p, so we look at all uu with RtuR t u and have to see if 𝔐,up\mathfrak{M},u\models p, but as RwtR w t and RtuR t u hold then RwuR w u holds, since RR is transitive, and we then know that 𝔐,up\mathfrak{M},u\models p.


The terminology “S4” in modal logic originates with:

The “possible worldsKripke semantics of S4 originates with:

See also:

A natural deduction-formulation and making explicit the modal operator as a comonad:

See also:

More on S4S4, S5 (m)S5_{(m)} and their applications in Artificial Intelligence can be found in

  • J.- J. Ch. Meyer and W. Van der Hoek, Epistemic logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995.

General books on modal logics which treat these logics thoroughly in the general context include

  • Patrick Blackburn, M. de Rijke and Yde Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.

  • Marcus Kracht, Tools and Techniques in Modal Logic, Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999.

Last revised on March 8, 2024 at 11:33:20. See the history of this page for a list of all contributions to it.