nLab
the logic S4(m)

Context

(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 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
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, (idemponent) 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

The epistemic logics S4S4 and S4 (m)S4_{(m)}

Idea

The flavor of modal logic called S4S4 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

  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)

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 \Box 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 denote S4(n)S4(n). Here ip\Box_i p is sometimes interpreted as “the iith agent knows pp.”

Properties

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.

Proof

(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.

References

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.

Revised on November 28, 2014 17:30:16 by Urs Schreiber (82.113.121.76)