nLab
the logic K(m)

The epistemic logics KK and K (m)K_{(m)}

Idea

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.

Properties

Axiomatisation

  • (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’.

Normality

This logic is the smallest normal modal logic.

Semantics

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: 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.

=–

Revised on November 5, 2012 19:22:26 by Urs Schreiber (82.113.98.246)