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## Kripke-Style Semantics for Kd

We define the frames for modeling Kd as follows:

 F is an Kd Frame: F = such that: 1) W is a non-empty set 2) R is a subset of W × W 3) DEM is a subset of W 4) ∀i∃j(Rij & j ∈ DEM).

A model can be defined in the usual way, allowing us to then define truth at a world in a model for all sentences of Kd (as well as for KTd):

M is an Kd Model: M = <F,V>, where F is an Kd Frame, <W,R,DEM>, and V is an assignment on F: V is a function from the propositional variables to various subsets of W.

Basic Truth-Conditions at a world, i, in a Model, M:

 [PC]: (Standard Clauses for the operators of Propositional Logic.) [□]: M i □p iff ∀j(if Rij then M j p). [d]: M i d iff i ∈ DEM.

Derivative Truth-Conditions:

 [◊]: M i ◊p: ∃j(Rij & M j p) [OB]: M i OBp: ∀j[if Rij & j ∈ DEM then M j p] [PE]: M i PEp: ∃j(Rij & j ∈ DEM & M j p) [IM]: M i IMp: ∀j[if Rij & j ∈ DEM then M j ~p] [GR]: M i GRp: ∃j(Rij & j ∈ DEM & M j ~p) [OP]: M i OPp: ∃j(Rij & j ∈ DEM & M j p) & ∃j(Rij & j ∈ DEM & M j ~p)

(Truth in a model and validity are defined just as for SDL.)

Metatheorem: Kd is sound and complete for the class of all Kd models.