#### Supplement to Deontic Logic

## Kripke-Style Semantics for K*d*

We define the frames for modeling K*d* as follows:

Fis an KdFrame:F= <W,R, DEM> such that:

Wis a non-empty setRis a subset ofW×W- DEM is a subset of
W- ∀
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 K*d* (as well
as for KT*d*):

Mis an KdModel:M= <F,V>, whereFis an KdFrame, <W,R,DEM>, and V is an assignment onF:Vis a function from the propositional variables to various subsets ofW.Basic Truth-Conditions at a world,

i, in a Model,M:

[PC]: (Standard Clauses for the operators of Propositional Logic.) [□]: M⊨_{i}□piff ∀j(ifRijthenM⊨_{j}p).[d]: M⊨_{i}diffi∈ DEM.Derivative Truth-Conditions:

[◊]: M⊨_{i}◊p: ∃j(Rij&M⊨_{j}p)[ OB]:M⊨_{i}OBp: ∀j[ifRij&j∈ DEM thenM⊨_{j}p][ PE]:M⊨_{i}PEp: ∃j(Rij&j∈ DEM &M⊨_{j}p)[ IM]:M⊨_{i}IMp: ∀j[ifRij&j∈ DEM thenM⊨_{j}~p][ OM]:M⊨_{i}OMp: ∃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: K*d* is sound and complete for the class of all
K*d* models.

Return to Deontic Logic.