Supplement to Deontic Logic
Kripke-Style Semantics for Kd
We define the frames for modeling Kd as follows:
F is an Kd Frame: F = <W, R, DEM> such that:
- W is a non-empty set
- R is a subset of W × 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 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] [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: Kd is sound and complete for the class of all Kd models.
Return to Deontic Logic.