#### Supplement to Deontic Logic

## Kripke-Style Semantics for SDL

We define the frames (structures) for modeling SDL as follows:

Fis an Kripke-SDL (or KD) Frame:F= <W,A> such that:

Wis a non-empty setAis a subset ofW×WAis serial: ∀i∃jAij.

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 SDL (and SDL+):

Mis an Kripke-SDL Model:M= <F,V>, whereFis an SDL Frame, <W,A>, andVis an assignment onF:Vis a function from the propositional variables to various subsets ofW(the “truth sets’ for the variables—the worlds where the variables are true for this assignment).

Let “*M*
⊨_{i}
*p*” denote
*p*'s truth at a world, *i*, in a model, *M*.

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

[PC]: (Standard Clauses for the operators of Propositional Logic.)

[OB]:M⊨_{i}OBp: “∀j[ifAijthenM⊨_{j}p]

Derivative Truth-Conditions:

[PE]:M⊨_{i}PEp: ∃j(Aij&M⊨_{j}p)

[IM]:M⊨_{i}IMp: ~∃j(Aij&M⊨_{j}p)

[GR]:M⊨_{i}GRp: ∃j(Aij&M⊨_{j}~p)

[OP]:M⊨_{i}OPp: ∃j(Aij&M⊨_{j}p) & ∃j(Aij&M⊨_{j}~p)

pis true in the model,M(M⊨p):pis true at every world inM.

pis valid (⊨p):pis true in every model.

Metatheorem: SDL is sound and complete for the class of all Kripke-SDL
models.^{[1]}

Return to Deontic Logic.