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Supplement to Deontic Logic

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

Fis an Kripke-SDL (or KD) Frame:F= <W,A> such that:1) Wis a non-empty set2) Ais a subset ofW×W3) Ais 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]:MiOBp: “∀j[ifAijthenMjp]

Derivative Truth-Conditions:

[PE]:MiPEp: ∃j(Aij&Mjp)

[IM]:MiIMp: ~∃j(Aij&Mjp)

[GR]:MiGRp: ∃j(Aij&Mj~p)

[OP]:MiOPp: ∃j(Aij&Mjp) & ∃j(Aij&Mj~p)

pis true in the model,M(Mp):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.

Paul McNamara mcnamara.p@comcast.net |

Stanford Encyclopedia of Philosophy