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Supplement to Deontic Logic

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

Fis an KdFrame:F= <W,R, DEM> such that:1) Wis a non-empty set2) Ris a subset ofW×W3) DEM is a subset of W4) ∀ 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.) [□]: Mi□piff ∀j(ifRijthenMjp).[d]: Midiffi∈ DEM.Derivative Truth-Conditions:

[◊]: Mi◊p: ∃j(Rij&Mjp)[ OB]:MiOBp: ∀j[ifRij&j∈ DEM thenMjp][ PE]:MiPEp: ∃j(Rij&j∈ DEM &Mjp)[ IM]:MiIMp: ∀j[ifRij&j∈ DEM thenMj~p][ GR]:MiGRp: ∃j(Rij&j∈ DEM &Mj~p)[ OP]:MiOPp: ∃j(Rij&j∈ DEM &Mjp) & ∃j(Rij&j∈ DEM &Mj~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.

Paul McNamara mcnamara.p@comcast.net |

Stanford Encyclopedia of Philosophy