modal logic D

The modal logic D (for deontic) is the smallest normal modal logic containing the schema D:


A binary relationMathworldPlanetmath R on W is serial if for any wW, there is a uW such that wRu. In other words, R is first order definable:


The Kripke frames corresponding to D are serial, in the following sense:

Proposition 1.

D is valid in a frame F iff F is serial.


First, assume D valid in a frame :=(W,R), and wW. Let M be a model based on , with V(p)={uwRu}. Then wp, so that wp. This means there is a v such that wRv, and hence R is serial.

Conversely, let be a serial frame, M a model based on , and w a world in M. Then there is a u such that wRu. Suppose wA. Then for all v such that wRv, we have vA. In particular, uA. Therefore, wA, whence wAA. ∎

As a result,

Proposition 2.

D is sound in the class of serial frames.


Since any theoremMathworldPlanetmath in D is deducibleMathworldPlanetmath from a finite sequencePlanetmathPlanetmath consisting of tautologiesMathworldPlanetmath, which are valid in any frame, instances of D, which are valid in serial frames by the propositionPlanetmathPlanetmath above, and applications of modus ponensMathworldPlanetmath and necessitation, both of which preserve validity in any frame, whence the result. ∎

In additionPlanetmathPlanetmath, using the canonical model of D, we have

Proposition 3.

D is completePlanetmathPlanetmathPlanetmathPlanetmath in the class of serial frames.


We show that the canonical frame 𝐃 is serial. Let w be any maximally consistent set containing D. For any AΔw:={BBw}, we have Aw, so that Aw by modus ponens on D. This means that ¬Aw since w is maximal. As a result, ¬AΔw, showing that Δw is consistent, and hence can be enlarged to a maximally consistent set u. As a result, Au, whence wR𝐃u. ∎

D is a subsystem of T, for any reflexive relation is serial. As a result, any theorem of D is valid in any serial frame, and therefore in any reflexiveMathworldPlanetmath frame in particular, and as a result a theorem of T by the completeness of T in reflexive frames.

Title modal logic D
Canonical name ModalLogicD
Date of creation 2013-03-22 19:33:54
Last modified on 2013-03-22 19:33:54
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 03B45
Related topic ModalLogicT
Defines D
Defines serial