# modal logic D

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

 $\square A\to\diamond A$

A binary relation  $R$ on $W$ is serial if for any $w\in W$, there is a $u\in W$ such that $wRu$. In other words, $R$ is first order definable:

 $\forall w\exists u(wRu).$

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

###### Proposition 1.

D is valid in a frame $\mathcal{F}$ iff $\mathcal{F}$ is serial.

###### Proof.

First, assume $D$ valid in a frame $\mathcal{F}:=(W,R)$, and $w\in W$. Let $M$ be a model based on $\mathcal{F}$, with $V(p)=\{u\mid wRu\}$. Then $\models_{w}\square p$, so that $\models_{w}\diamond p$. This means there is a $v$ such that $wRv$, and hence $R$ is serial.

Conversely, let $\mathcal{F}$ be a serial frame, $M$ a model based on $\mathcal{F}$, and $w$ a world in $M$. Then there is a $u$ such that $wRu$. Suppose $\models_{w}\square A$. Then for all $v$ such that $wRv$, we have $\models_{v}A$. In particular, $\models_{u}A$. Therefore, $\models_{w}\diamond A$, whence $\models_{w}\square A\to\diamond A$. ∎

As a result,

###### Proposition 2.

D is sound in the class of serial frames.

###### Proof.

We show that the canonical frame $\mathcal{F}_{\textbf{D}}$ is serial. Let $w$ be any maximally consistent set containing D. For any $A\in\Delta_{w}:=\{B\mid\square B\in w\}$, we have $\square A\in w$, so that $\diamond A\in w$ by modus ponens on D. This means that $\square\neg A\notin w$ since $w$ is maximal. As a result, $\neg A\notin\Delta_{w}$, showing that $\Delta_{w}$ is consistent, and hence can be enlarged to a maximally consistent set $u$. As a result, $A\in u$, whence $wR_{\textbf{D}}u$. ∎

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 reflexive  frame in particular, and as a result a theorem of T by the completeness of T in reflexive frames.

Title modal logic D ModalLogicD 2013-03-22 19:33:54 2013-03-22 19:33:54 CWoo (3771) CWoo (3771) 13 CWoo (3771) Definition msc 03B45 ModalLogicT D serial