# modal logic D

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

$$\mathrm{\square}A\to \diamond A$$ |

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 $\mathrm{F}$ iff $\mathrm{F}$ 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 ${\vDash}_{w}\mathrm{\square}p$, so that ${\vDash}_{w}\diamond p$. 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 ${\vDash}_{w}\mathrm{\square}A$. Then for all $v$ such that $wRv$, we have ${\vDash}_{v}A$. In particular, ${\vDash}_{u}A$. Therefore, ${\vDash}_{w}\diamond A$, whence ${\vDash}_{w}\mathrm{\square}A\to \diamond A$. ∎

As a result,

###### Proposition 2.

D is sound in the class of serial frames.

###### Proof.

Since any theorem^{} in D is deducible^{} from a finite sequence^{} consisting of tautologies^{}, which are valid in any frame, instances of D, which are valid in serial frames by the proposition^{} above, and applications of modus ponens^{} and necessitation, both of which preserve validity in any frame, whence the result.
∎

In addition^{}, using the canonical model of D, we have

###### Proposition 3.

D is complete^{} in the class of serial frames.

###### Proof.

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

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 |
---|---|

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 |