# modal logic T

The modal logic T is the smallest normal modal logic containing the schema T:

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

A Kripke frame $(W,R)$ is reflexive^{} if $R$ is reflexive on $W$.

###### Proposition 1.

T is valid in a frame $\mathrm{F}$ iff $\mathrm{F}$ is reflexive.

###### Proof.

First, suppose $\mathcal{F}$ is not reflexive, say, $(w,w)\notin R$. Let $M$ be a model based on $\mathcal{F}$ such that $V(p)=\{u\mid wRu\}$, where $p$ is a propositional variable. By the construction of $V(p)$, we see that for all $u$ such that $wRu$, we have ${\vDash}_{u}p$, so ${\vDash}_{w}\mathrm{\square}p$. But since $w\notin V(p)$, ${\vDash \u0338}_{w}p$. This means that ${\vDash \u0338}_{w}\mathrm{\square}p\to p$.

Conversely, let $\mathcal{F}$ be a reflexive frame, and $M$ any model based on $\mathcal{F}$, with $w$ a world in $M$. Suppose ${\vDash}_{w}\mathrm{\square}A$. Then for all $u$ such that $wRu$, ${\vDash}_{u}A$. Since $wRw$, we get ${\vDash}_{w}A$. Therefore, ${\vDash}_{w}\mathrm{\square}A\to A$. ∎

As a result,

###### Proposition 2.

T is sound in the class of reflexive frames.

###### Proof.

Since any theorem^{} in T is deducible^{} from a finite sequence^{} consisting of tautologies^{}, which are valid in any frame, instances of T, which are valid in reflexive 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 T, we have

###### Proposition 3.

T is complete^{} in the class of reflexive frames.

###### Proof.

We show that the canonical frame ${\mathcal{F}}_{\text{\mathbf{T}}}$ is reflexive. For any maximally consistent set $w$, if $A\in {\mathrm{\Delta}}_{w}:=\{B\mid \mathrm{\square}B\in w\}$, then $\mathrm{\square}A\in w$. Since T contains $\mathrm{\square}A\to A$, we get that $A\in w$ by modus ponens and the fact that $w$ is closed under modus ponens. Therefore $w{R}_{\text{\mathbf{T}}}w$, or ${R}_{\text{\mathbf{T}}}$ is reflexive. ∎

T properly extends the modal system D, for $\mathrm{\square}A\to A$ is not valid in any non-reflexive serial frame, such as the one $(W,R)$, where $W=\{u,w\}$ and $R=\{(u,u),(w,u)\}$: just let $V(p)=\{w\}$. So ${\vDash}_{w}p$ and ${\vDash \u0338}_{u}p$, or ${\vDash \u0338}_{w}\mathrm{\square}p$. This means ${\vDash \u0338}_{w}\mathrm{\square}p\to p$.

Title | modal logic T |
---|---|

Canonical name | ModalLogicT |

Date of creation | 2013-03-22 19:33:58 |

Last modified on | 2013-03-22 19:33:58 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 03B45 |

Related topic | ModalLogicD |

Defines | T |