# modal logic T

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

 $\square A\to A$

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

###### Proposition 1.

T is valid in a frame $\mathcal{F}$ iff $\mathcal{F}$ 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 $\models_{u}p$, so $\models_{w}\square p$. But since $w\notin V(p)$, $\not\models_{w}p$. This means that $\not\models_{w}\square p\to p$.

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

As a result,

###### Proposition 2.

T is sound in the class of reflexive frames.

###### Proof.

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

T properly extends the modal system D, for $\square A\to A$ 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 $\models_{w}p$ and $\not\models_{u}p$, or $\not\models_{w}\square p$. This means $\not\models_{w}\square p\to p$.

Title modal logic T ModalLogicT 2013-03-22 19:33:58 2013-03-22 19:33:58 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 03B45 ModalLogicD T