# modal logic B

The modal logic B (for Brouwerian) is the smallest normal modal logic containing the following schemas:

• (T) $\square A\to A$, and

• (B) $A\to\square\diamond A$.

In this entry (http://planetmath.org/ModalLogicT), we show that T is valid in a frame iff the frame is reflexive   .

###### Proposition 1.

B is valid in a frame $\mathcal{F}$ iff $\mathcal{F}$ is symmetric.

###### Proof.

First, suppose B is valid in a frame $\mathcal{F}$, and $wRu$. Let $M$ be a model based on $\mathcal{F}$, with $V(p)=\{w\}$, $p$ a propositional variable. Since $w\in V(p)$, $\models_{w}p$, and $\models_{w}p\to\square\diamond p$ by assumption  , $\models_{v}\diamond p$ for all $v$ such that $wRv$. In particular, $\models_{u}\diamond p$, which means there is a $t$ such that $uRt$ and $\models_{t}p$. But this means that $t\in V(p)$, so $t=w$, whence $uRw$, and $R$ is symmetric.

Conversely, let $\mathcal{F}$ be a symmetric frame, $M$ a model based on $\mathcal{F}$, and $w$ a world in $M$. Suppose $\models_{w}A$. If $\not\models_{w}\square\diamond A$, then there is a $u$ such that $wRu$, with $\not\models_{u}\diamond A$. This mean for no $t$ with $uRt$, we have $\models_{t}A$. Since $R$ is symmetric, $uRw$, so $\not\models_{w}A$, a contradiction   . Therefore, $\models_{w}\square\diamond A$, and $\models_{w}A\to\square\diamond A$ as a result. ∎

As a result,

###### Proposition 2.

B is sound in the class of symmetric frames.

###### Proof.

Since B contains T, its canonical frame $\mathcal{F}_{\textbf{B}}$ is reflexive. We next show that any consistent normal logic $\Lambda$ containing the schema B is symmetric. Suppose $wR_{\Lambda}u$. We want to show that $uR_{\Lambda}w$, or that $\Delta_{u}:=\{B\mid\square B\in u\}\subseteq w$. It is then enough to show that if $A\notin w$, then $A\notin\Delta_{u}$. If $A\notin w$, $\neg A\in w$ because $w$ is maximal, or $\square\diamond\neg A\in w$ by modus ponens on B, or $\square\neg\square A\in w$ by the substitution theorem on $A\leftrightarrow\neg\neg A$, or $\neg\square A\in\Delta_{w}$ by the definition of $\Delta_{w}$, or $\neg\square A\in u$ since $wR_{\Lambda}u$, or $\square A\notin u$, since $u$ is maximal, or $A\notin\Delta_{u}$ by the definition of $\Delta_{u}$. So $R_{\Lambda}$ is symmetric, and $R_{\textbf{B}}$ is both reflexive and symmetric. ∎

Title modal logic B ModalLogicB 2013-03-22 19:34:11 2013-03-22 19:34:11 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 03B45 B