Can it be the case $p$,$\lnot p$ are true at $\mathbf w$ And $\mathbf w'$ respectively with $\mathbf w$ and $\mathbf w'$ have access to each other?

Question

Given a model $\mathbf{M} = (\mathbf{W}, \mathbf{R}, \mathbf{V})$ for a set of atomic formulae $\Omega$. We have possible worlds $\mathbf{w}, \mathbf{w'} \in \mathbf{W}$, access relation satisfies $(\mathbf{w}, \mathbf{w'}),(\mathbf{w'}, \mathbf{w})\in \mathbf{R}$, $p \in \Omega$.

Moreover, we define necessity operator and possibility operator as follows:Given $(\mathbf{w_1}, \mathbf{w_2}) \in \mathbf{R}$. If a proposition $\phi$ is true at $\mathbf{w_2}$ then $\Diamond{\phi}$ is true at $\mathbf w_1$. If $\Box \phi$ is true at $\mathbf w_1$, then $\phi$ is true at $\mathbf w_2$.

Question Can it be the case $p$,$\lnot p$ are true at $\mathbf w$ and $\mathbf w'$ respectively with $\mathbf w$ and $\mathbf w'$ have access to each other?

I think there is no incoherency, but remark 2 on page 29 of this note suggests otherwise.

Answer 1

This is a typo. (At first I thought transitivity might be assumed in the background, but looking through the note I see that restricting to specific classes of models is not introduced until a few pages later on.)

What the author certainly meant to say in Remark 2 was: "Notice that the picture would become incoherent were $\mathbf{w}$ accessible from $\mathbf{w}$."