Euclidean frame validity

Question

I have trouble understanding why $$\Diamond P \to \square\Diamond P$$ is valid in Euclidean frames.

I found a proof online which is detailed as follows:

Proof. Suppose $F$ is a Euclidean frame, and $M$ a model based on $F.$ Suppose $\models_w \Diamond A$. Then there is a $v$ such that $w\mathrel{R}v$ and $\models_v A.$ Now, for any $u$ with $w\mathrel{R}u$, we have $u\mathrel{R}v$ since $R$ is Euclidean. So $\models_u \Diamond$A. Since $u$ is arbitrary, $\models_w \Box\Diamond A,$ and therefore $\models_w \Diamond A \to \Box\Diamond A.$

My question is what if the arbitrary $u$ is chosen as world $v.$ My understanding of necessarily true is that from every world from the current world should satisfy the condition. However, at world $v,$ there is no path to another world where $p$ is true since $p$ is only true at $v$ itself. Doesn't that imply that $\Diamond A$ is false at $v$ and hence $\Box\Diamond A$ is false as well? If that's the case, why is the axiom valid?

Thank you in advance for any explanation!

Answer 1

Maybe it is worth addressing precisely the question asked: what if the arbitrary $u$ is chosen as world $v$.

One thing to note here is this: Euclideanness implies that the endpoint of any path has a loop. That is, $wRv$ implies $vRv$ (indeed $wRu$ and $wRv$ together imply $uRv$; now take $u=v$).

Thus although there might be no path from $v$ to another world, there is a path from $v$ to itself, $v$ being an endpoint of a path (the one starting in $w$). Thus also $v$ possesses a path to a point where $p$ is true - namely a path to itself.

Answer 2

$u$ is an arbitrary world such that $w R u$, not just any old arbitrary world. If $v$ happens to be one of these worlds, then we know that $w R v$ must hold.

Suppose that $\models_w \Diamond P$. Then we get a world $v$ with $w R v$ and $\models_v P$. We then consider an arbitrary world that can be reached from $w$ (as required by the semantics of $\Box$) and call this $u$. This means that $w R u$, and since the frame was assumed to be Euclidean, $w R u$ and $w R v$ implies that $u R v$, from which we conclude that $\models_u \Diamond P$. Since the $u$ represents any world that can be reached from $w$, we have thus shown that $\models_w \Box \Diamond P$.

Answer 3

My question is what if the arbitrary $u$ is chosen as world $v.$ My understanding of necessarily true is that from every world from the current world should satisfy the condition. However, at world $v,$ there is no path to another world where $p$ is true since $p$ is only true at $v$ itself.

Why do you say that? $v$ need not be the only accessible world which witnesses the truth of $\vDash_w\Diamond A$.

In any case, in an Euclidean frame, by definition, any world $v$ accessible from $w$ is also accessible from any world $u$ that is itself accessible from $w$ . $(wRv~\& ~wRu)\implies uRv$. This inlcudes includes $v$ itself. $(wRv~\& ~wRv)\implies vRv$ . Since $v$ is the witness to $\vDash_w\Diamond A$, it is accessible from $w$, so therefore $vRv$ and hence $\vDash_v \Diamond A$ is okay.

But more, $\vDash_u\Diamond A$ is true for all arbitrary worlds, $u$, accessible from $w$ when $\vDash_w\Diamond A$, since they can each access some world $v$ where $\vDash_v A$ when $\vDash_w \Diamond A$ and the frame is Euclidean.

Hence $\vDash_w\Box\Diamond A$ if $\vDash_w \Diamond A$, and so therefore $\Diamond A\to \Box\Diamond A$ is a theorem in an Euclidean frame