Validity of: ◻◇◻p → ◇p in transitive frames

Question

How can I show the (in)validity of: ◻◇◻p → ◇p in a transitive frame?

Proof method: I want to do this by showing in a transitive frame that ◻◇◻p holds and that ◇p doesn't hold.

(I think that ◻◇◻p would never hold in a transitive frame and that the implication is then automatically right, is this correct?)

Answer 1

HINT: any sentence of the form "$\Diamond \varphi$" is not true in any world (in any frame) with no arrows coming out of it. Does transitivity imply that the accessibility relation is nontrivial?

You should think about the simplest frame possible, here; this frame $F_0$ will be transitive, and will have the property that

• $\neg\Diamond p$ is valid in $F_0$

but

• $\Box\Diamond\Box p$ is also valid in $F_0$,

so it is a counterexample to the claim that $\Box\Diamond\Box p\implies\Diamond p$ in transitive frames.

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What happens if we add reflexivity - which does imply that the accessibility relation is nontrivial - to the mix?