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In the book modal logic for open minds by johan van benthem there is on page 169 a statement that the sentence the first formula bellow is valid and the second one is invalid (* means iteration)

So why is this valid: $[ (R\lor S)^* ] \phi \rightarrow [R^*]\phi \land [S^*] \phi$

And why is this one invalid: $[ (R\lor S)^* ] \phi \leftarrow [R^*]\phi \land [S^*] \phi$

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I will give a hand-wavy general argument for the contraposition of the first validity: $\langle R^\ast\rangle \phi \vee \langle S^\ast \rangle \phi \rightarrow \langle (R \vee S)^\ast \rangle \phi$. The main thing to notice is that the set of $R-S$-paths $(R\vee S)^\ast$ includes the sets of $R$-paths and $S$-paths. So, if some $\phi$-state is reachable via an $R$-path, then it is trivially reachable via an $R-S$-path. The same holds for a $\phi$-state reachable via an $S$-path. In general, $\langle R^\ast\rangle \phi \rightarrow \langle (R \vee S)^\ast \rangle \phi$ is valid.

Now, let us consider the contraposition of the second formula: $\langle (R \vee S)^\ast \rangle \phi \rightarrow \langle R^\ast \rangle \phi \vee \langle S^\ast \rangle \phi$. In order to demonstrate that this formula is invalid, it is enough to notice that even though there is an $R-S$-path that reaches a $\varphi$-state, it may be neither $R$- nor $S$-path (i.e. the path alternate $R$ and $S$). A simple counterexample would be a three-state model, where a $p$-state is reachable via two steps, one $R$- and one $S$-step, and the initial and the intermediate states have $\neg p$.

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  • $\begingroup$ What does $R-S$-paths mean? And why did you change the box with label to diamond with label inside it? $\endgroup$ Mar 12, 2019 at 15:07
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    $\begingroup$ By an $R-S$-path I mean any sequence of $R$- and $S$-arrows in a model. I changed box to diamond because I think that using the contrapositive ($\phi \rightarrow \psi \leftrightarrow \neg \psi \rightarrow \neg \phi$) of the presented formulas makes reasoning more straightforward. So, $[R^\ast] \phi \wedge [S^\ast] phi \rightarrow [(R \vee S)^\ast]\phi$ is equivalent to $\langle (R \vee S)^\ast \rangle \neg \phi \rightarrow \langle R^\ast \rangle \neg \phi \vee \langle S^\ast \rangle \neg \phi$. Note that box and diamond are duals. Anyway, the reasoning and counterxample work for both cases. $\endgroup$ Mar 12, 2019 at 17:27

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