How to prove that " necessarily (A OR ~A) " does not imply " necessarily A OR necessarily ~A". In _De Interpretatione_ , Aristotle criticizes logical fatalism ( a metaphysical doctrine professed in the Megarian School, in particular by Diodorus Cronus). Aristotle reconstructs the reasoning of logical fatalists as follow : (1) Necessarily ( there will be a sea battle tomorrow or there will not be a sea battle tomorrow). (2) Therefore, necessarily a sea battle will happen tomorrow or necessarily it will not happen. One can say informally that the logical mistake consists in the fact that " the necessity of the disjunction does not imply the disjunction of the necessities". However, how to prove rigorously using the tools of modal logic that the reasoning is not valid?

You can provide a semantics for 'necessarily $A$' by saying that in all worlds out of some collection of worlds, $A$ is true.

As such, we can show that we can make $(1)$ true, but $(2)$ false: create a collection of two worlds, $w_1$ and $w_2$. Then assume that $A$ is true in $w_1$, but $A$ is false in $w_2$.

Now, $(1)$ is true, since in $w_1$, we have that $A$ is true, and hence $A \lor \
eg A$ is truew in $w_1$ as well. In $w_2$ we have that $\
eg A$ is true, and hence $A \lor \
eg A$ is true in $w_2$ is true as well. So, $A \lor A$ is true in all worlds, and hence it is necessarily true. So, $(1)$ is true.

$(2)$, however is false. We don't have that $A$ is necessarily true, since $A$ is not true in all worlds: $A$ is not true in $w_2$. Likewise, $\
eg A$ is not true in $w_1$, and hence $\
eg A$ is not necessarily true. Hence, both disjuncts from $(2)$ are false, so $(2)$ is false.