This is the informal proof of Drinker's paradox
The proof begins by recognising it is true that either everyone in the pub is drinking (in this particular round of drinks), or at least one person in the pub isn't drinking.
On the one hand, suppose everyone is drinking. For any particular person, it can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking — because everyone is drinking.
Suppose, on the other hand, at least one person isn't drinking. For that particular person, it still can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking — because that person is, in fact, not drinking.
I can agree with the first case of the proof. But how is the second case true ? How can they apply material implication in the second case when the material conditional itself has not been proved yet or given to us ?