I am looking for help understanding a problem of modal logic. In his 1963 essay 'Hume on evil', the philosopher Nelson Pike raises the following inconsistent triad of propositions (where S, W, and L stand for swans, white, and large respectively):
(7) All S are W. (8) Some S are ~L. (9) All W are L.
Here is the start of Pike's analysis of the set:
Suppose (9) is true, but not necessarily true. Either (7) or (8) must be false. But the conjunction of (7) and (8) is not contradictory. If the conjunction of (7) and (8) were contradictory, then (9) would be a necessary truth (p.184 – 185).
That last sentence in particular troubles me. Given that the conjunction of (7) and (8) isn't contradictory (conceded in the second-last sentence of the quotation), I'm having a hard time understanding how the counterfactual state of affairs where (7) and (8) is contradictory is supposed to be conceived of. Presumably that antecedent isn't simply to be interpreted as the truth of the negated conjunction of (7) and (8); for the negated conjunction of (7) and (8) doesn't imply (9)'s truth, and I take that to mean it doesn't imply (9)'s necessary truth either.
Perhaps the antecedent of the conditional that puzzles me refers to a state of affairs where the negated conjunction of (7) and (8) is taken to be not just true, but necessarily true. But if the truth of –[(7) & (8)] doesn't imply (9), I'm not clear on how the necessary truth of –[(7) & (8)] is supposed to imply (9), never mind imply the necessary truth of (9).
Is there something obvious, or even something not-so-obvious, that I'm missing here? Any insight here would be greatly appreciated.