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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.

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The supposition that (9) is not necessarily true is the critical thing here: if (9) were necessarily true, that would mean that (7) and (8) are necessarily not both true (because the failure of either (7) or (8) is a consequence of (9)) so the conjunction would be necessarily false. A "contradiction" is a sentence that is not just false but necessarily false.

If the conjunction of (7) and (8) were contradictory, it would be necessarily false. But that means that in any universe in which (7) holds, (8) must not hold. Consider a universe in which there is at least one W which is not L. Let S be the same as W (so something is S if and only if it is W). In this universe, (7) and (8) are both satisfied; if (7) and (8) are supposed to be contradictory, this universe can't exist, so we can't allow one W which is not L. In other words, it is necessarily the case that all W are L.

Based on the quote, I suspect that a similar argument is outlined on pages 184 and 185 of the text.

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  • $\begingroup$ Dear Reese, Thank you very much for your reply. I am concerned/confused about the line "Let S be the same as W". This stipulation would appear to make (7) (All S are W) analytically true. And that, combined with the necessary falsehood of the conjunction of (7) and (8), would imply (8)'s necessary falsehood. But (8)'s necessary falsehood, when S and W are logically equivalent, implies (9)'s necessary truth, which is the conclusion we're seeking. $\endgroup$ – Confused Philosopher Mar 6 '17 at 16:39
  • $\begingroup$ @ConfusedPhilosopher Yes, that was exactly the intent. The goal of that paragraph is to demonstrate that, as claimed in the passage you quoted, if (7) and (8) are contradictory then (9) is necessarily true. $\endgroup$ – Reese Mar 6 '17 at 17:57

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