All of None

Some might deny that "all ravens are black" means the same thing as "there are no non-black ravens". The latter claim is true even if there are no ravens at all. But it seems odd to say that the former would be true in such a case. After all, it would be just as true to say "all ravens are entirely non-black", since there are no black ravens to contradict it. From these we can infer that "all ravens are both black and entirely non-black", which seems very odd indeed. Logicians mutter that it's just another way of saying that no ravens exist at all. But it clashes sorely with common linguistic usage. We might instead hold that universal statements have existential import: to say "all ravens are black" implies that there are some ravens.

This would provide a simple resolution of the raven paradox - for while randomly sampled red herrings might confirm that there are no coloured ravens, they do nothing at all to confirm the existence of black ravens, so if the universal claim is committed to such existence, then the red herrings do not confirm that "all ravens are black" after all. Our intuitions are saved! (Not really: read on.)

It's worth noting just what the randomly sampled red herrings really show. They don't specifically confirm that there are no coloured ravens. This implication is very much implicit -- red herrings don't say anything direct about ravens. Rather, the fact that our random sample turned up a red herring indicates that red herrings are probably quite common. (If you pick an object at random, chances are you're going to pick an object of a common rather than rare type, for obvious reasons.) We might say that it directly confirms the claim "everything is red herrings". This in turn then implicitly confirms the logical entailments, "there are no ravens", and particularly, "there are no non-black ravens", which the modern logician takes to be equivalent to "all ravens are black".

But the red herring doesn't just confirm that all ravens are black (without existential import); it equally confirms that all ravens are green, and that no ravens exist at all. It is only from our background knowledge that some black ravens exist that we rule out the other equally "confirmed" hypotheses, and say that the observation supports the specific claim that all ravens are black (rather than, e.g., that no ravens exist at all).

I think this helps explain why the herring/raven confirmation seems paradoxical to people. The observation in itself says nothing specific about ravens, and that's why we suppose that it cannot confirm claims about ravens (e.g. that all ravens are black). But it can confirm that there are no (or relatively few) ravens. And while this alone cannot support "all ravens are black" over "all ravens are green", the latter can be ruled out by our background knowledge.

Through most of this I've been assuming, with modern logicians, that universal statements have no existential import. "All X are Y" is true if no X's exist, no matter what Y might be (it can even be a contradiction, as we saw above). But in fact the herring confirmation of "all ravens are black" goes through regardless once we factor in background knowledge. Given that some black ravens exist, the existential condition is already satisfied, so all we need to confirm in addition is that there are no non-black ravens, which the randomly sampled red herring does help to confirm.

But enough about ravens already. I'm wondering: do universal statements have existential import?

Let's take an example: (M) All mermaids are green.

Is M true? False? Neither? I don't know what to make of it. The problem with stipulating that empty universals are vacuously true was mentioned above: it implies the truth of apparent contradictions such as "all mermaids are not mermaids". But existential import fares no better, for that implies that empty universals are all vacuously false, which is implausible for tautologies like "all mermaids are mermaids" (which is surely true even if no mermaids exist).

Perhaps what we need is for the quantifier to range over non-existent possibilia in such cases. We go to some appropriate possible world where mermaids exist, and assess the truth of the universal in that world.

Are all unicorns white? I think so. Are they all black? Nope! Is it possible to get a coloured unicorn? Maybe in some distant possible world, but that need not contradict our judgment that all unicorns are white in the appropriate world.

Hempel actually discusses this issue in his original raven paradox paper ('Studies in the Logic of Confirmation', pp.16-17). One reason he offers, to think that universal statements don't have existential import, is that some just obviously don't. An astronomer's universal hypothesis about what happens in extreme conditions "need not imply that such extreme conditions ever were or will be realized".

But in fact this supports my view, not his. We want to say the astronomer's hypothesis is true only if his prediction would happen were the extreme conditions to occur. In other words, the hypothesis must hold true in the nearest possible worlds where the extreme conditions obtain. Otherwise we would hold it to be false. Logicians, by contrast, would hold the universal statement to be vacuously true if the conditions never actually obtained. They would say the opposite prediction would be just as (vacuously) true. But this is clearly wrong. His prediction is not a vacuous one, and it can be either true or false, no matter whether the conditions actually obtain.

(One way to take my point here is to say that such claims are not properly analyzed in terms of universal quantifiers and material conditionals. They are better understood as counterfactual claims, if they refer to nothing actual.)