[QUOTE=Frylock]
(I’m not comfortable using this kind of example to motivate the view you’re trying to motivate. Because “All horses with one horn are not unicorns” is also true, and “All horses with one horn are reptiles” is even true. For most people, the truth of “All horses with one horn are unicorns” intuitively rests on the fact that unicorn is defined as “a horse with one horn.” So they’ll accept the truth of the statement, but will balk at generalizing from the truth of a statement to a view that any statement predicating something of an empty referent is also true, because for the same reason they acknowledge your statement is true, they’ll tend to balk at agreeing that the other, weirder statements I mentioned are true.
Certainly, you were strictly speaking just trying to show that a referent can be empty yet be such that true predications can be made of it. But really what we’re after is explaining why any predication made of an empty referent is true. And the kind of example you gave will actually present a stumbling block towards that goal.
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Certainly reasonable. I suspect that many people’s ordinary language treatment of universal quantification actually does correspond to their treatment of implication, with their position on vacuous truth over empty domains in the former corresponding to their position on vacuous truth over false antecedents in the latter. Just as an accurate model of the latter may accurately require wrapping material implication under a suitable modality of necessity, so may an accurate formal model of the former require wrapping universal quantification, as traditionally formally analyzed, under the same kind of modality (thus, “All horses with one horn are unicorns” will be true because it holds necessarily, but “All horses with one horn are reptiles” will not, because it presumably holds only contingently).
But, my own intent in the last several posts has not been to motivate the view “You must approach universal quantification this way” (which I have explicitly disavowed), but merely “You should see why it is not as manifestly ridiculous as you apparently think, and is actually even useful for many purposes, to approach universal quantification this way; it is glib to dismiss this out of hand as untenable”. I cannot speak for Lemur866’s motivations, but I suspect they are similar; the point is not to attack alternative ways of speaking, but merely to defend this one against automatic rejection.
With that in mind, once one admits the plausibility of reasonably construing “All horses with one horn are unicorns” as true, enough of a foothold presents itself as to allow argument defending reasonably construing all vacuous universal quantification as true. (As might be expected from above, a natural argument proceeds the same way as one might for defending the material conditional) To wit:
A very attractive, perhaps even intuitive, principle is that of (a very weak) truth-functionality: that the truth or falsehood of a large statement should depend only on which of its constituent “atomic” claims are true. That is, in this particular case, that the truth or falsehood of “All Xs have property P” should depend only on which particular Xs have property P. But, as far as horses with one horn go, the collection of those which are unicorns is precisely the same as the collection of those which are reptiles; empty, in both cases. Thus, if we are to hew to this principle, we must take “All horses with one horn are unicorns” and “All horses with one horn are reptiles” as equally true or false. At this point, if one is inclined to accept the former as reasonably considered true, one hopefully can understand why the latter, perceived as equivalent on this account, can also be reasonably considered true.
