In the nineteenth century Sir William Hamilton caused some controversy by suggesting that the predicate as well as the subject term should be understood as quantified. Thus, we should understand "All dogs go to heaven" as saying "All dogs are some things that go to heaven". This gives the following schedule of propositions
Afa U All S is all P
Afi A All S is some P
Ifa Y Some S is all P
Ifi I Some S is some P
Ana E No S is any P
Ani η No S is some P
Ina O Some S is not any P
Ini ω Some S is not some P
The three-letter names at the beginning are Hamilton's own; the letters in bold immediately afterward are Thomson's (who rejected η and ω), which became more widely used in discussing predicate quantity.
Predicate quantity quickly fell out of favor, in part because the new propositions are very difficult to interpret, and the proponents of it never developed a consistent account of it, in part because they kept getting tripped up by the particular quantification of the predicate. If we take 'some' in the predicate to work the same way as 'some' in the subject, we can link each of the new propositions to information conveyed classically in the following way:
U = (SaP and PaS)
Y = PaS [i.e., Only S is P]
η = PoS [i.e., Not only S is P]
ω turns out to be a bit of a puzzler. It has no ordinary contradictory, and would have to be true, always, except where the P in question is the very same S we find in the subject (in which case it would say that that S is not that S).
Adding these propositions, therefore, doesn't convey more information than we could convey with the ordinary forms (with the possible exception of ω, which seems difficult to find a use for); and it complicates the rules of syllogism a bit, since the standard rules with U propositions will fail to rule out some syllogisms as invalid. The new propositions are messing with the distribution of terms; which complicates matters quite a bit, and for very little gain. There are, in fact, only two points of note: it allows us directly to handle U and Y propositions, which do show up in ordinary discourse quite often, without treating them as exponible; and (although this was not, as far as I recall, discussed at the time) it makes it easier to do some things if we are predicating singular terms. Singular terms are curious because they are predicable but also seem to carry their quantity around with them, so to speak. For instance, if I say,
Tully is Cicero,
I am predicating a singular term; and it is crucial for understanding what I intend that we recognize that 'Cicero' is a singular term. This is, in fact, the root of a number of longstanding puzzles about singular terms: unlike other terms, they seem always to be quantified. You can, in fact, treat them as always quantified and get viable (if sometimes confusing) results; I have done so in the past. For instance, in discussing singular terms in my series on Sommers-Englebretsen term logic, I treated singular terms as always quantified. This is not quite orthodox, and reading over the posts again I should have been more clear than I was that it wasn't (since I find that I didn't indicate at all that it wasn't!); but it simplifies discussion of singular terms quite a bit, because the only place it can cause problems is with interpretation, since in SETL singular terms have wild quantity. But while this has its conveniences, I have become less satisfied with it of late, and have begun to think it should be handled differently. In SETL, subscripts makes more sense, I have come to think; and, outside SETL, it seems reasonable to treat propositions with a predicate that apparently has singular quantity (however one understands singular quantity) as exponible.
But I don't know for sure. And there is a (small) part of me that hopes that someone will come up with a better account of quantified predicates for categorical propositions than Hamilton's; it was an ingenious idea, and worth trying, although it seems, having done the experiment, to be more trouble than it could possibly be worth. (It would have been better, I think, to conceive of quantity not as attaching to the subject or the predicate but as being the quantity of the proposition, and arguing that the quantity of the proposition is actually the proposition's pattern of distribution. Then quality becomes simply affirmation and negation, having nothing to do with distribution at all. But this doesn't seem to solve all the problems with predicate quantity.)