I'm reading into categorical propositions and there seems to be lots of different opinions on what they are, and what their existential import is.
Why are there so many different variations? Shouldn't Aristotle's original definitions be fine as they are?
It is not uncommon when reading authors who wrote long ago, to find that they were unaware of distinctions that we now consider important. It is pretty much inevitable that as a subject advances it becomes more complex and nuances emerge that were previously overlooked. It is also usually pointless to try to read back into an ancient author what they would have agreed with if they had been presented with a modern account of some subject.
This is true of Aristotle's logic. It is not as clear as it might seem to a novice reader, and it is not suitable for the demands of modern logic.
You specifically mention categorical propositions and the issue of existential import. The four categorical propositions in Aristotle are
A. All S is P.
E. No S is P.
I. Some S is P.
O. Some S is not P.
These were arranged into a square of opposition by later writers. A and E are contrary, meaning they cannot both be true. I and O are subcontrary, meaning they cannot both be false. A implies I, and E implies O, by subalternation. A and O are contradictory, as are E and I, meaning one is always true and the other false.
But this square is problematic when we consider the issue of existential import. Existential import is concerned with the question of which of the categorical propositions implies the existence of some S. Or, equivalently, which would automatically be false if there were no S. In modern logic, I and O have existential import, while A and E do not. But this is not a possibility for Aristotelian logic, since A implies I and E implies O. However, that does leave several options. The main three are:
All of these have been defended by some commentators or other, though the first is the most common. At the cost of making this answer really long, let's examine the options in a little more detail.
Option 1. A, E, I, O all have existential import. The main problem here is that on the face of it, it breaks the square of opposition. If there are no S, then all are false, and hence I-O cannot be subcontraries, and A-0 and E-I cannot be contradictories. The usual move made by the defenders of this position is to say that Aristotle is assuming the existence of S but not actually asserting it. Another problem is that it is unsatisfactory to assume that A and E are always instantiated. Sometimes we wish to reason hypothetically about concepts, without committing in advance to there being instances of them. Another problem is that it is unnatural to take E statements to have existential import.
Option 2. A and I have existential import. This makes much better sense of the square of opposition. But it is unnatural to understand O statements as lacking existential import. "Some S is not P" on this account must be understood disjunctively as either there exist some S that are not P or there exist no S at all. Also, this account requires that we reject the form of immediate inference called obversion, which is normally taught as part of Aristotelian logic. In particular, it means that "Some S is not-P" has existential import, while "Some S is not P" does not, so they are not equivalent. This seems pretty dubious to me. Nevertheless this approach appears to be have been held by William of Ockham, and in modern times has been defended by Stephen Read.
Option 3. None have existential import. On this interpretation, the issue of existence is separate from that of predication. The categorical propositions speak only of how concepts overlap, not whether any of the concepts are instantiated. It allows that statements like "Some dragons breathe fire" might be true, even if there are no dragons.
Option 1 can be mapped into modern logic by using typed variables that range over a non-emtpy domain. This was done by Timothy Smiley and John Corcoran. Smiley, T. “Syllogism and quantification”, Journal of Symbolic Logic, 27, 58–72 (1962). Corcoran, J. “Aristotle’s many-sorted logic”, Bulletin of Symbolic Logic, 14, 155–156 (2008).
Option 2 can be mapped into first order logic. This position is defended by Stephen Read “Aristotle and Łukasiewicz on Existential Import” Journal of the American Philosophical Association 1 (3):535-544 (2015).
Option 3 can be mapped into free logic. This interpretation is defended by Edward Buckner in his article on Existential Import in the The Logic Museum.
