Another difference is that the species/genus schema is not predicated on the bare particulars of modern logic — Leontiskos
Judging by your quote I'd say that this is really just where I disagree with Aristotle -- there's no such thing as essences.
So while I understand that within his system of philosophy he believes that induction is valid, and this is due to things having an essence -- i.e. there is some needed common traits or relationships which are shared by each member of the genera.
But, OK -- yes, that's a difference in terms of the philosophy. I don't think Aristotle is a Quinean or anything of the sort when considering his whole system -- but conceptually the various species and their genera look a lot like sets.
For Aristotle I would just add that there are ontological conditions to the sets, whereas today we'd prefer to abstract to the logic alone and leave the ontology undefined so that we could then speak clearly about what exists.
Make sense?
Simplified, Aristotle is basically saying that familiarity breeds understanding. If we become familiar with swans then we will begin to understand swans. There is no guarantee for Aristotle that there are no black swans. There is no Humean induction. — Leontiskos
I'm only using that as a familiar example.
You would agree that there is an essence which a swan possess which makes the swan a swan according to Aristotle?
Whatever the essence is for Aristotle there is an essence that can be discovered, and I think that's what secures his knowledge via the empirical method. He can know about every member of the set because of these necessary and sufficient conditions which make a tiger what a tiger is.
Whereas I would say that it's in the very logic itself that makes the move from species to genera invalid. There is no essence that holds all tigers together, a what-it-isness which makes the tiger a tiger.
So I'm not talking about being wrong in the sense of error as much as I'm saying there is no valid construction of induction because there is nothing to universalize. This is a big difference between my understanding of Aristotle, vs. Darwinian, biology. The species aren't as distinct as what Aristotle's method indicates -- they slowly morph over time and we update our taxonomies the more we learn, but this isn't a logically valid move.
Familiarity with swans will help us to understand swans, and if we happen to notice that all birds have wings then we might say that the essence of birds is "has wings", and since all swans are birds all swans have wings since that is what holds for all birds.
In such a world, if you could correctly identify an essence -- what holds for all the tigers, or whatever species/genera is under discussion -- then moving up to a more encompassing category would appear entirely valid.
So, in my understanding of Aristotle at least, I can understand why he believes it's valid. It's not like he didn't know what validity was. However, I think he is wrong about essence, and what you end up with for any process of induction is never a logically valid move. It's a guess. Hopefully an educated guess, but a guess all the same -- and the taxonomies we write about animals are our way of understanding life rather than the essence of life.
I don't think I've said anything Humean here -- if I were I'd be talking about relationships between events or the wash of perception or the emotional grounding of inference or something. But I'm just saying that Aristotle is wrong about essences, and that's what masks the invalidity of going from a particular to a general. The allusion to mathematical induction is to say that it appears to me that Aristotle treats inductive knowledge of particulars in the same manner that we treat inductive knowledge of math -- i.e. there is some conditions which hold for all tigers, and once we know that from some tigers we will have knowledge of all tigers in their essence, if not in their particulars/accidents.
Or I could be wrong about the role of essence in Aristotle. But at least this is how I'm understanding it. Does it make sense to you?