| gallo,
In traditional logic, one doesn't need two stated premises to prove something. For example, identity follows without any premises at all! Less trivially, we can "prove" the law of excluded middle without any premises, because it follows from only the basic axioms of logic (and logic differentiates arbitrarily between premises and axioms).
But all proofs have unstated premises/axioms necessary to actually make the proof. Take the "correct" form of your first syllogism:
1. All horses have 4 legs
2. That animal is a horse
3. Therefore, that animal has 4 legs.
Prove that 1 and 2 imply 3! What logical basis do you have for believing that 1 and 2 imply 3? You need another premise, of the form: If all horses have have 4 legs, then any animal that is a horse has 4 legs. (You might find this premise silly given the conclusion, and try to come up with a more general axiom relating the specific to the general, but that is irrelevant towards are essential problem).
Our new proof:
1. All horses have 4 legs
2. That animal is a horse
3. If all horses have 4 legs, then any animal that is a horse has 4 legs
4. Therefore, that animal has 4 legs
Ah, now prove that 1 and 2 and 3 imply 4! You need another premise. :) This can go on forever, hence you need infinite premises to make a proof. The argument that any proof also needs unstated premises will wait for later. |