So in the logic course I am currently teaching we are about to start talking about the contrast between the modern and traditional square of opposition. The main difference, of course, is that poor Aristotle thought that if it was true that All A’s are B’s that made it the case that Some particular A was in fact B, and so he would be forced to exclude syllogisms as valid that have fictional terms. Consider the following proposition,
1. Some humans are mammals
Now, I often find that students think that 1 is false. They think that it is false because they think that it implies 2
2. Some humans are not mammals
but in the traditional square of opposition this is not the case. 1 and 2 are subcontraries, which means that they can’t both be false. Now this means that it is a possibility that both of them are true, but there is also the possibility that only one of them is true and the other is false. Since it is true that all humans are mammals by subalternation we know that 1 is true . Thus, we can say that 1 is true because we know that 3 is true and 2 has to be false because it is the contradiction of 3.
3. All humans are mammals
But what are we to make of this on the modern interpretation? Can we no longer say that 1 is true because of the fact that all humans are mammals? It is still the case that 2 is false, as it is still the contradiction of 1. But what about the so-called existential fallacy?
But if we take this discussion out of the realm of categorical propositions and start talking in terms of truth-functional semantics don’t we get the traditional square back? So, 1 is written as 1* with
1* Ex(Hx & Mx)
Whereas 2 and 3 turns into 2* and 3*
2* Ex(Hx & ~Mx)
So to get the traditional square back we would need to show that 3* implies 1*.
Consider the following proof,
A. (x) (Hx –>Mx) [assumption]
B. Ex (Hx) [assumption]
C. Hy [Existential Instantiation of B]
D. Hy –> My [Universal Instantiation of A]
E. My (Modus Ponens D, C)
E. Hy & My [conjunction introduction]
F. Ez (Hz & Mz) [existential generalization]
So, if B is correct and there are humans then 3 does entail 1 just like Aristotle said. Am I missing something?