I am reading about categorical syllogism in Copi's Introduction to Logic. I was wondering what differences are between the following statements:
Am I correct that
Does the book answer my question somewhere?
It depends.
The propositions E ("No S is P") and A* ("All S are not-P") may or may not be logically equivalent, depending on your interpretation of the existential import of E.
That is:
Bumble discussed this matter in another post, which provides a fuller treatment, showing three options.
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....
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...
Option 3. None have existential import. On this interpretation, the issue of existence is separate from that of predication.
I am squarely in Option 2, with the qualifier that I would formulate O as "Not [all S are P]", rather than the typical "Some S are not P": the latter formulation is an affirmation (" It is the case that 'some S are not P' "), rather than a denial (" It is not the case that 'all S are P' "), and seems to imply O has existential import when it does not.
So, read my view as assuming propositions A ("All S are P") and I ("Some S are P") have existential import, but E ("No S are P" / "Not [some S are P]") and O ("Not [all S are P]") do not.
No, they are not the same thing in traditional/Aristotelian logic, although they may both be true in some cases.
However, in the Brentano-Venn thesis, which denies existential import for universals (both positive and negative) they are; but the propositions ought to be re-written as:
There is a lot of confusion about this, even from the likes of Bertrand Russell, who uses the traditional Aristotelian form of these propositions:
"The proposition 'No Greeks are men' is, of course, the proposition 'All Greeks are not-men.'"
—Bertrand Russel, The Philosophy of Logical Atomism (1918–19), Lecture 5: General Propositions and Existence
Given the labels:
E is a denial, while A* is an affirmation in the Aristotelian system.
Aristotle spent the entirety of Prior Analytics I 46 discussing their difference:
In establishing or refuting, it makes some difference whether we suppose the expressions 'not to be this' and 'to be not-this' are identical or different in meaning, e.g. 'not to be white' and 'to be not-white'. For they do not mean the same thing, nor is 'to be not-white' the negation of 'to be white', but 'not to be white'.
—Aristotle, Prior Analytics I 46
I show it visually as follows:
To say 'A*' ("All S are not-P") is to affirm something about all of S: that all S lies outside of P and is not-P as a result.
In this case, S must clearly exist for this proposition, A*, to be true, and so the diagram for A* has two circles (S and P).
Contrarily, to assert 'E' ("No S are P") is to deny something about all of P, to claim that nothing within the confines of P is S, and so the diagram for E has only one circle (P).
You'll notice that E is true when S does not exist, while A* is false (since there is no S to apply the label "not-P" to).
The typical logic diagram for E shows two non-overlapping circles, S and P, rather than the single circle, P, I used to depict E. You can see this in Hamilton's Lectures on Logic:
I would argue two non-overlapping circles show A* (All S is not-P) rather than exclusively E (No S is P)—of course, E is still true in the case of A* being true—and that this construction further confuses people as to the difference between E and A*.
