If you have ever seen any syllogism, chances are it’s this one:
All men are mortal;
Socrates is a man;
therefore, Socrates is mortal.
Aristotle believed that there were 19 valid syllogisms. Today — since the work of George Boole (and possibly of De Morgan) — we take it that there are only 15 valid syllogisms.
As I have alluded to before, this distinction arises because we consider the possibility of an empty set. This affects the valid conclusions from such statements as “all unicorns are mortal”.
We take that statement to be true, even though we believe that unicorns do not exist; we take it that all zero of them are mortal.
We do not conclude, however, that some unicorn is mortal.
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