A problem in classical logic

This is from my friend, Dominic Foo.

In Aristotelian logic, if you said that “All roses are red.” the “contrary” of this is “No roses are red”, and since the contradictory of “no roses are red” is “some roses are red”, there is a straight forward inference from “All roses are red” to “some roses are red”. Statements of the form “All x is P” implies that “Some x are P”.

Now you might think this is “common sense”, but this is actually fallacious on modern symbolic and mathematical logic. Consider the following proposition: “All unicorns have horns” and “Some unicorns have horns.” In symbolic logic the latter statement is *not* entailed by the former unlike Aristotelian logic. “All unicorns have horns” is translated in symbolic logic as “For all x, if x is a unicorn x has a horn” and for “Some unicorns have horns” this is translated as “There is some x such that x is a unicorn and x has a horn.” However it is a logical fallacy to infer that there is some unicorn from the universal statement that all unicorns have horns. In symbolic logic you have to be able to make true statements about empty sets. For example, “For all x, if x is the largest prime number then x is divisible only by itself and 1.” Of course the set of largest prime numbers is zero, there is no largest prime number, but in mathematics you have to be able to make statements like these on empty sets without implying that there is a member of the set in order to perform reductio ad absurdum proofs.

It is this “Aristotelian logic” thinking which caused some of the objectors to think “No female nature is assumed” which is equivalent to “All female natures are not assumed” implies or entails, “There are female natures”. Whereas on symbolic or mathematical logic, which I am a lot more used to, it seems pretty obvious to me that one can make universal statements on a set without implying that the set is non-empty.


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