Topics Logic

Question I am reading a by book by the great logician Raymond Smullyan. In this book he says that any statement of the form, "All As are Bs" are true if there are no "As". That is, these statements are vacuously true. He gives the following example, "All Unicorns have 5 legs" is true since there are no unicorns. So is "All unicorns have 6 legs", and "All unicorns are purple", etc. But this strikes me as obviously false. For example, "All unicorns have two horns" and "All unicorns are necessarily existing" are false statements. The first is false in virtue of the fact that unicorns are by definition one-horned. The second is false in virtue by the fact that it is impossible for something to be both necessarily existing and nonexistent. Am I missing something here or misreading Smullyan? Or are these counterexamples sufficient in refuting the claim that any statement of the form "All As are Bs" is vacuously true if there are no "As"? For reference the book is, "Logical Labyrinths" from pages 99-101. Thanks for your reply.

Accepted:
November 27, 2019

Comments

I don't know that book in

I don't know that book in particular, but I can give you a standard explanation that at least makes sense of the view you find puzzling.

In Aristotle's logic, any statement of the form "All S are P" implies that at least one S is P, so the statement comes out false (rather than vacuously true) if nothing is S. By contrast, in contemporary logic, "All S are P" is interpreted as saying "For anything at all, if it is S, then it is P": it is interpreted as a universal quantification applied to a conditional statement.

Crucially, the conditional statement "If it is S, then it is P" is standardly treated as a truth-functional conditional that is equivalent to the disjunction "It is not S, or it is P." Now suppose that nothing is S, so that "It is not S" is true of everything. Then the disjunction "It is not S, or it is P" will come out true no matter what we substitute for "it," because a true disjunction needs only one true disjunct. In that case, the truth-functional conditionals "If it is S, then it is not P," "If it is S, then it is not S," etc., also come out true for the same reason.

I think it's the decision to treat "If-then" statements as truth-functional conditionals that produces the counterintuitive results. But here we can perhaps distinguish the truth-conditions for an "If-then" statement from the conditions in which it would be informative or appropriate to assert the statement. A statement can be true without being, in the context, informative or appropriate to assert.