Formal logic and English conversational logic are quite different things; everybody knows that when the waitress asks whether you want ‘soup or salad’, ‘both’ is not a valid answer. But even when they have different rules, they can still come to the same conclusion. Consider the statement ‘not all of my children are mass murderers’. Is it true for me, given that I do not actually have any children? According to the usual rules of conversational English’s logical quantifiers (‘all’ and ‘there is’ and their various rephrasings), the answer is no: ‘not all of my children are mass murderers’ implicates that I have at least one child, and furthermore entails that that child is not a mass murderer. But what about if we interpret the statement in formal logic? The answer is still no, but for a completely different reason: it is the negation of the statement ‘all of my children are mass murderers’, and that statement is true.

The reason that I can state truthfully in formal logic ‘all of my children are mass murderers’ without actually having any children is for the same reason that the sum of an empty set is zero and the product of an empty set is one: if it were any other way, then several useful properties of those operations would not hold. For sum, we would lose the fact that the sum of the two sets A and B is the sum of the set A plus the sum of the set B, and similarly for products. These are true because 0 and 1 are identities of addition and multiplication, respectively. For the case of ‘for all’, we can think of it as first applying the predicate to the set, and then taking the logical and of the result; we then must have that is true when S is empty, or else we would lose the theorem that . While it could certainly be special-cased to take into account the cases where A or B is the empty set, it would detract from its simplicity, and would require proofs involving it to first show that neither of the two is empty (which can be a challenge, and might even require additional axioms if dealing with infinite sets, choice functions, etc.)

One of the interesting things that this implies is that for any predicate , , even when is false for all , much like how . But returning to the linguistics theme, it seems rather odd; this is because the quantifier ‘all’ has an implication that the entities that are being talked about exist; in other words, the statement ‘all x are y’ implies that at least one x exists. If this is known to be false by the speaker, it sounds odd, much like the statement ‘John has two children’ sounds odd if the listener knows that John actually has exactly four children; the statement ‘John has two children’ implicates that he has exactly two.