23.3 Vacuous Generalizations
Remember back to PROP, when we learned about the conditional.
A funny fact is that conditionals are always true when the antecedent is false.
We called the vacuously true conditionals.
Since All Ps are Q, Ax(P(x)->Q(x)), has a conditional in it, the same thing happens here.
If there are no Ps, then All Ps are Q is vacuously true.
It is called a vacuously true generalization.
A lot of people find that counter intuitive.
For example, imagine you asked me how the animals at the shelter are doing, and I say:
All the cats are fed.
If there are no cats, then what I said is true. But if you then learned that there are no cats, you might accuse me of lying!
The way logicians think about it: I wasn’t lying, because it was technically true; I was just misleading you, because a norm of conversation is that what I say should be relevant to your question.
But if there aren’t any cats, then my answer isn’t really relevant. All the lions, blue whales, and unicorns at the shelter are fed too!
Logic separates this issue about misleading uses of words, which is about pragmatics, from whether my sentence is technically true, which is about semantics.
If you still don’t like vacuously true generalizations, we can still use FOL to translate generalizations with existential force:
But you should assume for the rest of this book that generalizations are to be translated in the standard way, Ax(P(x)->Q(x)), without the existential.