Source: p 165. Sweet Reason: A Field Guide to Modern Logic (2010 2 ed) by Henle, Garfield, Tymoczko.
I read this on Math SE; please advise if it pertains to my simpler question.
One property of the empty universe [hereafter EU] is that every existential statement [hereafter ES] is false. That should sound reasonable. ∃xPx is true in a universe only when you can find a thing with property P. But since you can't find anything in an empty universe, you can't find anything with the property P in an empty universe — no matter what property P denotes. That means a pretty innocent statement such as ∃x(Px ∨ ¬ Px) is false in the empty universe (but it's true everywhere else).
The flip side of this is that every universal statement [hereafter US] is true. Remember that the negation of a universal, ∀xPx, is an existential, ∃x ¬Px (see above and p.33) and since every existential statement is false in the empty universe, every universal statement is true. Think about it. is wearing a purple shirt." Let's say P__ means "__ is wearing a purple shirt." What is the truth value of ∀xPx in the empty world? Is everyone wearing a purple shirt? Well, do you see anyone who's NOT wearing one? What's that? You don't see anyone at all? Well then you don't see anyone not wearing a purple shirt, right? So everyone is wearing a purple shirt!
I'd like a direct, intuitive explanation. Please do not answer with formal proofs that I already understand. I understand the above, but it
- is indirect because it starts from the Falsity of an ES in an EU, and then ¬ES = US
- and fails to convince, because the last paragraph can equally but falsely be said of ∃x¬Px.