# Intuitively, why are Universal Statements true in the Empty Universe?

Source: p 165. Sweet Reason: A Field Guide to Modern Logic (2010 2 ed) by Henle, Garfield, Tymoczko.

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.
• To put it short, "all none of them meet the condition", it holds without exception, it covers everything in the universe (all none of it). Like the truth table of the material conditions, it is not really intuitive, but it should be intuitive that it is safe to assign all ambiguous or meaningless statements a truth value of true because no unexpected contradiction can arise.
– user9166
Commented Aug 9, 2016 at 2:29
• @jobermark Negations of ambiguous/meaningless statements are presumably also ambiguous/meaningless, so blanket assignment of "true" won't work. Commented Aug 9, 2016 at 3:22
• For me, the direct, intuitive explanation is that everything in the empty universe satisfies any property. I suppose it's one of those "it's obvious once you're used to it" sorts of things.
– user6559
Commented Aug 9, 2016 at 13:56
• @Hurkyl Aren't things you have to get used to exactly the opposite of what people mean by 'intuitive'?
– user9166
Commented Aug 9, 2016 at 20:49
• @jobermark: But intuition is developed through experience. Not everything becomes intuitive with experience, but (for me) this is. So much so that I actually have trouble understanding why people have problems with it. Intellectually, the explanation appears (to me) to be some combination of unfamiliarity with and learned avoidance of vacuous lines of thought -- e.g. that people are used to rejecting a vacuous question rather than answering it one way or the other.
– user6559
Commented Aug 9, 2016 at 22:54

Think of ∀xP(x) as an implicit conditional: ∀x(xϵU → P(x)), where U is the universe. In an empty universe the antecedent is always false, hence the conditional is vacuously true. In contrast, ∃xP(x) is an implicit conjunction ∃x(xϵU ∧ P(x)), so it is vacuously false. This is in line with the standard way of transcribing "all humans are liars" with a conditional, but "some humans are liars" with a conjunction (here all humans form the universe of discourse), and the closest predicate calculus can approximate intuition, see Why can't we use implication for the existential quantifier?

If we wish to drop the universes from notation, but to end up with the same truth values for the empty universe, we are forced to adopt the universal-statements-are-true convention. The same is needed for uniformity of notation with restricted quantifiers in mathematics, like ∀x>0 P(x), which by convention is interpreted as ∀x(x>0 → P(x)). The intuitive qualms about this convention are of the same nature as those about the material conditional being vacuously true for false antecedents, so this is not a quantifier specific issue, see Why are conditionals with false antecedents considered true?

The point is that there logical operators need a set that they work on. In mathematics this is clearer. There you always say "∀x∈X : Px" where X is the set you talk about. Now this is important, because logic does not make sense if you do not operate on sets. If you do not operate on sets some statements might be both true and false.

Philosophers like to avoid the problem of defining the set by talking about the universe [U]. This Universe is the set you operate on. Every universal statement you make, only concerns elements of the universe.

Now if you make a universal statement, you calculate the truth value by having a look at every single element and checking if the statement is true. If the universe contain 10 elements, you check 10 times and each time the statement must be true for the universal statement to be true. 10 checks, 10 times true. Now in an empty universe you have 0 elements, therefore zero checks to do. You can't be wrong!

Another way to think of it, could be formulating the universal statement as "every time x is in X, it has property P": every time we find something (x) that exists in the universe (U), this thing has the property (P). When nothing exists, the statement is true.

It becomes clearer why this way of thinking is necessary, if you think of logic as set theory. A property gives you a subset of a set. "Every child in the classroom older than 10" will give you a subset of "children in the classroom". A universal statement is true, if this subset is the whole set again. If the original set is empty, this is always the case.

You can say anything about "nothing", and it won't be false.

But possibly the problem is that an "empty universe" is counter-intuitive in itself, so anything that is said about it is counter-intuitive?