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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.
Community
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    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. –  Aug 09 '16 at 02:29
  • @jobermark Negations of ambiguous/meaningless statements are presumably also ambiguous/meaningless, so blanket assignment of "true" won't work. – Conifold Aug 09 '16 at 03:22
  • @Conifold Take it in context. We assign true to the positions in material implication that are ambiguous. It is not something that can be done willy-nilly, but it is a reasonable description of what happens in both of these circumstances. –  Aug 09 '16 at 03:30
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    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. –  Aug 09 '16 at 13:56
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    @Hurkyl Aren't things you have to get used to exactly the opposite of what people mean by 'intuitive'? –  Aug 09 '16 at 20:49
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    @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. –  Aug 09 '16 at 22:54
  • @Hurkyl Then what is left outside intuition? Learned effectiveness eventually becomes expertise, but expertise is not intuition, for many phil's intuition is 'human' and humans do not share the same experiences. Besides, it is not clear that proposition like "all of the zero balls in this bag are black" or "if kittens are furry then the sky is blue" really should be given truth values, or that we *learned* to avoid them. I am sticking with this just not being intuitive, in any normal sense, and its acceptance just being a result of logicians liking artificial simplicity. –  Aug 10 '16 at 17:04
  • @jobermark : I never said expertise is intuition (although intuition can be part of expertise) -- I said intuition is developed through experience. If you don't have experience with something, you can't develop intuition about it -- e.g. I have little to no intuition about the behavior of dogs, but I grew up around cats and have a pretty good idea about those. I really do think this is a matter of language shaping thought. (and more precisely falls into the same category as quantifying with zero or working with empty collections) –  Aug 11 '16 at 00:55
  • @Hurkyl By the standards of Descartes and most folks after him, intuition does not require proof, evidence, or conscious reasoning -- you just get the idea naturally upon first exposure. Experience is evidence, hence this is not intuition. It is expertise or 'mastery' -- when conscious thinking becomes unconscious and we no longer have access to the internal content of our own thought without making an effort. –  Aug 11 '16 at 09:36
  • @jobermark: From http://plato.stanford.edu/entries/intuition/ : `research has shown that agents with sufficient experience in a given domain (e.g., neonatal nursing, fire-fighting, or chess) make decisions on the basis of a cognitive process other than conscious considerations of various options and the weighing of evidence and utilities. Such expert “intuitions” that some infant suffers from sepsis, that a fire will take a certain course, or that a certain chess move is a good one, appear immediately in consciousness.` –  Aug 11 '16 at 10:09
  • @Hurkyl Note the quotes, implying this is not the proper use of the term. You give evidence to my point. –  Aug 11 '16 at 10:12
  • @jobermark In a paragraph beginning `It appears clear that ordinary usage includes...` –  Aug 11 '16 at 10:14
  • @Hurkly. Did you note the quotes? Do you listen when people speak? Clearly the author thinks this is an inappropriate spreading of the meaning, and he goes on to keep the more narrow one for the remainder of the article. I am done with this. Be wrong if you like. –  Aug 11 '16 at 10:15
  • @jobermark: If inappropriate, it's still the ordinary usage. I think the OP is using "intuitive" as ordinary English. Do you think the OP means it as a technical term? –  Aug 11 '16 at 10:23
  • @Hurkyl.. i think that he wants something that invokes intuition -- **his** intuition and that only the classical definition makes any sense in this context, since you cannot give him the remainder of what makes this 'intuitive' to you. So whether he is conscious of the distinction or not, he means the definition that *has meaning* in this context. –  Aug 11 '16 at 14:30

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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?

Conifold
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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.

don-joe
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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?

Luís Henrique
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