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I'm stuck with the concept of vacously true statement. For example, I know that the statement "Every element of the empty set is a zebra" is a vacously true statement because we can't find an element of the empty set which IS NOT a zebra.

However, there is one point that I can't understand. That is we also can't find an element of the empty set which IS a zebra.

So in some sense, the statement is both "true" and "false" (which is quite disturbing). So why we claim that the vacously true statement TRUE even when we can't prove it is true ?

I have heard something like "In mathematics, everything is true unless proven false" as an excuse for the vacously true statement. But i think it is quite false, for example in the case of many hypothesis (for example Riemann hypothesis) we wait and we try to prove or disprove it without automatically supposing it is true (until someone prove it is false).

Could you please explain me the philosophy of the "vacuously true" statement and the logic of the stament "in mathematics, everything is true unless proven false" ?

Mauro ALLEGRANZA
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InTheSearchForKnowledge
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  • The negation of every element is a zebra would be that there is some element that is not a zebra – J. W. Tanner Sep 19 '21 at 21:08
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    The statement that "everything is true unless provably false" is nonsense. "Every element of the empty set is a zebra" means "for all things $X$, if $X$ is in the empty set, then $X$ is a zebra" and this is true, since a conditional with a false antecedent is true. – Rob Arthan Sep 19 '21 at 21:12
  • @RobArthan could you please explain why the statement "if $x$ is in the empty set, then $x$ is a zebra" is True ? I think we just can't prove or disprove it since there is nothing we can use to prove or disprove the statement – InTheSearchForKnowledge Sep 19 '21 at 21:15
  • @RobArthan Isn't "for all $x$ in the empty set, $x$ is not a zebra" also true? I believe that's where the confusion is. – John Douma Sep 19 '21 at 21:17
  • It's by the definition of material implication. See numerous MSE posts for explanations of why we use that definition in mathematics. Note also that "$X$ is a zebra" could be replaced by any statement, even "$X$ is not a zebra" and the vacuously true statement would still be true. – Rob Arthan Sep 19 '21 at 21:17
  • @JohnDouma: then please help the OP out by offering your attempt to clear up the confusion. – Rob Arthan Sep 19 '21 at 21:19
  • @RobArthan : what does "MSE" stand for please ? – InTheSearchForKnowledge Sep 19 '21 at 21:21
  • MSE means math.stackexchange.com (this forum). – Rob Arthan Sep 19 '21 at 21:22
  • It's vacuously true that every element of the empty set is a zebra and it is also vacuously true that ever element of the empty set is not a zebra. My take (which may not be correct) is a vacuously true statement is a statement about a condition of non-existent elements. As they do not exist any possible condition applies. The issue is why must impossible to apply conditions be true? As opposed to must be false which is more intuitive? Well, Making them always false will lead immediately to contradictions. It's similar to why we have $False\implies True$ for any statements. – fleablood Sep 19 '21 at 21:51
  • ""In mathematics, everything is true unless proven false"" I don't think that is true. It should be "In mathematics, everything is true that can't not be false". We don't know of the Reiman hypothesis is true or false. But we do know that "Every element of the emptyset is a zebra" can not be false. If it were false there would be an element that was not a zebra.... and there is not. – fleablood Sep 19 '21 at 22:02
  • " That is we also can't find an element of the empty set which IS a zebra." That is not a contradiction. "Every element we can find is a zebra" and "We can't find any zebras" are not contradictory if we can't find any elements. – fleablood Sep 19 '21 at 22:04

3 Answers3

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Both

$(A)$ "Every element of the emptyset is a zebra"

and

$(B)$ "Every element of the emptyset is not a zebra"

are true. However, they are not the negations of each other - the negation of $(A)$, for example, is

$(\neg A)$ "Some element of the emptyset is not a zebra."

We may reflexively expect $(\neg A)$ to be a consequence of $(B)$ but that's based on a glitch of intuition. The intuitively-obvious implication $$\forall x\in U(P(x))\rightarrow \exists x\in U(P(x))$$ only holds if the scope of quantification $U$ is nonempty, which in this case it isn't.

So in fact there is no simultaneous truth and falsity going on here. $(A)$ and $(B)$ are each true, and their negations $(\neg A)$ and $(\neg B)$ are each false.

Noah Schweber
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  • Hi, thank you very much for your answer, so, do you mean that because we don't know how to prove "every element of empty set is a zebra", we try to use its negation which is "some element of the empty set is not a zebra" which is defined to be false, to claim that the former statement is true right ? – InTheSearchForKnowledge Sep 19 '21 at 21:28
  • @InTheSearchForKnowledge I don't really understand that question. In what sense do we not know how to prove that every element of the emptyset is a zebra? – Noah Schweber Sep 19 '21 at 21:28
  • I also don't understand your claim in the OP that "in mathematics, everything is true unless proven false." This seems to be an exaggeration of the equivalence between "$\forall$" and "$\neg\exists\neg$," but the latter is just how the universal quantifier is defined (the "no-counterexample" interpretation). – Noah Schweber Sep 19 '21 at 21:30
  • what i mean is that we just don't know how to prove or disprove the statement just because the set is empty, we don't have anything to look at. I mean, how can we PROVE that every element of the emptyset is a zebra when there is no element of the emptyset ? The same goes for its negation "Some element is not a zebra". It confuses me a lot because the vacously true reason is used many times as a base case in inductive proofs of many theorems. – InTheSearchForKnowledge Sep 19 '21 at 21:32
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    @InTheSearchForKnowledge "how can we PROVE that every element of the emptyset is a zebra when there is no element of the emptyset ? " By definition, "Every element of $U$ has property $P$" means "there is no element of $U$ which does not have property $P$." If we know that $U$ is empty, we're really really confident of this. Having fewer things to check makes verifying the non-existence of a counterexample easier, not harder. – Noah Schweber Sep 19 '21 at 21:33
  • Ah Ok, i see the point. Thank you very much for your help! – InTheSearchForKnowledge Sep 19 '21 at 21:40
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Statements like the ones above are true because we require that all statements be true or false. So, if "for all in the empty set, is a zebra" is false, then we are saying that there exists an in the empty set such that is not a zebra. That can never happen.

In general, all statements of the form $$\forall x P(x)$$ are true for the empty set because their negations are $$\exists x\lnot P(x)$$

John Douma
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I think to understand how this comes about you need to appreciate two things.

  1. Sets are conceived as abstractions of properties. Two properties might have quite different meanings, but if every object with each property also has the other, then the properties correspond to the same set. Properties are a complicated idea, and sets are intended to simplify this complicated idea: if two sets contain the same elements, they are the same set, even if they were defined in very different ways.

    One important consequence of this is that there is only one empty set. The property of being an even prime number greater than 10 is nothing at all like the property of being a living crown prince of the Ottoman Empire. But the two sets are the same set.

  2. Consider the following common-sense claim:

    If all rubies are red, then all rubies belonging to MJD are red.

    As it happens, I don't own any rubies. The set of rubies belonging to MJD is empty. If you don't agree that all the rubies in the empty set are red, then you should deny the common-sense claim above. You should want me to qualify it:

    If all rubies are red, then all rubies belonging to MJD are red, if there are any.

    We would have to to include this unimportant qualification every time we made a claim about elements of any subset of anything. This would be unhelpful. So we agree that such claims can be vacuously true: if all rubies are red, then all my rubies are red, whether I have any rubies and even if I don't.

When these two ideas come together, one gets the odd-sounding claim that every element of the empty set is red. (I own no rubies, so the set of my rubies is empty, and every ruby in that set is red, because every ruby is red.) But if we want sets to work as abstractions of properties, we have to accept that every element of the empty set is red.

By the same reasoning, every element of the empty set is blue, because all my blue diamonds are blue.

If you don't like that every element of the empty set is both red and blue, you need to find a way to say that the set of all my rubies is different from the set of all my blue diamonds. But this is exactly what what we don't want to do; it's exactly the complicated situation that sets were invented to simplify.

When you simplify anything, you necessarily lose some of its nuance. There is a philosophical distinction between the property of being one of my rubies and the property of being one of my blue diamonds. Set theory intentionally discards this distinction.

And anyway what is wrong with the idea that every element of the empty set is both a red ruby and a blue diamond? There aren't any things that are both red rubies and blue diamonds, so the set of all such things ought to be exactly the empty set.

MJD
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