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I've ben struggling with the concept of vacuous truth, as used (1) in proving implications, (2) as base cases for induction proofs.

To help me understand, it would be useful to understand if the concept of vacuous falsehood exists, and if so, what simple examples might be.

Discussion

This question comes from the notion that truth is defined where something cannot be proved false - which to me has always felt insufficient.

So I ask myself, if something can't be proved true, is it vacuously false?

I expect the answer is that there is no such thing as vacuously false. Explaining this will shed light on my original confusion.

MJD
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Penelope
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5 Answers5

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Here's my favorite example of vacuous truth. First let's agree that

all rubies are red.

This is true, by definition; the same gemstone, when not colored red, is called a sapphire.

Since all rubies are red, we can conclude

all the rubies in my vault are red.

This makes sense because if all rubies are red, then the ones in the vault are certainly red.

"Vacuous truth" just means that we agree that we will consider "all the rubies in my vault are red" to be true even if there are no rubies in the vault. This is mostly for convenience. Otherwise we find ourselves saying a lot of things like

all the rubies in my vault are red, if there are any rubies in my vault.

which makes mathematical discussion more confusing and complicated for no real benefit.

Now to your question. What would a vacuous falsity be? Tian Vlašić's suggestion in the comments is just what we need:

If a statement is vacuously true, then its negation is vacuously false.

The negation of

all the rubies in my vault are red

is

there is a ruby in my vault that is not red.

One way this could be false is if there are some rubies in my vault and they are all red.

But another way this could be false is that there might not be any rubies in my vault at all. This is the vacuously false case.

And in the vacuously false case we don't even need to read the second half of the sentence:

there is a ruby in my vault that …

It doesn't matter what “…” is. We know immediately this is vacuously false, because there is no ruby of any sort in my vault.

The irrelevance of the “…is not red” part is mirrored exactly in the irrelevance of the “… are red” part in the vacuously true statement:

all the rubies in my vault are …

Again, if my vault has no rubies, then it doesn't matter what “…” is. We know the statement is vacuously true, because there is no ruby of any sort in my vault.

MJD
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  • By the way, speaking of philosophical statements, like vacuous true, how do we know you actually have a vault? Does it exist in solely in your mind? If it's in only as a picture in your mind, how do we verify it and these supposed rubies? :-) (-; :-) (-; :-) (-;
    Hopefully, at least one person laughed.     – nickalh Sep 28 '23 at 22:37
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    All the rubies in my vault laughed at your comment. – MJD Sep 29 '23 at 07:09
  • @MJD thanks for an excellent reply. It has helped me understand why "vacuous" is applied as a label - the required conditions set out in the statements don't apply (room has no rubies).

    This has helped me move forward. But I'm not there yet.

    Why is "there is a ruby in my vault that …" vacuously FALSE, but "all the rubies in my vault are …" I vacuously TRUE?

    Why can't "there is a ruby in my vault that …" be nauseously FALSE?

    If I can get my head around this I think I will have finally cracked it.

     – Penelope Sep 29 '23 at 14:49
  • @Penelope I think what you meant to ask in that comment is not what you actually wrote, so can you please look it over carefully, and if it doesn't say what you wanted, post a new comment that does say what you meant to? – MJD Sep 29 '23 at 17:15
  • @MJD yes there was an auto-carrot typo in the previous comment. Why is "there is a ruby in my vault that …" vacuously FALSE, ...but "all the rubies in my vault are …" is vacuously TRUE? ... Why can't "there is a ruby in my vault that …" be vacuously FALSE? If I can get my head around this I think I will have finally cracked it. – Penelope Sep 29 '23 at 20:13
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    I'm assuming you meant to ask 'Why can't "there is a ruby in my vault that …" be vacuously TRUE?' (changed word in boldface) and the answer is, it can't be vacuously true because it is false. The precondition ('there is a ruby") is not satisfied, so it has to be false. – zwol Nov 24 '23 at 19:31
  • I have an alternative motivation for empty foralls being true. Suppose I have two vaults, and I want to know whether all the rubies in both vaults are larger than 3 carats. This should be true when all the rubies in vault 1 are large and all the rubies in vault 2 are large. But for this computation to be correct, I have to choose to say that empty vaults have all their rubies large. More generally, for any monoidal accumulation, you should probably choose the identity to be the result of the empty accumulation; this explains empty exists=>false, sums=>0, and products=>1 among others. – Daniel Wagner Nov 28 '23 at 16:06
  • Yes, this is how I always explain it to computer programmers. Let $\oplus$ represent list concatenation (or set union or any other monoidal operation, except I don't say “monoidal”). Surely $\operatorname{all}(\Phi, a\oplus b)$ should equal $\operatorname{all}(\Phi, a) \land \operatorname{all}(\Phi, b)$. But for this to work we need $\operatorname{all}(\Phi, empty) = \top$. This persuades most people. A few more are won over by the observation that if you don't do this, your recursive all function will need a… (gasp) special case. – MJD Nov 28 '23 at 18:46
  • None of this was enough to persuade the author of the Perl List::Util::sum function, however. He dug in his heels and nobody could convince him that sum() shouldn't return undef instead of zero. I see there is now a sum0 function that works around this. – MJD Nov 28 '23 at 18:48
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MJD's "rubies in my vault" discussion is an excellent example of a vacuously true statement, but I don't think it confronts the source of your confusion head on.

You wrote:

This question comes from the notion that truth is defined where something cannot be proved false - which to me has always felt insufficient.

Being unable to prove something false is indeed insufficient to say it's true. But that's not the meaning of the term "vacuously true".

Only statements of a certain type are said to be vacuously true. The two most common cases (already mentioned in the comments, and in MJD's example):

  1. "For all $x$ with property $P$, the property $Q$ also holds", where there are no $x$'s with property $P$.
  2. $A\to B$, where $A$ is false.

According to the standard conventions used in mathematics, these are both vacuously true. You can see how (1) and (2) are closely related: if "$x$ has property $P$" is false, then "if $x$ has property $P$ then it also has property $Q$" is vacuously true.

MJD's discussion explains clearly why we use this convention in mathematics and logic. But for a statement like, say, "there are no odd perfect numbers", just because we haven't been able (so far) to prove it false, doesn't mean we regard it as vacuously true. It doesn't fit the pattern. Likewise for the negation, "there exists an odd perfect number".

Let me dig a little deeper into the form "all $x$ with property $P$ have property $Q$". It will be vacuously true if there are no $x$'s with property $P$. But to prove that it's vacuously true, you have to prove that there are, in fact, no $x$'s with property $P$. The statement "all odd perfect numbers are the product of at least 3000 primes" might be vacuously true, but we don't know. If tomorrow someone proved that there are no odd perfect numbers, then the vacuous truth of that "3000 primes" assertion would follow.

You asked why statements of the form (1) are vacuously true rather than vacuously false.

The answer is "convenience", but I mean that in a very strong sense---in the same sense, for example, that it's convenient to have a number 0 with the property that $x+0=x$ for all $x$, or that negative times negative is positive. These are mathematical conventions, and you can construct consistent mathematical systems with different conventions, but they're usually awkward. You could say they offend against some indefinable sense of mathematical "rightness".

Consider the statement "all odd perfect numbers have at least 3000 prime factors". If it turns out that there are no odd perfect numbers, what truth-value should we assign to this? Three options:

  1. True.
  2. False.
  3. It doesn't have a truth-value.

Option (3) would be a so-called "truth-value gap" (a term coined by Strawson). Classical logic relies heavily on the law of the excluded middle, which forbids truth-value gaps. (In everyday language, we're not so rigid.)

Option (2) would be very awkward. It's been proven that an odd perfect number must have at least 101 prime factors. I haven't seen the proof, but I'm sure it looks something like this: "Assume $n$ is an odd-perfect number. Then $n=2m+1$ for some $m$, and $\sigma(n)=2n$...". Then some standard reasoning, concluding with at least 101 prime factors. We can check the reasoning for validity without knowing if there are any odd perfects, but only because we regard the assertion as true if there aren't any. With option (2), the reasoning becomes somehow invalid in the absence of odd perfect numbers, even though it would be fine if thare is one.

And that's a general phenomenon. We want to be able to prove conditional statements "If $P$ then $Q$" without knowing if $P$ is true. The classical truth table for $P\to Q$ has F only on the "TF" line ($P$ true and $Q$ false). If you make the "FT" and "FF" lines also F, then you get the truth table for $P\wedge Q$. In other words, to prove that $P$ implies $Q$ you'd have to prove that $P$ is true. That would be most inconvenient.

Michael Weiss
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  • thanks for the useful reply. I'm asking myself why your example (1) can't be vacuously false? What "reductio ad absurdum" would happen if we said that (1) was vacuously false? – Penelope Sep 29 '23 at 20:23
  • I think this answer comes closest to addressing the source of my confusion. You addressed the "truth value gap" which was also part of my thinking.

    I am 90% there and will mark this as answered in the hope this great answer helps others. Myself I am closer than ever but not quite there - perhaps I just need a little time. The equivalence of (1) with the implication (2) is what helped me most as I am happy to think abut the truth of the implication (rationale that goes from P to Q) if P is true. If all statements of the form (1) can be recast as form (2) then I am happy. Is this the case?

     – Penelope Oct 03 '23 at 20:27
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    At least in the context of formal logic, yes. The formal expression of “All P are Q” is $\forall x(P(x)\to Q(x))$. – Michael Weiss Oct 03 '23 at 23:13
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Expanding on the comment by @TianVlašić: If proposition $A$ is false, then $A\to B$ is said to be vacuously true, regardless of the truth value of $B$. We could also say that $A \land B$ is vacuously false regardless of the truth value of $B$.

Example

Consider the implication, if is raining ($R$), then it is cloudy ($C$).

$~~~~~~R\implies C$

If it is not raining ($R$ is false), then it may or may not be cloudy.

The truth table:

enter image description here

If it is not raining (lines 3-4), then it may or may not be cloudy. The implication would be true in either case. We say that the implication $R\implies C$ is vacuously true when $R$ is false.

Such arguments are rarely if ever used in daily discourse. Rarely is consideration given to the implications of a proposition that is known to be false. As such, meaningful "real-world" examples are hard to come by. They are often, however, used in very technical arguments, e.g. mathematical proofs.

As for "vacuously false," consider the truth table for $A\land B$:

enter image description here

When $A$ is false (lines 3-4), $A\land B$ is (vacuously?) false regardless of the truth value of $B$.

Dan Christensen
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@MJD already has an excellent answer to the primary question. This response deals primarily with "So I ask myself, if something can't be proved true, is it vacuously false?"

There is a third possibility. In math, there are statements in math which are cannot be proven true and cannot be proven false. Godel's Incompleteness theorem establishes this fact. Technically, it only applies in the context. "The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers."

The Collatz conjecture may be a true hypothesis which is unprovable. That does not make it vacuously false. A possibly oversimplified statement of the Collatz conjecture follows.
if n is even, then the next integer in the sequence is $n/2$. If n is odd, then the next integer in the sequence is $3n+1$.

This is a specific example of a recursive sequence, where the next value is determined solely by the previous value.

Suggestion that it may be unprovable from Quanta Magazine "Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades."

nickalh
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  • This goes to the hear of my confusion. That something can't be proved false does not mean it is true .. in my head anyway. But this isn't how most logic textbooks discuss logic. – Penelope Sep 29 '23 at 14:52
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I think

  • A proposition is vacuously true when it's true because it concerns an empty class of objects (and says that something is true about all of them).
  • A proposition is vacuously false when it's false because it concerns an empty class of objects (and says that something is true about one of them).

This is pretty much what MJD says, but I'm making the principle explicit whereas he focuses on giving an example. In MJD's example, the class in question is "the rubies in my vault".

It's not quite the same as what Dan Christensen says. Dan's answer is in some sense more general, but I don't think I would use the word "vacuously" in cases that can't be fitted into the schema above. (Think about what "vacuously" means, namely "emptily". Things are vacuously true/false when their truth/falsity is because some class is empty.)

The general thing that's going on here (whether you prefer my presentation, or MJD's, or Dan's) is that in logic there is a duality where if you take some universally-true thing it remains universally true when you go through it and exchange "true" and "false", "and" and "or", "for all" and "there exists", etc.

Gareth McCaughan
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