Talk:Vacuous truth

message: :The character formatting is not working on my browser (Netscape 4.7).  What's the problem?  --John Knouse
: That Netscape 4.7 is old, cruddy and obsolete. Try Opera or Mozilla -- Tarquin
: You can't make netscape 4.7 work with the right fonts? (What's the problem anyway? Are math symbols ignored altogether? Do they come out as little squares?) Speaking of fonts, anyone know what fonts to install to make IE 6 work right with the math symbols? As shipped, most of the math symbols are displayed as little squares. As IE is probably the most popular browser, it would be good to get the math symbols working on it. --Ryguasu
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Dude! If you're still using MSIE6.x, you desperately need to get a better browser! Because it's so outdated, not only may it not recognize more current versions of HTML, but you'll be hideously susceptible to malware exploits.
You don't have to go with MSIE; other browsers are available, especially Chrome and Firefox.  — Preceding unsigned comment added by TheGrandRascal (talk • contribs) 09:59, 9 May 2020 (UTC)
replies: text: This is a low-traffic article and does not attract a lot of attention on the talk page, so as far as I can tell it has never been archived.  Often good to check how old a comment is before responding to it.  The comments you are responding to are from August 2002.  --Trovatore (talk) 18:32, 9 May 2020 (UTC)
thread_title: Issues with mathematical symbols

message: Is the statement in the first paragraph about even primes greater than 2 true in any sense?  --rmhermen
:Yes, of course; the English phrase "even primes greater than 2" translates into the mathematical expression "The intersection of the sets (even integers), (prime numbers), and (integers greater than 2)"; that is, the empty set. The statement is then a vacuously true one of the third form. --LDC
::Ha! I knew that this list of "every form" of vacuously true statements was incomplete. The problem with LDC's reduction of this example to the 3rd form is that the statement can be made in a logical system that does not have sets (such as Peano arithmetic). Of course, we can reduce it the second form by similar tricks:
:::For all x, if x is even and x is prime and x is greater than 2, then ....
::but the same is true for the 3rd form after all, yet we list it. Of course, the 3rd form is in formal language, while "even primes greater than 2" is in informal language; LDC and I have simply described two different ways to formalise it. But my point is that there are yet more possibilities for formal language. I'm going to at least add an example for typed logic (which the "even primes greater than 2" example can also be reduced to if one wishes), but the important thing is that we shouldn't insist that these are the only possibilities. The imaginations of logicians are endless. — Toby 08:27 Sep 28, 2002 (UTC)
:Do any of AxelBoldt's new additions clarify anything? If not, perhaps some further examples or restructuring are in order. --Ryguasu
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Who thinks that the empty set case is in some sense the most basic?
Most logics have no notion of set in the first place, yet any classical logic has vacuous truth!
(Ironically, this may be based on something that I did months ago, but redirecting Vacuous truth to Empty set on the grounds that the empty set version of vacuous truth was discussed there.
That was when I was unskilled with redirects; I never should have done that.)
— Toby 08:33 Sep 28, 2002 (UTC)
I made the empty set example follow the same form as the others.
That is, we never put in a contradiction for P, but instead an arbitrary statement that happened to be false.
Similarly, we needn't put in the symbol {} for (what I have called) A, since we can use an arbitrary set that happens to be empty.
That change, I just noticed, makes the only reason given for the primacy of the empty set example vanish; I'll remove that reason so as not to make the alleged supporters of that position look silly ^_^.
But I'd still like to know if any actually exist, or if this was simply a misinterpretation of my poor redirecting skills.
— Toby 10:28 Sep 28, 2002 (UTC)
I admit it - I put in the idea that some view the "empty set version" as more canonical in order to appease the authors of the empty set page. I never had any particularly good reasons for believing it myself. --Ryguasu
Since it was just based on a misunderstanding of me, I'll remove it.
— Toby 09:59 Sep 29, 2002 (UTC)
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Vacuous truth is not limited to two-valued logics; only some of the arguments in favour of it (the ones that say "Well, it can hardly be false!") are so limited.
In particular, intuitionistic logic has the same concept of vacuous truth.
— Toby 08:35 Sep 28, 2002 (UTC)
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I changed "two big questions" to "one big question", since there didn't seem to be any attempt to answer the 2nd.
Indeed, the 2nd is largely discussed prior to the listing of the questions, while this part of the article is identified by the header as being about the 1st.
(I suppose that this was the result of a rearrangement of the article.)
— Toby 10:28 Sep 28, 2002 (UTC)
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The New York argument and its cousin seem quite tenuous to me.
For the case of the commutative law (called "symmetric" in the article), this problem disappears if you assume that F → T = T, even if you still pick F → F = F.The other example didn't even make sense as written, since both statement simply said that I'm sane (given that 3 indeed equals 3).
I put a new example in there.
But both of these appeals to inuition have more problems than just disbelieving that F → p = T.
For example, the New York argument works on the assumption that people want to reject "If I'm in New York State, then I'm in New York City".
Yet that is precisely what we want the vast majority of humanity (residents of Buffalo being a principal exception) to accept!
I myself am in California, so this statement is as true its converse.
To avoid taking advantage of false assumptions (thereby reinforcing them), we need to focus on a specific person, such as one of the aforementiond residents of Buffalo.
But when we do that, now people will start to doubt that "If I'm in New York City, then I'm in New York State" is really true, since it's only vacuously true for the Buffaloan.
The real way to deal with the statement is to use universal quantification to rephrase it as "For any person x, if x is in New York City, then x is in New York State".
(Arguably, this is what the original phrasing about "being in New York city" was supposed to mean.)
But now we're no longer dealing with any of the forms of vacuous truth discussed in the article.
(Alternatively, we could keep my "I" and quantify over possible worlds, or speak of future probabilities, but that doesn't fit the templates any better.)
My new mathematical example succumbs even more obviously to a missing quantifier, and I suspect that any mathematical example would do so, since it's hard for mathematics to avoid being precise.
I would get rid of these entirely; they're fairly obscure arguments anyway (you did what with a truth table, thinks the reader?).
— Toby 10:28 Sep 28, 2002 (UTC)
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:"Vacuously true" is also sometimes also used as a synonym for tautological. This article, however, attempts a more technical analysis of a more limited concept of vacuous truth.
In what sense is the concept of vacuously true discussed in this article "more limited" than the concept of tautological? Is the idea that all vacuously true statements are tautological? If so, could someone please flesh out the concept of tautological? If not, saying the one is a subset of the other seems almost misleading. --Ryguasu
You're right; I'll fix this.
— Toby 09:59 Sep 29, 2002 (UTC)
While we're on the subject, who uses "vacuously true" as a synonym for "tautological".
(The external reference mentions the possibility of such use, but even the person that mentions it there disparages it.)
For example, I would call F → p vacuously true, and tautological if F is a contradiction (or more generally an explosive statement in Brazilian logic); but I'd call p → T trivially true instead, and tautological if T is a tautology.
Thus, the notion of vacuous truth is quite independent of the notion of tautology; as you were quite right to point out above, neither is stronger or weaker than the other.
And I would regard somebody that said that "If the sky is blue, then if the grass is green, then the grass is green" vacuously true as simply mistaken.
— Toby 10:10 Sep 29, 2002 (UTC)
I vote for the removal of the usage of tautological==vacuous. There may be some people who use the word in this sense, but they are probably confused. No need to reinforce or justify this mistake. AxelBoldt 00:08 Sep 30, 2002 (UTC)
I hope nobody is hurt by my removing the whole "New York" and "crazy" discussion. I don't think it can convince anybody. AxelBoldt 00:25 Sep 30, 2002 (UTC)
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I don't get the first two examples - in fact, I intuitively say the statements are false. "All elephants inside loaves of bread are pink" - why? If you said "...have ears", that would be true because all elephants - loaf-inhabiting or not - have ears. "All even primes >2 are multiples of 3" cannot be true because 3 itself does not satisfy the even requirement and any greater multiple of 3 does not satisfy the prime requirement. If I theoretically had an even prime >2 (not a multiple of two, an even number - I don't know how) it could not be divisible by 3.
I could say, however, "All non-exponential functions that are their own integrals are also their own second derivatives" because that can be proven for any function f(x) - exponential or not. Here it is of no consequence that only e^x, an exponential function, is its own integral. The examples given cannot be proven true and have no basis for truth. Or, say, "Many of the registered users on the Pig Latin Wikipedia are hopeless Wikipediholics" can be assumed true because, despite there being no Pig Latin 'pedia, many users of other language Wikipedias (myself included) are hopeless Wikipediholics. But the two statements given are only absurd, as far as I can tell. --Geoffrey 01:04 24 Jun 2003 (UTC)
:That's the point, really. You can't have such a prime such as the one descibred in the article - that's what makes the truth vacuous. Show me an elephant in a loaf of bread that isn't pink. Show me an even prime >2 not divisible by 3. You can't, because such things don't exist. The statements are thus true, but vacuously true (as the article says). --Camembert
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I removed the paragraph on the contrapositive. It tried to prove the truth of vacuous truths using the following argument
:If P is false, then for any Q
:\neg Q \Rightarrow \neg P
is also true, apparently using the fact that an implication is automatically true if its conclusion is true. Some vacuous truths have a true conclusion however, so we use what we want to prove in the proof. Furthermore, the equivalence of every implication with its contrapositive is arguably much harder to justify than the truth of vacuous truths. AxelBoldt 13:20, 24 Nov 2003 (UTC)
thread_title: Additional comments

message: This is a Featured Article that doesn't have a picture. This would stop it appearing as a Main Page feature, for example. Is there a useful picture possible? A Venn diagram or similar? - David Gerard 23:25, 8 Jul 2004 (UTC)
replies: text: Hate to nix it, but a Venn diagram would communicate precisely the wrong thing.  How about a larger form of that "P with a double arrow going to Q"? Meelar (talk) 23:29, 8 Jul 2004 (UTC)
replies: text: Anything that'd be informative and helpful in explaining the concept but would nevertheless look good on the main page ;-) - David Gerard 23:31, 8 Jul 2004 (UTC)
replies: text: I couldn't do it myself; not so handy with the images. But anybody who can, should. Meelar (talk) 23:44, 8 Jul 2004 (UTC)

text: A pink unicorn, perhaps? With text in front that says if X is a unicorn, then X is pink?  —Preceding unsigned comment added by Mike40033 (talk • contribs) 06:04, 23 April 2008 (UTC)
thread_title: Picture

message: See Wikipedia:Featured article removal candidates. --mav 21:40, 4 Sep 2004 (UTC)
thread_title:  FA removal candidate 

message: From the article: Consider the implication "if I am in Boston, then I am in Massachusetts." [...] There is something inherently reasonable about this claim, even if one is not currently in Boston. [...] Thus at least one vacuously true statement seems to actually be true.
The only problem is that there are a number of people in Britain who are managing, without any great effort, to be in Boston without being in Massachusetts.
Might I therefore suggest that the example of an "actually true" vacuous statement be changed, and the first step, in selecting a better one, should be to check whether it is in fact actually true?
:But I think the context makes it clear that one particular city called Boston was intended.  (Besides, that's not the only problem, since besides the Boston in England (the original Boston) and the one in Massachusetts, there are, after all, various other Bostons.) Michael Hardy 02:09, 19 Mar 2005 (UTC)
replies: text: I think much of the confusion can be avoided by realizing that in formal/symbolic logic the term "true" means "valid". So an argument with false premise(s) and conclusion(s) is valid but not sound (true premises and conclusions). Moreover, let's take the Boston example (without the fallacy of language, by knowing we mean Boston, Massachusetts) in a new direction: Say you live in California. So: If you live in Boston, you live in New York. It's valid, because you cannot deal with rules outside of reality, so you cannot verify (perhaps related to Aristotle's future contingents). So when someone says, "Well that argument cannot be valid because if one lived in Boston, one would live in Massachusetts," the problem is that the premise has been switched from False to True, thus the argument becomes invalid T->F is false. That's because when one assumes a certain premise, one isn't interested in looking at it from the point of view of it being false, which serves no purpose. So there are two subtleties at work here: the definition of true/false in logic versus semantics and the subtle switch of the validity/truth value of the premises from false to true. This could perhaps clarify some things, though I'm sure I wrote it in the most confusing manner possible. Cornelius (talk) 10:28, 1 January 2016 (UTC)
thread_title:  Boston example is flawed 

message: "the fact that the result of multiplying no numbers at all is 1 -- which is useful in a variety of mathematical fields"
The only utility I can think of is that it saves us the bother of writing "except zero" repeatedly in our proofs.24.64.166.191 04:06, 6 Jun 2005 (UTC)
thread_title:  useful in a variety of mathematical fields 

message: I added a sentence about how vacuously true statements can be used to mislead people. I feel that should be expanded on, and include quotes from famous movies (I would have included one but I couldn't think of any!). The reason is that these statements are very important when speaking, especially in situations such as giving testamony. 163.192.21.43 18:57, 21 July 2006 (UTC)
thread_title:  Vacuously true statements in real life 

message: Not that it matters, but Note that lying could be defined as knowingly making a false statement is wrong; lying should be defined as knowingly making a false statement with the intention of deceit. Albmont (talk) 17:41, 16 October 2008 (UTC)
thread_title:  Lying 

message: Not necessarily. In set theory, much of the definition and existence of the null set is pretty much determined by the vacuous truth that, for every member x of the null set, x is also a member of any other given set (which is of course a vacuous truth.) MarcelB612 (talk) 22:54, 10 August 2009 (UTC)
thread_title:  Devoid of content? 

message: By the same arguments in this erroneous article, there is such a thing as a vacuous falsehood. 61.132.87.130 (talk) 03:45, 14 April 2010 (UTC)
thread_title:  Scrap this anti-logical, anti-mathematical nonsense 

message: The correct resolution for the conundrums raised in this article is that vacuous truths are especially false: they are false because they would be false, were the first part true the second part would not follow or be false, or the statement as a whole would be false but they are even more especially more so false because the first part is false.  So not only is a vacuous truth false, it's ESPECIALLY false.  "Were present King of France goes prancing around naked in pubic, nobody would even bothers to take a photograph" is ESPECIALLY false because it is false enough to say that were the king of France to prance around naked in public nobody would even bother to take a photograph, but it is even MORE ESPECIALLY false because there is no present king of France.  Thus the article should be written thusly:
Confused mathematicians have called a "vacuous truth" any especially false statement that is not only false in its conclusion but especially false in that its premise is not even close to being an appropriate one to draw conclusions from.
For example, the statement "all cell phones in the room are turned off" is especially false if there are no cell phones in the room: it is false in that no cell phones are turned of, but it is even more especially false in that there are not even any cell phones in the room.
To confused mathematicians, who have historically called this a vacuous "truth", the statement would be true;  however, tge "all cell phones in the room are turned on" would also be considered "true" -- and not 'especially false', which is the correct resolution.  Likewise mathematicians have not traditionally accepted as even more especially false the statement: "all cell phones in the room are turned on and turned off".
And so on.  Please update the whole article to reflect the correct resolution.  You may also publish this correct resolution first.
Mathematicians need to be careful that when they make a general statement, such as "if x is a positive integer, then insofar as x is even, x + 2 is even", the last part can be ESPECIALLY FALSE.  It is ESPECIALLY FALSE to say "insofar as x is even, x + 2 is even" when x is odd, for example when x is 7.
Let's go through it: "Insofar as (if) 7 is even, then 9 is even" is ESPECIALLY FALSE, because the whole statement is baloney, but even more so the whole premise about seven being even is even more hogwash.  Thus it is an even more especially false statement as regards odd x's.  (A point to be careful on.)
Statements "If the present king of france is bald..." are ESPECIALLY false (whatever the final part of the sentence).
Finally, correcting this article enables you to resolve correctly the exam paradox (execution paradox). 
http://en.wikipedia.org/wiki/Unexpected_hanging_paradox
realizing that the statement "if the teacher will give the exam on Friday, then we will know this fact come thursday night" is ESPECIALLY FALSE if the teacher is not giving the exam on Friday enables you to see the correct resolution to the hanging (non)-paradox.
if the teacher does not give the exam on Friday, all the deductions following thereafter are not only false, they are ESPECIALLY false.
It is ESPECIALLY false to say, then, in this case, the teacher must give it by thursday if the students have not had it by thursday, and so on, growing ever more especailly more especially false with each ESPECIALLY false especially bad deduction.
The teacher can give the test on any day except Friday.  on Thursday morning the students would wonder if their statement "If the teacher does not give the exam today, he will give it tomorrow and yet surprise us" is ESPECIALLY false, or merely false.
likewise on Wednesday the students will wonder if their statement "if the teacher does not give the exam today, it is either false or ESPECIALLY false that if he does not give it tomorrow (Thur), he will give it Friday and yet surprise us" is false or ESPECIALLY false
likewise on Tuesday the students will wonder if their statement is false or ESPECIALLY false that, if the teacher does not give the exam today, it is either false or ESPECIALLY false that if he does not give it tomorrow (Wed), it is either false or ESPECIALLY false that if he does not give it the next day (Thur), he will give it Friday and yet surprise us."
And early Monday morning, the first chance to talk before possibly being given the test, the students can furiously debate whether it is merely false or ESPECIALLY FALSE to say that if they do not get the test tomorrow (Tuesday), it is either false or ESPECIALLY false that if he does not give it the next day (Wednesday), it is either false or ESPECIALLY false to say that if he does not give it the next day (Thur), he will give it Friday and yet surprise us."
Naturally whether these statements are false or ESPECIALLY false has no bearing on whether the studetns are being given a test, and, resigned, they must simply await the teacher to come in and tell them whether they are being tested.
(In case you think this is semantics, consider the two-day case.  The teacher will give the test Monday or Tuesday and yet surprise the students.  The student could reason "if the test is not given today, then if tomorrow it is not given the condition that it is a surprise is broken".
This contains an ESPECIALLY FALSE condition either in the first part (the test is not given today) or in the second (if tomorrow it is not given).
This resolution is clearly the correct one, and you should change the whole article (or any other relevant mathematics) to reflect this.  You may publish something in some glossified paper form first if you need to reference it. 188.157.169.36 (talk) 22:57, 20 March 2012 (UTC)
replies: text: So you say that for "if A then B" to be true, A must be true. Usually, when someone (including mathematicians) says that "if P then..." he does not know whether P is actually true, he just says what happens in the case that P is true --84.229.25.134 (talk) 14:08, 21 March 2012 (UTC)
thread_title:  correct resolution 

message: A set is delimited by the empty element & infinite element in that set, those being the boundary conditions onto that set, and therefore no statement made about the elements of a set include the empty set & infinite set. Inside the set, the intersection of the elements with the empty set element & infinite set element pertain to the boundary limitations but never reach those boundary limitations, therefore those are limits towards, and never on.
That mistake in logic is very similar to the assertion that division by zero, is the exception, not the rule, where in fact, multiplication is the exception, not the rule, the rule being that multiplication does NOT include zero nor infinite as elements of the set.
That same error in logic is derived from educational sessions where it is assumed that numbers stand on their own, without the presence of operators. That is not true, a description of a mathematical process contains two element subsets, the set of numbers onto which it is applicable, AND the subset of operators which are applicable. Defacto, the operators themselves, are proper elements of that set, due the union of both into the set.
The reason for not allowing zero in multiplication, is because the inflection point of addition & subtraction is NOT the same as the inflection point for multiplication & division [0 for +-, 1 for */], therefore the intersection at that point does not exist for all elements consisting of (number;operator) pairs as elements, and the intersection at that point becomes the empty set [of which the infinite set is a particular result in inflection].  — Preceding unsigned comment added by 201.209.217.235 (talk) 13:25, 8 February 2013 (UTC)
replies: text: The set {1, 2, ⑀} has the elements 1, 2, ⑀. Which of those is the "empty element" and which the "infinite element"? Are you confusing "sets" with "elements", among other category errors?  — Preceding unsigned comment added by 125.239.154.159 (talk) 00:27, 23 January 2021 (UTC)
thread_title:  empty element & infinite element in a set. 

message: Does anyone know of a term for a statement P->Q where Q is a tautology?
I believe that if such term exists, it should be presented in this article.  — Preceding unsigned comment added by 46.121.232.249 (talk) 12:41, 4 April 2014 (UTC)
replies: text: Well, tautology? P → ⊤ is equivalent to ¬P ∨ ⊤ which is itself a tautology. Unless you refuse to believe in material conditionals. Or you could say that (P → ⊤) ↔ (⊥ → ¬P), making it yet another example of a vacuous truth. Unless you refuse to believe in contrapositives. — Keφr 19:30, 4 April 2014 (UTC)

text: Isn't a vacuous truth also a tautology? I was looking for a term analogous to vacuous truth, in the sense that just as vacuous truth speaks of a implication whose antecedent is a contradiction (or always false), the term I'm looking for speaks of a proposition whose consequent is a tautology (or always true). Both are tautologies.  — Preceding unsigned comment added by 213.57.104.221 (talk) 21:42, 6 April 2014 (UTC)
replies: text:  I doubt a specific term for this exists, because such implications are not very interesting to logicians — if you already know that Q is true, why bother investigating if P implies it? (But if you insist, this trivially follows from the deduction theorem.) Whereas vacuous truths arise naturally in mathematical induction and are somewhat counterintuitive, making them worthy of an own name. — Keφr 09:09, 7 April 2014 (UTC)
thread_title:  A similar term for when the consequent is a tautology? 

message: Removed this:
:Such statements can also be used sarcastically, as in the line "If she's a lady, then I'm a Vermicious Knid!" spoken by the character Grandpa Joe from the film Willy Wonka & the Chocolate Factory.
This is a humorous application of modus tollens that has nothing to do with vacuous truths: the speaker is supposed to infer that, since Grandpa Joe isn't a Vermicious Knid, then "she" must not be a lady. Although the two statements have no logical connection, we're supposed to infer the antecedent must be false, which is distinct from it being obviously false on its own ("if Uluru is in France") and rendering the consequent irrelevant. 31.169.57.1 (talk) 12:30, 9 November 2015 (UTC)
thread_title:  If she's an X, then I'm a Y 

message: The vacuous truth article has become poor quality. At one time, this article was even a featured article (way back in January 2004, admittedly) but the long chain of edits since then have stagnated the article, causing its quality to not keep pace with other Wikipedia articles. There are at least two major problems with the article as it stands.
1. The page is inconsistent in its stance on what vacuous truth is, but doesn't make this explicit (potentially causing confusion). The top section of the article gives two different definitions of what vacuous truth is. The first paragraph says that vacuous truth is "a statement that asserts that all members of the empty set have a certain property". The second paragraph says that vacuous refers to  "a conditional statement with a false antecedent". These are not the same thing, although both are indeed sometimes referred to as "vacuous truth". This should be explained to the reader implicitly, rather than giving two conflicting definitions in a poorly phrased way like the article currently does.
2. Little to no explanation for why vacuous truth exists is given in the article, even though such information is readily available. Previous attempts by editors (dating back years) of explaining why vacuous truth has to apply have been deleted based on false/misguided notions by other editors that it amounts to "essay-like" content or "original research" etc. However, there is no other way to explain the reason why vacuous truth exists except by walking through the truth tables and illustrating that the other possible truth tables one might choose don't work, and doing so amounts to *direct evidence* rather than research. Direct logical proof of extremely simple facts like this should not require citation, especially in the math sections. It isn't original research when it is trivially obvious from the basic truth tables.
This would be like the article on water not giving any explanation of the molecular chemistry of H20, or like a physics article giving no mention of the underlying principles. It is absurd. Explaining why vacuous truth exists is not original research. It is the bare minimum of what should be in the article, but isn't because misguided/rigid-minded editors keep deleting it. Without an explanation for the underlying why behind vacuous truth, the article is massively less valuable to the public, to the point of bordering on being relatively worthless. 
Stop being so actively hostile to clarity and comprehension. There is a reason why the math pages on Wikipedia are regarded as the most incomprehensible subset of Wikipedia by the general public. The community has a culture of deliberate obscurity and systematic hostility to anything that is easy to understand. It's almost as if you don't want the public to understand, just so that you can feel smarter than them by pretending to know more by being deliberately obscure. Huge swathes of the math section of Wikipedia could use being re-written, but the editing culture is currently too hostile for any editors to do so. The community scares away all of the new editors who care about clarity, thus causing the articles to accumulate lots of cruft and superficial obscurantism that adds no value.
The most recent example of this (for this article) was when Wcherowi removed my recent edit (https://en.wikipedia.org/w/index.php?title=Vacuous_truth&oldid=783859142) where I resurrected someone else's text from years earlier (with some minor edits) that explained very clearly why vacuous truth exists, at least for the case of classical truth dealing with simple true-false values. By not including a section that explains the why behind vacuous truth like this, you are massively reducing the value of this article for the users. It's like if the article on water had no content whatsoever on the chemistry of water (as I said earlier). It is absurd. 
Just because someone from years ago labeled the content as "essay-like" doesn't mean you should continue blindly assuming that it is. Including an explanation for the underlying why of vacuous truth is ESSENTIAL to the article being useful, just as H20 chemistry (the underlying why) would be essential to the article on water. The exact wording and title of the explanation can vary, but it DEFINITELY should be in the article in at least some form.
In addition to the need for adding back in some kind of explanation for why vacuous truth exits, this article in general seems like it may even need a complete re-write. Originally, I might have been willing to, but it seems to me that the community is too hostile to clarity for it to be worth me investing any more time to making these kinds of major contributions to Wikipedia articles (for example, I wrote pretty much the entire article on Trapezoidal Distribution recently, which used to be a much less useful stub). No wonder you people have trouble keeping new editors. If this is how the community acts, where even the most basic most fundamentally important aspects of what an article on a piece of subject matter needs to cover is viewed with contempt, then editing here on Wikipedia probably isn't a very good use of time, in most cases. Attempts at clarifying articles to even a basic standard of usability are frequently destroyed. Many of the deleted edits in many articles are actually better than the current versions of them. I see similar problems a lot on StackOverflow, where the moderators often "close as not constructive" some of the most useful threads (in fact, the majority of the most useful pages I've seen on StackOverflow were "closed as not constructive" by mods, as I seem to recall).
Anyway, this time round, if you want the article to improve you'll have to do the work yourself. I know better than to invest in a community that can't even value the most basic and fundamental essentials of what a piece of content should have. Not including the underlying why in many of the other subsets of Wikipedia would never be tolerated in many of the other subsets of Wikipedia, but apparently in math (where people are in love with obscurantism) including the fundamentals of any concept in a way that is easy for the general public to understand is viewed with nothing but hostility. How very counterproductive. As long as this culture continues to dominate on Wikipedia, it will never reach its full potential, and more and more editors will continue abandoning the site (which is what the current trends are if you look at the charts).
Perhaps, however, there is some faint chance that me writing this here will reduce this toxic cultural trend at least somewhat. Maybe at least someone out there is sane. It's ludicrous that something so essential and fundamental to understanding a subject would be deleted. Wikipedia really has become a victim of its own culture and policies.
--MagneticInk (talk) 17:16, 5 June 2017 (UTC)
replies: text: Perhaps you would have had a better understanding of why you have run into these problems had you read the essays WP:HOW and WP:5P. Pointing out the deficiencies of this article has been constructive and there is nothing wrong with the content of the material you tried to add to fix those problems. The issue lies with the fact that only verifiable content is acceptable. Content that has appeared elsewhere and can be directly referenced via reliable secondary sources. Editors are not allowed to espouse their own views on a topic or create new and different ways to look at something. One hallmark that an editor might be doing so is essay-like writing, especially if there is no accompanying documentation. It takes a considerable amount of work by other editors to change that into something with an encyclopedic tone replete with sources–so much so that it is often better to scratch the whole thing and start over again. If you are familiar with the literature you can contribute to this project by referencing this material and indicating what is said there. That is the, albeit limited, role of a Wikipedia editor. Contributions of this ilk are always welcome. --Bill Cherowitzo  (talk) 18:00, 5 June 2017 (UTC)
thread_title:  Article quality has become poor. Has inconsistencies and also needs explanation of why vacuous truth exists. 

message: So why delete my submit and not add a source / adjust it? Aren't the empty set and its properties the most common vacuous truth example in mathematics? If find the example regarding the closed and open property very important for basic topology and maybe more important than the other two ones.   — Preceding unsigned comment added by (Tensorproduct (talk) 23:42, 8 June 2017 (UTC)
replies: text: I did have the option to correct the statement, but chose not to do so since I don't think that the example was a very good one for the level of readership of this article. Being familiar with topological concepts would be a prerequisite to understanding this example and if you have that knowledge you are probably mathematically sophisticated enough to not need to look at an article on vacuous truth. Also, I believe that if you want to include a statement, you have the responsibility of providing a citation ... asking others to do your work for you is a bit gauche. --Bill Cherowitzo  (talk) 03:51, 9 June 2017 (UTC)
thread_title:  Vacuous truth example on empty set 

message: These are called lies and we already have a page Lie. --Bill Cherowitzo  (talk) 18:13, 7 November 2017 (UTC)
thread_title:  Make page: [[false truth]] (= pseudotruth; a false truth, data or information) 

message: A post of mine was correctly deleted from here, and I moved the discussion about it to User talk:Pashute#From Talk:Vacuous truth on my personal talk page.פשוט pashute ♫ (talk) 23:31, 13 December 2021 (UTC)
thread_title:  erased post 

message: In the example given at the top of the page, the sentence "her cellphone is off" implies that "she has a cellphone", or mathematically speaking, "exists (her cellphone) such that is_off(her cellphone)". 
This is clearly false, and so should not be a good example of a vacuous truth 176.12.196.63 (talk) 13:37, 10 November 2022 (UTC)
thread_title:  Implied falshood in example statement 

message: How was this reasoned in mathematical history? I've heard that Greek philosophers got it backwards, as vacuously false. DAVilla (talk) 18:12, 30 November 2023 (UTC)
thread_title:  Historic context 

message: The introduction ends with the following lines:
In addition, a vacuous truth is often used colloquially with absurd statements, either to confidently assert something (e.g. "the dog was red, or I'm a monkey's uncle" to strongly claim that the dog was red), or to express doubt, sarcasm, disbelief, incredulity or indignation (e.g. "yes, and I'm the King of England" to disagree with a previously made statement). 
Are these actually vacuous truths though? Neither of them is a conditional statement as-written. The first one seems to be closer to disjunctive syllogism ((A∨B) ∧ ~B = A), and the second represents a contradiction ((A∧B)∧~B = F). Anerdw (talk) 00:23, 5 April 2024 (UTC)
thread_title:  Accuracy of colloquial examples