Today I encountered a vacuously true case. But I am not so sure, so please help examine it.

It was from a question form that asks:

Have you taken part in your department's open discussion session already?

And below the question, there were a "yes" and "no" boxes for me to check.

I think both the "yes" and "no" are vacuously true in my case, and why I think like that:

First of all, there aren't any open discussion sessions from my department.

According to Wikipedia's explanation about the subject,


The set of my department's open discussion session is empty, so I conclude that both the "yes" and "no" are vacuously true in my case, and that I could've in fact checked both boxes.

I am not an experienced logic or philosophy learner, so I am not sure if my understanding and deduction is correct.

Would you guys help me out?

The following part was posted on 09/04 2020 to further explain my question

Here is a more detailed process of my understanding of this case.

Step 1

The "yes" means "I have taken part in my department's open discussion session already."

Step 2

The "no" means "I have not taken part in my department's open discussion session already."

Step 3

Given that there are no open discussion sessions from my department, both the "yes" and "no" are true because the set of "my department's open discussion" has no representatives.

which is also mentioned on the Wikipedia page.

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A case mentioned in Wikipedia, it says "All the cell phones in the room are turned off" and "All the cell phones in the room are turned on" are both true when there are no cell phones in the room.

Another case mentioned in Wikipedia tells us that "I ate every vegetable on my plate" is true, when there were no vegetables on the plate to begin with.

The Wikipedia page: https://en.wikipedia.org/wiki/Vacuous_truth

So, I wonder whether both "yes" and "no" are true. But I am not confident in my understanding and deduction.

Hope you guys would help me examine the case.

  • 1
    As is, the question is ambiguous and that is why it seems that both truth values obtain, but, in fact, ambiguous questions are ill-posed and have no answer. In practice, ambiguities are resolved by context. Translated into the predicate calculus to eliminate the ambiguity the sentence looks like this: ∃x(D(x) Λ P(I,x)) (there is something which is today's discussion session and I participated in it). Since there was no discussion session the sentence is false and the answer to the (disambiguated) question is no. It is not yes, and it is not vacuous, there is no implication involved.
    – Conifold
    Sep 3 '20 at 19:30

It's not really a case of vacuous truth; it's a matter of the pragmatics of language. It is similar to the classic, "Have you stopped beating your wife?" The question makes an incorrect assumption, so in the event that you never started and hence never stopped beating your wife, to answer 'no' is true but potentially misleading.

If your department has had no open discussion sessions then a fortiori you have not participated in any, so the correct answer is 'no'. But it is understandable that you are reluctant to check 'no' since this gives the misleading impression that such sessions occurred and you declined to take part. If the question had been worded, "Have you taken part in every open discussion session that your department has held?" then a 'yes' answer might qualify as vacuously true.

  • "answer 'no' is true but potentially misleading" How could a true answer be at all misleading? Sep 3 '20 at 19:17
  • 2
    @Speakpigeon Because language has more that fifty shades of grey and the truth value does not at all determine the meaning.
    – Philip Klöcking
    Sep 3 '20 at 20:02
  • 2
    True statements can easily mislead and often do. A statement can be a partial truth, or it can obscure a truth by providing too much information, or it can play on a known false belief of the audience, or it can invite an incorrect inference. The theory of conversational implicature is concerned with explaining how pragmatic features of language allow true statements to avoid being misleading or inappropriate.
    – Bumble
    Sep 3 '20 at 20:34
  • 2
    Universals can be trivially true because in modern logic they are understood hypothetically. All P's are Q's is taken to mean that for any x, if x is a P then x is a Q. Hence if there are no P's this comes out true. But a claim about a specific P would come out false. So, if there are no mobile phones, "all mobile phones are switched on" is true. But, "I made a call on my mobile phone" is false. The latter is an existential claim: it means there exists a thing that is my mobile phone and I made a call on it, and this is false.
    – Bumble
    Sep 4 '20 at 21:37
  • 1
    I would go along with that. Russell agrees too, so you are in good company. Russell's examples are "the present king of France is bald" and "the present king of France is not bald". Both are false according to Russell because they should be understood to mean there exists one and only one thing that is the present king of France and that thing is bald (or not bald).
    – Bumble
    Sep 5 '20 at 16:59

On the notion of "vacuously true", it is a novelty introduced in mathematical logic. In logical terms, its operational value is zero. That is to say, the truth value of a "vacuously true" proposition is "true", not "vacuously true". In "classical" mathematical logic, there are only two truth values, and there is no room to accommodate "vacuous truth" as a third truth value. Further, the logical calculus does not depend on whether a proposition is "vacuously" true. The result of a logical operation is exactly the same whether a proposition is true or "vacuously true". In other words, "vacuously true" is a dummy. A vacuous truth value.

So, why bother with the dummy?

Very simple. In "classical" mathematical logic, a logical implication φ → ψ is true if the antecedent φ is false. This notion is of course abhorrent, and it is abhorrent because it is obviously false. Mathematicians are most of them adamant that this is nonetheless correct, but there is constant denigration and calling "vacuously true" any implication with a false antecedent at least goes some way towards alleviating the pressure. It provides psychological relief.

This is not the only example of mathematical logic toying with the lexicon of logic: not only "vacuous" truth, but "validity", "tautology", "material" implication... and more.

Yet, in this particular case, I guess the official answer is not "vacuous truth", but straightforward falsity. The justification is that the question implicitly assumes the existence of something that doesn't in fact exist, like saying "God loves you", or, to use Bertrand Russell's quaint example, "The king of France is bald": The king of France can neither be bald or not be bald because he doesn't exist.

Of course, it is not apparent that it is a proper logical explanation, but this is all that mathematicians can say in this case, and if not, then I'll be happy to be corrected.

  • 2
    If you weren't so focused on your crusade against the material conditionals you'd notice that this particular case does not even involve a conditional. The Russell's paraphrase of "The current king of France is bald" is "There is someone who is the current king of France and he is bald".
    – Conifold
    Sep 3 '20 at 19:41
  • @Conifold 1. It is apparent that I "notice that this particular case does not even involve a conditional".My answer has two parts. The first part is about "vacuous truth", which is the main point, and vacuous truth only concerns the so-called "material" implication. The second part concerns false assumptions. 2. You are so myopic that you don't even realise that your own interpretation falsifies your explanation. Sep 3 '20 at 19:49
  • In other words, the first part of your post should be deleted, and replaced with "this is not a case of vacuous truth because implication is not involved". That would make it much clearer.
    – Conifold
    Sep 3 '20 at 19:53
  • 2
    It does not ask about it, it asserts (erroneously) that it is the case here. And if you are going on excursion about it then, at least, do not make things up to match your pet preconceptions. "Vacuous" as in empty of substance is a colloquial expression, not a novelty introduced by mathematical logic, and is used there in informal commentary in that very sense. And I doubt that you can point to anything backing up your "abhorrent" or "psychological relief" other than your own armchair musings.
    – Conifold
    Sep 3 '20 at 20:28
  • 1
    It is not introduced as a truth value, an implication is said to be vacuously true when its premise is false. As such, it is a perfectly well-defined shorthand. The reason for using it is to emphasize the difference between the material and colloquial conditionals, which is perfectly legitimate when teaching those used to the latter.
    – Conifold
    Sep 6 '20 at 18:31

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