# Why vacuously true and not vacuously false?

Let a room is empty. Consider a statement.
Modified [4:04 PM, 26 March 20] Every mobile phone in this room is working.
: This is called vacuously true because there is no mobile phone in the room.
Let I say that this statement is vacuously false, If you think it is not, show a mobile phone (in this room) which is working. You cannot do this.
When we can choose both the options, why have we chosen Vacuously True and not Vacuously False?
Is it a convention?

Well, the statement you just gave is not even false or true, it doesn't refer to anything.

The statement which is vacuously true is "every mobile phone in this room is working". For that to be false, you would have to show that there exists a mobile phone in the room which is not working. No mobile phone in the room exists, so in particular, there exists no mobile phone which is not working. Therefore, the statement is true.

Now, the statement you seem to be attacking is "there exists a mobile phone in the room which is working". This statement is false. There is, again, no mobile phone around, so in particular none that is working.

So you could say it's a convention, but it's completely natural in the sense that it's the only convention consistent with the general negation rule $$\neg(\forall x:p)=\exists x:\neg p$$. In other words, it's as much of a convention as the convention that $$a^{-n}=(a^n)^{-1}$$.

• Thanks. I have modified the question. Commented Mar 26, 2020 at 10:51
• You're not really reading what I'm writing if you think that this doesn't contain andanswer for your new phrasing. It is true that every mobile phone in the room is working, at least if you subscribe to the logical inference rule that $p$ is true if and only if $\neg p$ is false. You seem to be conflating "every mobile phone" with the statement of the existence of some mobile phone, but that's not what "every mobile phone" means. It means, "if you were to give me a mobile phone from the room, then it's my job to show you that it's working". Commented Mar 26, 2020 at 10:54
• However, you cannot give me a mobile phone from the room, so I don't need to do anything, so my statement is true. Commented Mar 26, 2020 at 10:55
• It's the same as the fact that the statement "If you become president, I will pay you a million dollars" can never be false if you never become president. It can only become false if you do become president and I refuse to pay. Commented Mar 26, 2020 at 10:56
• Yes, but that's just a question of semantics. For instance the truth of $p\vee \neg p$ is a convention. Every rule of logical inference is a convention. But not all conventions are born equal. Commented Mar 26, 2020 at 13:33

Short answer : you are right when you say that " No mobile phone is working " is true. But you're wrong when you claim it implies that " all mobile phones are working" is false , assuming, of course that the set of mobile phones under consideration is empty.

• Your reasoning is as follows :

(1) If the negation of a sentence is true, then this sentence is false

(2) " No mobile phone is working " is the negation of " All mobile phones are working"

(3) But " No mobile phone is working" is true ( since no counterexample can be pointed out).

(4) Therefore, " All mobile phones are working" is false.

• However, proposition (2) is not correct. So, the conclusion does not hold( although the other premises are correct).

• The sentence " no mobile phone is working" is not the contradictory ( i.e. the pure negation) of " All mobile phones are working" but the contrary statement.

• The actual negation ( i.e. the contradictory sentence) of " All mobile phones are working" is " There is some mobile that is not working" . For the pure negation of " all" is simply " not-all...". In symbols :

$$\exists (x) [ M(x) \land \neg W(x)]$$.

Note : in traditional logic, contraries cannot be true at the same time; but if set M is empty, then " all M are W" and " all M are not-W" are both true, vacuously.

https://plato.stanford.edu/entries/square/

Below a diagram showing that any " convention" would have the same effect on both sentences : it's not up to us to decide that one is true and that the other is false.

• This answer would be even better if you added what the negation for proposition 2 should be instead; you only say what it’s not. Commented Mar 26, 2020 at 21:11
• Post edited in order to take your comment into account. Thanks for the suggestion.
– user655689
Commented Mar 29, 2020 at 12:50

Premising that there are no mobile phones in the room.

"Every mobile phone in the room is working," is vacuously true.

"Every mobile phone in the room is not working," is vacuously true.

To prove either statement false requires finding a mobile phone in the room to contradict that they all have the claimed status, but there are none to be found.

Likewise claims of existence of phones will be fallacious when there are no phones in the room.

"There is a mobile phone in the room that is working," is vacuously false.

"There is a mobile phone in the room that is not working," is vacuously false.

• I am getting an impression that being vacuously true or vacuously false is related to the structure of the sentence. "Every mobile phone in the room is working," is vacuously true. "There is a mobile phone in the room that is working," is vacuously false. Am I correct slightly? Commented Mar 26, 2020 at 13:24
• If it is related to “For all $x∈A$” it will be vacuously true. If it is related to “There exists some $x∈A$” it will be vacuously false. Am I correct slightly? Commented Mar 26, 2020 at 13:30
• It is related to "For all $x\in\emptyset$" and "There is some $x\in\emptyset$" indeed. Because there is nothing in the empty set, all things in the empty set do satisfy any predicate -- exactly because there are none that do not satisfy that predicate. Because there is nothing in the empty set, all things in the empty set do not satisfy any predicate -- exactly because there are none that do satisfy that predicate. This is the nature of vacuity. Commented Mar 26, 2020 at 22:47
• Another consideration. The empty set is the subset of all sets, including sets which are complements. So all elements of the empty set are in any set, despite that there is not an element in the empty set that is in any set. Commented Mar 26, 2020 at 22:56
• One more question. You have termed these two statements as vacuously false:- "There is a mobile phone in the room that is working," is vacuously false. "There is a mobile phone in the room that is not working," is vacuously false. Now, after all these exposures, my opinion is that these two statements are just straight forward false and not vacuously false. Am I correct? Commented Mar 27, 2020 at 4:25

It's down to the semantics that we assign to the word "every" in mathematics.

It would be perfectly reasonable to define the phrase "for all" to mean "there are no counterexamples and at least one example". Indeed, in plain English this is more or less how we do use that phrase, hence your confusion. But in math we choose to define the phrase "for all $$x$$, $$Px$$" to mean only the first half of that sentence: there are no counterexamples to $$Px$$.

The reason we do this is because then we get a nice symmetry with another phrase, "there exists", expressed by these formulae:

$$\neg\forall x Px\iff\exists x\neg Px$$ $$\neg\exists x Px\iff\forall x\neg Px$$

In other words, logical negation is a kind of isomorphism between the two operators $$\forall$$ and $$\exists$$.

Whatever we choose to call them, these two operators with their symmetrical relationship exist, and are fundamental in expressing the rules of logic. Since their meanings almost correspond to the English phrases "for all" and "there exists", we draw inspiration from those phrases in naming these operators. But even if you don't like those names, the operators themselves are in some sense the natural ones to use in logic, so whatever we call them they should be the ones we use in mathematics. In some sense they are "more elemental" than their English equivalents. This elementalness is seen in the beautifully simple symmetrical laws I already mentioned, and in the fact that the colloquial English "for every" can be expressed in terms of them as $$\exists xPx\wedge\forall x Px$$

Let $$U$$ be the universal set (room) containing elements (objects) denoted by $$x$$. The hypothesis says that $$x$$ is anything other than the mobile.

Note that a conditional statement is $$p\implies q$$ which is equivalent to $$\lnot p\lor q$$. Your statement is indeed a conditional statement which can be formulated as

"For every $$x\in U$$, if $$x$$ is a mobile then $$x$$ is working" which can be written using symbols as "$$\forall x\in U, p\implies q$$ where

$$p: x$$ is mobile

$$q: x$$ is working

So the conditional is equivalent to saying "$$\forall x\in U, \lnot p\lor q$$ ". Rewriting this in language, we get

"For every $$x$$, either $$x\in U$$ is not a mobile or $$x$$ is working"

Now the last statement stands true by the hypothesis that "$$x$$ is anything other than a mobile". Hope it helps!

• Do you mean that being vacuously true is a logical conclusion and not a convention? Commented Mar 26, 2020 at 11:52
• Yes its more than just a convention if verified using logic. Commented Mar 26, 2020 at 13:19

Why vacuously true and not vacuously false?

Suppose that proposition $$A$$ is false.

Then $$A\implies B$$ is true for any proposition $$B$$ regardless of its truth value. In this case, we say that $$A \implies B$$ is vacuously true.

On the other hand, $$A \land B$$ is then false for any proposition $$B$$ regardless of its truth value. In this case, we might reasonably say that $$A\land B$$ is "vacuously false" (my proposed terminology).

Note that if $$P\implies Q$$ is vacuously true, then $$\neg(P\implies Q)$$ would then vacuously false in the above sense of the term.