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all cell phones in the room are turned off" will be true when no cell phones are in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true, as would the statement which merges them: "all cell phones in the room are turned on and turned off

Why will be true when no cell phones are in the room? I can’t understand the meaning of this.

Asaf Karagila
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Guilherme Woolley
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    Because you cannot prove otherwise. – Andrew Chin Sep 07 '22 at 19:27
  • This is the first example given in Vacuous truth on Wikipedia. Are you asking why such statements may feel misleading outside mathematics? – peterwhy Sep 07 '22 at 19:27
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    A "for all" statement is vacuously true when there is no actual examples of what the "for all" statement refers to. In your case, there are no cell phones in the room, so any statement about "all cell phones in the room" will be true, but vacuously. Similarly for any statement about "all odd numbers divisible by $12$". It's to do with how we define truth of "for all" statements: for a "for all" statement to be false, there must be some counterexample, i.e. specific instance of something that does not have the desired property. If there are no examples, then there are no counterexamples. – Theo Bendit Sep 07 '22 at 19:29
  • Its vacuously truth because the antecedênce like: there are no cell phones in the room,can’t be satisfied by :all the cell phone in room are turned off ,because there’s no cell phones in the room? – Guilherme Woolley Sep 07 '22 at 20:05
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    @GuilhermeWoolley Pretty much: formally, we write it with a bounded quantifier as in $\forall(x\in\textrm{cell phones in room})(x\textrm{ is off})$. But at the "down-to-the-metal" level, we can only quantify variable symbols, so $\forall(x\in y)\phi(x)$ is introduced as an abbreviation for $\forall x(x\in y\rightarrow\phi(x))$. So really what we wrote is shorthand for $\forall x(x\textrm{ is a cell phone in the room}\rightarrow x\textrm{ is off})$, and the antecedent is never true, so this is true for all $x$. – C7X Sep 07 '22 at 20:34
  • What’s “x is off”? – Guilherme Woolley Sep 07 '22 at 21:04
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    @GuilhermeWoolley If you want someone to respond to your comments, you need to tag them (like I did to you), otherwise they won't know you're asking them a question. In this case, "$x$ is off" is a predicate: a true-false statement depending on an unknown $x$. For example, if $x$ is my TV, then "$x$ is off" would be true. But if $x$ is my laptop, then "$x$ is off" would be false. The statement "$x$ is a cell phone in the room $\rightarrow$ $x$ is off" means, "if this thing $x$ happens to be a cell phone in the room, then it is switched off. – Theo Bendit Sep 07 '22 at 21:28
  • Note that in real life such statements would be considered different as in mathematics. If I mention "all planes I have are pipers" , almost everyone would assume that I have planes although "technically" I do not lie if I say this and have no planes at all. – Peter Sep 08 '22 at 09:56

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