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The wiki article on vacuous truth says:

a vacuous truth is a conditional or universal statement that is only true because the antecedent cannot be satisfied.

I'm familiar with identifying the antecedent and consequent in a conditional, but I'm unsure what part of a universal proposition would be the antecedent.

The article gives the following example:

For example, the statement "all cell phones in the room are turned off" will be true when there are no cell phones in the room.

Would the subject class "cell phones in the room" be the antecedent since it being empty results in the statement being true? In general, is the subject class the antecedent in a universal?

Slecker
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    Yes, your guess is correct. – Natalie Clarius Jun 25 '21 at 20:16
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    To translate "all cell phones in the room are turned off" into first order logic, you'd have to turn it into something like "for all x, cellphoneintheroom(x) -> turnedoff(x)", so the antecedent would be cellphoneintheroom(x) and this would be vacuously true if there is no x in your domain of discourse that satisfies cellphoneintheroom(x). – Hypnosifl Jun 25 '21 at 20:47
  • How can a ceĺl phone be turned off if there are none to be turned off? The antecedent can be there but if it has nothing to cause a change on it becomes vacuous. – Deschele Schilder Jun 26 '21 at 00:11
  • So its not the subject class thats the antecedent but the cause. The turning of is the subject class and the turned off phones the belong to the consequense class. The phones themselves belong to the conditional class (a tv can be turned off also). So the subject class is "turning off", the conditional class is "phones", and the consequence class is that of " turned off phones". So without phones in the room the conclusion that all phones in the room are turned off cant be made. – Deschele Schilder Jun 26 '21 at 00:28
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    The antecedent is the noun or noun clause BEFORE the main verb in the alleged sentence. In this case Cell Phones would be the noun clause BEFORE the main VERB which is ARE. You can make a conditional statement from the All statement: if there are any cell phones in the room, then they must be turned off. Now your objection to the case where there are no cell phones in the room never arises.I never stated there were cell phones in the room.The statement only applies if there were a cell phone in the room.The instuctions in the scenario do not apply if there are no cell phones. No falsification. – Logikal Jun 26 '21 at 15:18
  • @DescheleSchilder I'm unfamiliar with the way you label the subject class and "consequence" class. Being new to logic I'm only familiar with the labeling conventions for standard categorical propositions. My understanding is the subject class is the noun that comes before the verb 'are' (the copula), and the predicate class is everything that comes after the 'are'. So in the example, the subject class would be 'all cell phones in the room' and the predicate class would be 'turned off.' – Slecker Jun 26 '21 at 17:53
  • @DescheleSchilder Also, I'm confused about the "conditional" class. Are you trying to render the All statement a conditional one? If so I think Logikal's comment achieves that. – Slecker Jun 26 '21 at 17:54
  • @Logikal *"if there are any cell phones in the room, then they must be turned off"* But "must be" is different in meaning from "are". Also, OP should be clear that in first-order logic, the fact that vacuous truths are true has nothing to do with things like how close they are to plausible normative statements like the one you mentioned, or to plausible modal claims like "if there are any cell phones in the room, they are smaller than a breadbasket". Vacuous truths which would sound absurd in ordinary language are also true, like "if there are cellphones in the room, they weigh over 100 tons". – Hypnosifl Jun 26 '21 at 21:17

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The guess is not correct. It's the turning off that is the subject class. The phones are the conditional class (you can turn off many things but we restrict to phones, ie, the turning off is conditioned). The turned off phones belong to the consequential class. If there are no phones in the room then they can't be turned off so the conclusion that if there are no phones in the room then they are tutned off is false.

Deschele Schilder
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    What do you mean by "subject class" and "conditional class", have you seen these terms used in any branch of formal logic? Normally "antecedent" and "consequent" in discussions of logic refer specifically to material implication, i.e. for a statement of the form P -> Q, P is the antecedent and Q is the consequent. So in this context, if you wanted to say turning off cellphones is the antecedent, you need to show how you could do a reasonably faithful translation of the statement into a more formal version w/ the material conditional, like my "for all x, cellphoneintheroom(x) -> turnedoff(x)". – Hypnosifl Jun 26 '21 at 23:36
  • @Hypnosifl In the quetion is asked for the subject class. The subject is the turning of itself. Turning of can be applied to many things. But we only apply it to phones. This is the condition n the turning off. So the antecedent class is automatically the phones. This means that the antecedent class (being the same as the condition class) has no need to be mentioned. It's already implied by the conditional class. The consequential class is the class containing the phones that are turned on and is a subclass of the antecedent class, depending on the subject class. – Deschele Schilder Jun 27 '21 at 01:12
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    The question uses the words "subject class" to describe the linguistic category "cell phones in the room", it's not *asking* if that's the subject class. It's asking if in logical terms "cell phones in the room" would be the antecedent, and whether more generally the subject class in any similar sentence would be the antecedent. – Hypnosifl Jun 27 '21 at 01:22
  • @Hypnosifl Thats exactly what I said in my last comment. The antecedent class contains the phones (in the room) and the consequence class is a subclass of this (containing the phones that are turned on). The more general question I havent answered because I didnt see that. – Deschele Schilder Jun 27 '21 at 02:10
  • @Hypnosifl If the antecedent class is empty (no phones in the room) then the consequnce class is empyu too. So no phones in the room cant imply that they are turned off because the class of turned off phones is empty. Just as the antecedent class. – Deschele Schilder Jun 27 '21 at 02:18
  • @Hypnosifl Of course you can make the turning on the conditional class but then you consider different categories. – Deschele Schilder Jun 27 '21 at 02:21
  • "So no phones in the room cant imply that they are turned off because the class of turned off phones is empty" But it does imply that in first-order logic, assuming you translate the sentence as "for all x, cellphoneintheroom(x) -> turnedoff(x)". For any given choice of x, the material conditional "cellphoneintheroom(x) -> turnedoff(x)" is *true* if "cellphoneintheroom(x)" is *false*, just based on the material conditional's truth table. So, if "cellphoneintheroom(x)" is false for every choice of x in our domain of discourse, then "cellphoneintheroom(x) -> turnedoff(x)" is true for every x. – Hypnosifl Jun 27 '21 at 02:26
  • @Hypnosifl How can it be logically speaking, that if there are no phones in the room that they are turned off? – Deschele Schilder Jun 27 '21 at 02:30
  • Because to translate the statement into first-order logic you have to use the material conditional. Keep in mind it's well known that the material conditional doesn't always match up with the meaning of if-then statements in ordinary language, see the discussion of the difficulties formalizing ordinary-language [indicative conditionals](https://plato.stanford.edu/entries/conditionals/) on SEP. The material conditional is defined *only* by its truth table, you have to discard any other notions you may have about if-then statements in ordinary language. – Hypnosifl Jun 27 '21 at 02:34
  • @Hypnosifl It might be a vacuous truth precisely because you doñt know if they sre turned on or not. I dont think it maked sense at all about being turned on or not if there are no phones at all. A senselesd truth so to speak. The class maybe empty or vacuous but then a class is senselesd. Only when there is somethimg in a class it makes sense. The empty class can contain anything. – Deschele Schilder Jun 27 '21 at 02:42
  • @Hypnosifl The material condition being turned off? But still. When no phones are there (the antecedent class being empty) this means there are no turned off phones either. Why does no phones being there imply that all are turned off? To what refers the all? To nothing? – Deschele Schilder Jun 27 '21 at 02:48
  • In first-logic you have a bunch of individual named objects in your domain of discourse, let's call them x1, x2, x3, etc. If our domain of discourse is the room, x1 might be a particular person, x2 might be a particular chair, x3 might be a particular desk, etc. If some particular named object, let's say x7, is *not* a cellphone, would you agree the atomic proposition "cellphoneintheroom(x7)" is false, i.e. that predicate doesn't correctly describe the object x7? If so, would you also agree the truth table of the material conditional says "P -> Q" is true for *any* false proposition P? – Hypnosifl Jun 27 '21 at 02:52
  • @Hypnosifl What does your proposition mean? – Deschele Schilder Jun 27 '21 at 03:14
  • Do you mean the atomic proposition cellphoneintheroom(x7)? In formal logic a statement of the form P(q) is an assertion that some [predicate](https://en.wikipedia.org/wiki/Predicate_(mathematical_logic)) P applies to the object q, so something like "red(q)" would translate in english to "q is red". Similarly, "cellphoneintheroom(x7)" is meant to be a formal way of saying "x7 is a cellphone in the room". So, if x7 is not in fact a cellphone in the room--if x7 is the label we use for a particular desk, for example--then the proposition "cellphoneintheroom(x7)" is false. – Hypnosifl Jun 27 '21 at 03:24
  • @Hypnosifl Ah I see what you mran. In that case if the propsition is false (x7 is no phone) than p-->Q is true if P is false. Minus times minus is plus so to speak. – Deschele Schilder Jun 27 '21 at 03:34
  • @Hypnosifl This problem is driving me crazy ( now already). How can it be that if there are no phones in the room, all phones are turned off is true? Because there are no phones in the room (my wife is going crazy too...because of me...)? – Deschele Schilder Jun 27 '21 at 03:50
  • Well, like I said I think it's just because the logicians have found it convenient to define the material conditional purely in terms of a [particular truth table](https://www.webpages.uidaho.edu/~morourke/202-phil/11-Fall/Handouts/Philosophical/Material-Conditional.htm), it wasn't intended to capture all the meanings of "if-then" statements in ordinary language, which philosophers label the [indicative conditional](https://plato.stanford.edu/entries/conditionals/). – Hypnosifl Jun 27 '21 at 15:27
  • (cont.) As for why logicians chose to define it using that particular truth table and not some other truth table, it seems like the natural choice if you want to combine the universal quantifier and a [binary operator](https://en.wikipedia.org/wiki/Truth_table#Binary_operations) in order to translate "All A are B" type statements of traditional logic. In the more typical case where there *are* some things with property A, the logical statement "for all x, A(x) -> B(x)" then is equivalent in meaning to "All A are B", see my comment [here](https://philosophy.stackexchange.com/a/69416/10780). – Hypnosifl Jun 27 '21 at 15:34