Veeky Forums is the idea of a "vacuous" truth just bogus? Would it not be more sensible to just call it "nonsensical statement"?
For example, how can "all the phones in this room are turned off" be considered true if the opposite statement is also true (if there are no phones in the room). I mean yes I understand they call it "vacuously true", and vacuous means meaningless/empty, but how is it even true technically?
I see how it is useful to prove that the empty set is a subset of all sets, but isn't that rather arbitrary?
You kind of answered your own question. There are a lot of statements that are trivially true logically. Those statements are vacuous because they don't tell us more than we already knew. We could just as well say all phones in the room are turned on. Then logically, all phones are both on and off simutaneously. Again, it is pointless to observe. But how do we know if it is trivial or not? Without the empty set as a subset of all subsets, the answer becomes ambiguous in more complicated examples.
Jack Foster
I don't think I follow.
If both the statements: all phones are turned off; and all phones are turned on
can be considered true, then yes I agree we should call them vacuous truths. I disagree on whether those statements can even be "trivially considered" to be true. Because of their nonsensical nature.
The two statements contradict each other, and so should be considered as vacuous statements, not vacuous truths because neither is true in any sense.
It's use in proving that the empty set is a subset of all sets could seems rather arbitrary, doesn't it?
The way I understand it, because a vacuous truth allows an empty statement to be true in any situation, even "A and Not A", the empty set can be a subset of every set.
However this comes after the vacuous truth is defined to be true. And I disagree with that notion, and by extension that invalidates that stuff about the empty set. It all seems very contrived to make something true while not making sense realistically.
Carter Brown
>Because of their nonsensical nature. What do you mean by "nonsensical"? They clearly are not meaningless, since we are capable of interacting with them as language-users. For example, if somebody made your statement, we might say "What are you talking about? There are no phones in this room."
What do you make of "The present king of France is bald"? Is it true? False? Neither? Both?
Jace Jenkins
Yes, that is what I mean by nonsensical. We can interact with the nonsensical, but they make no sense i.e "The present king of France is bald". It is neither true nor false, it is nonsensical.
Zachary Gutierrez
What do you mean it's "nonsensical"? You clearly understand what it's supposed to refer to and what it's supposed to say about its referent. You and I both know what that sentence means. It's not like "Julius Caesar lightly dressing harsh and" which is a meaningless combination of words, and it's not like "Gbbrh fchfk iekkdkd" which is gibberish.
Xavier Ward
I see your grievance. By nonsensical I mean the statement is absurd, as in the conclusion cannot be stated because the condition of "there are phones in the room" has not been met.
For example, "There are no phones in this room and all the phones in this room are turned off." The first statement does not allow the second statement to make any sense, as there are no phones to be in the off position. In this sense the statement is nonsensical, or absurd.
It cannot be true because the condition on which it is predicated does not exist, i.e there are no phones to be in the off position, so it is a default absurd statement.
Evan Stewart
>how can "all the phones in this room are turned off" be considered true if the opposite statement is also true The opposite statement is not true. The opposite statement of "all the phones in this room are turned off" is "not all the phones in this room are turned off", i.e. "there is at least one phone in this room that is turned on". Which is false, in an empty room.
>all phones are turned off; and all phones are turned on >The two statements contradict each other Why?
>but how is it even true technically? It is true because "all the phones in this room are turn off" means "there is NOT a phone in this room that is turned on". Which is clearly true.
Wyatt Johnson
Yes but it is also clearly true that there are NOT any phones that are turned off because there are no phones. There are no phones for them to be in any position.
Despite this, it is a vacuous truth to say "all the phones are turned off" OR "all the phones are turned on". That is what they teach in my friends first year logic class, which I have a contention with. They say both those statements are vacuously true, i say neither are because there are no phones and the statements just make no sense.
Henry Edwards
>Yes but it is also clearly true that there are NOT any phones that are turned off because there are no phones. There are no phones for them to be in any position. Yes. And?
>Despite this, it is a vacuous truth to say "all the phones are turned off" OR "all the phones are turned on". Indeed.
>i say neither are because there are no phones and the statements just make no sense. They do make sense. "all the phones are turned off" means "there are no phones that are turned on". Similarly, "all the phones are turned on" means "there are no phones that are turned off". When there are no phones at all, then clearly "there are no phones that are turned off" and "there are no phones that are turned on" are both true, aren't they?
Jonathan Roberts
Forgot to reply to your "why?". I am not sure if you are seriously asking that. Both "A" and "Not A" cannot be true at the same time, as I can't be both 2 years old and 3 years old at the same time. That's just simple logic.
Alexander Clark
No, the statements do not mean that. They mean that all the phones occupy a position, which they do not. They assert a positive claim, not a negative one.
Isaiah Morales
But the two statements are not the negations of each other. The negation of "forall x, P(x)" is "not forall x, P(x)". The statement "forall x, not P(x)" is very different from "not forall x, P(x)".
>They mean that all the phones occupy a position, which they do not. No? Then please point me to one that doesn't. If you can't point me to an x such that not P(x), then "forall x, P(x)" holds.
Cooper Cook
Are you trolling or something? They don't need to be the negation of each other. They just need to be two statements that can't co-exist. I.e I am both 12, and 13.
Noah Moore
>They don't need to be the negation of each other. They just need to be two statements that can't co-exist. Yes, but if they are not the negation of each other, you will need to provide some other argument as to why the two statements cannot coexist. Hence "why?" in .
(There is no such reason, for the two statements CAN coexist. But feel free to prove be wrong here, if you are not convinced.)
Carter Lopez
Then your contention is with the number 0 and the empty set. The argument is whether or not 0 phones should be considered valid. Modern logic says it is. Like I said before, when you accept the existence of the empty set, you necessarily encounter this issue. But having the empty set is more convenient than not as long as you keep in mind that these situations can occur.
Why is 0 phones as an object any less sensical than 1 phone or 10 phones or 10^10^10 phones? We choose to represent each as an object. Then it follows that the multiple number of phones doesn't matter in logic. Only how the logic treats the object.
Wyatt Bennett
Because one literally makes the other one false I see what you mean, but I would object on the grounds that "0 phones" does not posses any property that could be used to identify a phone. I'm speaking realistically here.
I understand its convenient for math, but is it not arbitrary? And if it is so, shouldn't the term vacuous truth be done away with in a t least logic/philosophy, and replaced with "absurd statement"?
Levi Miller
For example, "If for A to be true it is necessary that B cannot be true (the phones are on, and the phones are off), then it is true that only one of those states can be in the true position at any given time, thus they are mutually exclusive states. Thus they cannot co-exist.
Landon Baker
>Because one literally makes the other one false Just repeating that doesn't make it true.
>I would object on the grounds that "0 phones" does not posses any property that could be used to identify a phone. For the purposes of this quantification, we are not *identifying* phones at all. Just discussing properties phones may or may not have.
>I understand its convenient for math, but is it not arbitrary? Not any more so than other math.
Liam Perry
>then it is true that only one of those states can be in the true position at any given time, thus they are mutually exclusive states. Thus they cannot co-exist. Indeed -- being on and being off are mutually exclusive states *for a given phone*. That does not say anything about collections of phones, though.
Justin Long
>Just repeating that doesn't make it true. You have not offered any counter argument, nor any useful analysis. You have not addressed how "all the phones are on" and "all the phones are off" can be true at the same time. You're being a pseud.
>For the purposes of this quantification, we are not *identifying* phones at all. Just discussing properties phones may or may not have.
I am objecting on the basis of its relevance in philosophy. To treat "0 phones" as being something that true and false statements can be made about in the same way 1 or more phones can be used to do is useless for empirical purposes.
Extend that exclusivity to the rest of the phones by making the statement for all phones.
"Every single phone is on. Every single phone is off". The statement can only be made if phones exist, and I posit that for phones to exist there needs to be 1 or more. And the statements can only be true exclusively.
Noah Brown
Statement >All phones in this room are turned off "Converse" >Everything in this room which is off is also a phone "Inverse" >All phones in this room are on "Contrapositive" >Nothing in this room which is on is also a phone
Now I'm sure you're irked that in the particular case where there are no phones in the room, what you will intuitively interpret as the statement, its converse, inverse and contrapositive, don't have their usual truth values. Namely, the statement and its inverse have the same "truth". But perhaps the statement can be rephrased, and with more precise language we can get to the bottom of this.
Statement >If there is a phone in this room, it is certainly off. Converse >If something is certainly off, it is a phone in this room. Inverse >If there is a phone in this room, its power state is unknown. Contrapositive >If something's power state is unknown, it is not a phone in this room.
Everything is now working as intended. The issue was found: the opposite of "off" isn't "on". It's "might be off". And "might be off" [math]\implies[/math] "might be on" [math]\implies[/math] "power state is unknown".
Daniel Hall
The reason I make that statement is because no properties of "0 phones" can be measured to be verifiable true, so true and false statements about something that doesn't exist are absurd.
>useless for empirical purposes Hence why it is *vacuously* true. That's the whole point.
Brayden Lee
>The statement can only be made if phones exist Why? I can make the statement quite happily without them.
James Barnes
The opposite of "all phones in this room are turned off" isn't "all phones in this room are turned on", which is also true, but "there is a phone in this room that isn't turned off", which isn't a true statement, so there's no contradiction here.
Isaiah Taylor
I love the fact that Veeky Forums takes this logic shit seriously, because it proves definitively that this is not just a "science & math" board but also a philosophy board, as evidenced by how much discussion of philosophy of language is going on ITT
Xavier Johnson
I am not talking about just the opposite case but any two mutually exclusive cases which can both be true in a vacuous sense despite being exclusive. Those are not "truths" vacuous or not but absurd statements. Everyone talking about opposites etc is misunderstanding my contention.
Jeremiah Bailey
Negations aren't the same as opposites necessarily. The statements are absurd in the sense that there's no physical reality to the statement but logic doesn't care about reality, only consistency. And, it's perfectly consistent in propositional logic for all phones of a given set to be simultaneously off and on. Namely, the empty set.
There's no definitions or theorems preventing this possibility. The great thing about logic is that you're free to disprove any claim. So you either need to accept the truth of the absurdity and fuck off or disprove it. At this point, your argument is circular.
Jacob Hall
your example is trivial to understand. is there a phone in the room that is turned on? no, right? are you black or something? you guys aren't as good at abstract logic.
Jacob Nelson
>implying there's anywhere else for smart people to talk about philosophy Veeky Forums is sophistry and Veeky Forums is for fucking nerds
