B Nonsense can be truth in logic?

[PeroK]

So, I look at this and the set $x^2<0$ isn't empty. $2i$ is a member for example.

There was an implicit assumption that $x$ is a real number.

 

[Paul Colby]

Ah, yes. It's not even implicit. I'm still mystified by this discussion, so I'll just not comment further.

 

[Yuras]

Looks like there is a whole branch of logic "requiring the antecedent and consequent of implications to be relevantly related": https://en.wikipedia.org/wiki/Relevance_logic

Isn't it all about the excluded middle? If I remember correctly (and I don't remember much) in intuitionistic logic "if $x^2<0$ then $x=23$" means that $\not\exists x: x^2<0 \land x\neq23$, which I think won't encounter any opposition. To interpret is as $\forall x. x^2<0\Rightarrow x=23$ one needs the excluded middle, which is not intuitive (pun intended.)

Nope, I think I was wrong, the law of excluded middle is not relevant here.

 

[pinball1970]

Of all the threads I have read on PF this is definitely one of them.

 

[Mark44]

The Wikipedia page has: "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia" as a vacuous truth. But, that doesn't hold by the rules of logic alone.

Sure it does. What you may not be comprehending is that there are 1) the truth values of the hypothesis and conclusion, and 2) the truth value of the implication defined by the hypothesis and conclusion.

Hypothesis: Tokyo is in Spain -- clearly false, but see note below
Conclusion: Eiffel Tower is in Bolivia -- also clearly false
Truth value of the implication: True -- see my table in post #7.

Note: although there are towns named Tokio in Washington State, North Dakota, and Texas here in the US, I am 100% certain that the "Tokyo" referred to is the large city in the Honshu province of a certain nation off the coast of Asia. Regarding the Eiffel Tower, I believe there is only one.

Logic itself is not supposed to depend on whether the statements themselves are true.

That's not correct. To be useful, the truth value of statements such as the implication need to address all possible pairs of truth values for the hypothesis and conclusion.

Of all the threads I have read on PF this is definitely one of them.

Looks like a true statement to me, unless you happened to have scrolled to the bottom of the thread without reading any of the intervening posts.

 

[Mark44]

Looks like there is a whole branch of logic "requiring the antecedent and consequent of implications to be relevantly related": https://en.wikipedia.org/wiki/Relevance_logic

Nope, I think I was wrong, the law of excluded middle is not relevant here.

Right, it's just stating that a statement can be either true or false, with no other values possible.

 

[PeroK]

Sure it does. What you may not be comprehending is that there are 1) the truth values of the hypothesis and conclusion, and 2) the truth value of the implication defined by the hypothesis and conclusion.

Hypothesis: Tokyo is in Spain -- clearly false, but see note below
Conclusion: Eiffel Tower is in Bolivia -- also clearly false
Truth value of the implication: True -- see my table in post #7.

Note: although there are towns named Tokio in Washington State, North Dakota, and Texas here in the US, I am 100% certain that the "Tokyo" referred to is the large city in the Honshu province of a certain nation off the coast of Asia. Regarding the Eiffel Tower, I believe there is only one.

If my television is switched on, then you're wrong!

 

[martinbn]

If my television is switched on, then you're wrong!

If Einstein said the Earth is flat, then it is flat. Something flat earthers and the rest can agree on.

 

[Hornbein]

Y'all are complicating the matter by implicitly bringing the element of time into it. In logic, statements and atoms don't change their truth value with the passage of time unless you specifically allow for this. If X is true it stays true and so forth.

When I was a kid there was a game called Wff N' Proof that partially involved reasoning with nonsensical statements. That is, something obviously false is declared true in the system you are working with. It helped in breaking reliance on intuition and meaning. You just follow the rules in a mechanical fashion. At the time I didn't get it and didn't play the game.

 

[Gavran]

An implication is a compound conditional statement, denoted by $A\implies B$ where A and B are two statements.
If red is blue then green is red is a true statement. Or, if $3=3$ then $7=8$ is a false statement.
In logic, if $x^2\lt0$ then $x=23$ where $x\in\mathbb{R}$ is not a statement because $x=23$ is not a statement. We are not able to say if $x=23$ is true or false.

 

[pines-demon]

An implication is a compound conditional statement, denoted by $A\implies B$ where A and B are two statements.
If red is blue then green is red is a true statement. Or, if $3=3$ then $7=8$ is a false statement.
In logic, if $x^2\lt0$ then $x=23$ where $x\in\mathbb{R}$ is not a statement because $x=23$ is not a statement. We are not able to say if $x=23$ is true or false.

What is the problem of ($x=23$)? You just have to go beyond propositional logic into first order logic, then you can make statements like "for all variables $x$ with property $P$, then $x$ has property $Q$".

 

[Gavran]

What is the problem of ($x=23$)? You just have to go beyond propositional logic into first order logic, then you can make statements like "for all variables $x$ with property $P$, then $x$ has property $Q$".

This is propositional logic and there is a problem with $x=23$. You can not state that something is equal to $23$ if you only know it is a real number.

 

[pines-demon]

This is propositional logic and there is a problem with $x=23$. You can not state that something is equal to $23$ if you only know it is a real number.

It is a real number with the property of having $x^2<0$. In terms of sets, it is in the intersection of real numbers and numbers $x^2<0$, which is basically the empty set.

Edit: it is equivalent to Fresh42 example above of "All elements of the empty set have brown eyes"

Edit2: wrong inequality

 

[PeroK]

This is propositional logic and there is a problem with $x=23$. You can not state that something is equal to $23$ if you only know it is a real number.

This is not correct at all. The conclusion $x = 23$ is perfectly valid mathematically.

 

[Mark44]

It is a real number with the property of having $x^2<23$. In terms of sets, it is in the intersection of real numbers and numbers $x^2<23$, which is basically the empty set.

Is this really what you meant to say? The set $\{ x \in \mathbb R | x^2 < 23\}$ is not empty. It contains all of the numbers in the half-open interval $[0, \sqrt{23})$.

 

[FactChecker]

if( x=y) then (y=x).
That is always true no matter if x=y or not.

 

[martinbn]

Is this really what you meant to say? The set $\{ x \in \mathbb R | x^2 < 23\}$ is not empty. It contains all of the numbers in the half-open interval $[0, \sqrt{23})$.

It contains even more, namely $(-\sqrt{23}, \sqrt{23})$.

 

[pines-demon]

Is this really what you meant to say? The set $\{ x \in \mathbb R | x^2 < 23\}$ is not empty. It contains all of the numbers in the half-open interval $[0, \sqrt{23})$.

Oops ! Edited.

 

[Mark44]

It contains even more, namely $(-\sqrt{23}, \sqrt{23})$.

Yes. I could weasel out and say that my statement was no incorrect, but the truth is, I neglected to include all of the negative numbers

 

[PeroK]

Yes. I could weasel out and say that my statement was no incorrect, but the truth is, I neglected to include all of the negative numbers

This is what happens when you become overly focused on vacuity!

 

[Mark44]

This is what happens when you become overly focused on vacuity!

No, this is what happens when you get on in years ...

 

[mcastillo356]

This is not correct at all. The conclusion $x = 23$ is perfectly valid mathematically.

Any conclusion holds. We are dealing with logic, IMO.

It is the fifth page of this link:

https://people.math.ethz.ch/~dkosanovic/24-FS/Munkres-Topology.pdf

 

[Mark44]

There's a subthread here that needs to be untangled.

In logic, if $x^2\lt0$ then $x=23$ where $x\in\mathbb{R}$ is not a statement because $x=23$ is not a statement. We are not able to say if $x=23$ is true or false.

Sure we can. "x = 23" is very much a statement whose truth value can be determined. x either is or is not equal to 23.
Also, as has been stated multiple times, the implication $x \in \mathbb R \land x^2 < 0 \Rightarrow x = 23$ is a valid implication albeit one that is true. The reason that the implication is true is because the hypothesis is false. There are no real numbers whose squares are negative.

What is the problem of (x=23)?

This is propositional logic and there is a problem with x=23. You can not state that something is equal to 23 if you only know it is a real number.

That's true, but you seem to be saying something in your first post I quoted that is incorrect; namely that "z = 23" is not a statement.

This is not correct at all. The conclusion x=23 is perfectly valid mathematically.

Yes, and its truth value can be determined.

 

[Mark44]

Any conclusion holds.

Yes, when the hypothesis part is false.

 

[Averagesupernova]

Not really taking a side, but I have a problem with: "If I sprout wings and fly, then I shall use them to fly to Mars."
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I cannot sprout wings, and even if I did I cannot use them to fly to Mars. But by definition it's a true statement unless I misunderstand post #7.

 

[phinds]

Not really taking a side, but I have a problem with: "If I sprout wings and fly, then I shall use them to fly to Mars."
-
I cannot sprout wings, and even if I did I cannot use them to fly to Mars. But by definition it's a true statement unless I misunderstand post #7.

It's in determinant as to whether or not you are going to spout wings, so I have a problem with that statement. If you say "If I have wings, I can fly to mars" then that is a true statement (unless, of course, you actually have wings).

 

[Averagesupernova]

It's in determinant as to whether or not you are going to spout wings, so I have a problem with that statement. If you say "If I have wings, I can fly to mars" then that is a true statement (unless, of course, you actually have wings).

I don't see the difference. If I have vs if I were to have. Anything with if in front is indeterminate.
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That aside, if I have wings then I would successfully use them to fly to Mars. True.
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If I have wings then I would fail flying to Mars. Also true.
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What kind of screwed up logic is that? Lol.

 

[phinds]

What kind of screwed up logic is that? Lol.

Well, you may also just be one of those folks who have a problem with vacuous truths, like my nephew. (see my post #6)

 

[Averagesupernova]

Well, you may also just be one of those folks who have a problem with vacuous truths, like my nephew. (see my post #6)

I may have a problem with the way it's presented. First statement=false then we don't care about the rest. It's the equivalent of a NOT gate.
-
First statement=true then there are other conditions to check.
-
This is how a latch or flip-flop works. That makes sense.

 

[Mark44]

I may have a problem with the way it's presented. First statement=false then we don't care about the rest. It's the equivalent of a NOT gate.
-
First statement=true then there are other conditions to check

More briefly, an implication is defined to be true when
a) the hypothesis is false,
OR*
b) the conclusion is true.
This is laid out in the table in post #7.
*Inclusive OR

Hard to believe that this thread now has 80 posts!

 

[martinbn]

Not really taking a side, but I have a problem with: "If I sprout wings and fly, then I shall use them to fly to Mars."
-
I cannot sprout wings, and even if I did I cannot use them to fly to Mars. But by definition it's a true statement unless I misunderstand post #7.

May be you can think about it this way. A false premise leads to a false conclusion. This is true, right? So if A and B are false then "if A than B" is true, right?

 

[mcastillo356]

"If I sprout wings and fly, then I shall use them to fly to Mars."

Two hypothesis: Spread wings & fly preceeded by a conditional. I cannot assign any truth or lie value

 

[phinds]

Two hypothesis: Spread wings & fly preceeded by a conditional. I cannot assign any truth or lie value

Which is what I said in #76

 

[phinds]

Hard to believe that this thread now has 80 posts!

Boy, howdy, ain't that the truth. I STILL say your post #7 should have ended the discussion.

EDIT: Mark, you're a mentor. How about you just tie this off? We are CLEARLY not going to get through to some folks.

 

[Mark44]

How about you just tie this off?

Done.