Nonsense can be truth in logic?

[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."
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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.

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.
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First statement=true then there are other conditions to check.
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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.
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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."
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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.

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.