B Nonsense can be truth in logic?

[mcastillo356]

TL;DR Summary

I'm curious about a statement said to be vacuously true

Hi, dear PF

One true statement about real numbers is the following:

If $x^{2}<0$, then $x=23$

The hypotesis is absurd, so I might also conclude $x=\mbox{me myself}$ or whatever.

PD: Post without preview

 

[phinds]

Yes, vacuous truths sound silly, but logically they are still true. If you start with a false premise, you can say that anything is true.

I would comment on our current political situation but that's against the rules.

 

[fresh_42]

I like to say that the elements of the empty set have purple eyes.

 

[FactChecker]

While your example is simple and obvious, there are many much more obscure examples where the condition is false and the full statement must be considered true in order to reach the proper conclusion. A vacuously true statement might be a small part of a large logic problem that is being solved in an automated system. The solution may not be correct unless the vacuous statement is assigned a True value.

 

[martinbn]

TL;DR Summary: I'm curious about a statement said to be vacuously true

Hi, dear PF

One true statement about real numbers is the following:

If $x^{2}<0$, then $x=23$

The hypotesis is absurd, so I might also conclude $x=\mbox{me myself}$ or whatever.

PD: Post without preview

I don't understand what the problem or the point is.

 

[phinds]

I don't understand what the problem or the point is.

The problem is that some people just naturally don't like vacuous truths because they can just seem weird.

EDIT: I had nephew who absolutely refused to believe that vacuous truths could be true statements. My example to him was "If you live in California (he lived in KY) then I live in Paris (I live in NY). He practically screamed at me "But that's just stupid !!! Neither one of those things is true". My response that "yes, but the overall statement is true" did not go over well.

 

[Mark44]

TL;DR Summary: I'm curious about a statement said to be vacuously true

The hypotesis is absurd, so I might also conclude x=me myself or whatever.

I don't understand what the problem or the point is.

A logical implication consists of two parts: a hypothesis and a conclusion. Each of these can be either true or false. The table below shows the four possible pairs of true/false values and the value of the implication.

Hypothesis	Conclusion	Implication
F	F	T
F	T	T
T	F	F
T	T	T

By definition, the only combination of the values of the hypothesis and conclusion that makes the implication false is when the hypothesis is true but the conclusion is false.

 

[PeroK]

TL;DR Summary: I'm curious about a statement said to be vacuously true

Hi, dear PF

One true statement about real numbers is the following:

If $x^{2}<0$, then $x=23$

The hypotesis is absurd, so I might also conclude $x=\mbox{me myself}$ or whatever.

PD: Post without preview

Athough that seems a pointless and absurd example, the key point is that it is valid to argue logically from a false premise. This is useful when you don't know whether your original premise is true or false - and, in fact, often you are trying to prove that it's false. If you start with a premise, argue logically and end up with something that you already know to be false (or that contradicts your original premise), then you have proved that the original premise is false. This is generally known as a proof by contradiction.

Note that in this case it is the entire statement that is vacuously true. Not accepting the concept of vacuously true statements undermines the concept of proof by contradiction.

 

[Mark44]

Adding to what I wrote, you can think of an implication as a contract in which the terms are met or not met.
Here's the hypothesis: You give me $500.
Here's the conclusion: I give you my 1960 Edsel (with title).

There are four possibilities:

1. You don't give me $500, and I don't give you the Edsel.
2. You don't give me $500, but I give you the Edsel anyway.
3. You give me $500, but I don't give you the Edsel.
4. You give me $500, and I give you the Edsel.

The only possibility in which you can claim that the contract terms were violated is #3, although you might be miffed if you really didn't want the car but I unloaded it in your driveway.

BTW, I don't have an Edsel.

 

[phinds]

1. You give me $500, but I don't give you the Edsel.
2. You give me $500, but I don't give you the Edsel.

Don't know why those are 1,2 instead of 3,4 but anyway I doubt you intended for them to be the same.

 

[martinbn]

A logical implication consists of two parts: a hypothesis and a conclusion. Each of these can be either true or false. The table below shows the four possible pairs of true/false values and the value of the implication.

Hypothesis	Conclusion	Implication
F	F	T
F	F	T
T	F	F
T	T	T

By definition, the only combination of the values of the hypothesis and conclusion that makes the implication false is when the hypothesis is true but the conclusion is false.

I know all this, that's why I don't see the point of the first post.

(You wrote the same first and second line in the table, but we know what you meant.)

 

[phinds]

I know all this, that's why I don't see the point of the first post.

AGAIN ... the point of the first post is that some people just don't like vacuous truths because they can seem weird. Why is that hard for you to understand? We are NOT talking logic here, we are talking emotional response to something that seems weird.

 

[martinbn]

AGAIN ... the point of the first post is that some people just don't like vacuous truths because they can seem weird. Why is that hard for you to understand? We are NOT talking logic here, we are talking emotional response to something that seems weird.

Which part is weird? And what do you call a vacuous truth? It is of the form "If A then B", and the truth of that shouldn't be confused with the truth of A or the truth of B.

 

[pines-demon]

I like to say that the elements of the empty set have purple eyes.

This is actually a pretty clever next-order vacuous truth.

 

[pines-demon]

Which part is weird? And what do you call a vacuous truth? It is of the form "If A then B", and the truth of that shouldn't be confused with the truth of A or the truth of B.

Vacuous truth is anything of the form (False $\Rightarrow Q$ ).

This is because (False $\Rightarrow Q)\equiv$ True.

 

[phinds]

Which part is weird? And what do you call a vacuous truth? It is of the form "If A then B", and the truth of that shouldn't be confused with the truth of A or the truth of B.

AGAIN, you are bring logic to an emotional argument. See my post #6.

 

[DaveC426913]

"If you live in California ... then I live in Paris"
...the overall statement is true...

I saw the truth table Mark44's post 7, yet this still rankles.

How is the above statement true?

The (if .. then) structure seems to imply a cause and effect relationship, to-wit: him living in California would instantly whisk you way to Paris. But there is no cause and effect relationship (at least, not that we know of until we try). So the statement does not seem to be true.

 

[PeroK]

I saw the truth table Mark44's post 7, yet this still rankles.

How is the above statement true?

The (if .. then) structure seems to imply a cause and effect relationship, to-wit: him living in California would instantly whisk you way to Paris. But there is no cause and effect relationship (at least, not that we know of until we try). So the statement does not seem to be true.

It's best, IMO, to stick to mathematical statements. I'm not sure I know how to deal with that statement about California and Paris. It may not stand up logically, I'm sorry to say.

A better example is: "Everytime I've entered the Gentlemen's Singles at Wimbledon, I've won the tournament." That's vacuously true.

 

[DaveC426913]

A better example is: "Everytime I've entered the Gentlemen's Singles at Wimbledon, I've won the tournament." That's vacuously true.

I see. That one makes more sense.

But it is qualitatively different from the OP's example. In your example, the answer is unambiguously zero.

In the OP's example, we're back to cause-effect.

If $x^{2}<0$, then $x=23$

 

[PeroK]

I see. That one makes more sense.

But it is qualitatively different from the OP's example. In your example, the answer is unambiguously zero.

In the OP's example, we're back to cause-effect.

It all boils down to sets. In both cases, $x^2 < 0$ or my Wimbledon entries, we are choosing from the empty set. That's what makes them logically similar.

The other example: "If I live in California, then you live in Paris" gets a big foggy and logically muddled. I'm not convinced by it.

 

[fresh_42]

The other example: "If I live in California, then you live in Paris" gets a big foggy and logically muddled. I'm not convinced by it.

Add "currently" and it is the same argument as Wimbledon. But this is semantics.

 

[PeroK]

Add "currently" and it is the same argument as Wimbledon. But this is semantics.

Perhaps, but if we say:

"If I lived in California, then you'd live in Paris" then it's definitely wrong.

 

[Mark44]

but anyway I doubt you intended for them to be the same.

(You wrote the same first and second line in the table, but we know what you meant.)

Apologies to both of you. I swapped the entries in the table and in the list but neglected to completely fix the appropriate table values/statements. I've fixed them in both places.

"If you live in California ... then I live in Paris"
...the overall statement is true...

I saw the truth table Mark44's post 7, yet this still rankles.
How is the above statement true?

You didn't copy enough of what @phinds wrote. In the full quote he stated that 1) you don't live in California and 2) he doesn't live in Paris. This makes both the hypothesis ("You live in California") and the conclusion ("I live in Paris") false, which makes the overall implication true.

The only way an implication can be considered to be false is if the hypothesis is true but the conclusion is false.

 

[PeroK]

Apologies to both of you. I swapped the entries in the table and in the list but neglected to completely fix the appropriate table values/statements. I've fixed them in both places.


You didn't copy enough of what @phinds wrote. In the full quote he stated that 1) you don't live in California and 2) he doesn't live in Paris. This makes both the hypotheses ("You live in California") and the conclusion ("I live in Paris") false, which makes the overall implication true.

The only way an implication can be considered to be false is if the hypothesis is true but the conclusion is false.

I'm not convinced by this. What precisely is the logical construction here? I don't see it.

 

[Mark44]

The (if .. then) structure seems to imply a cause and effect relationship, to-wit: him living in California would instantly whisk you way to Paris.

I don't think cause and effect are the right way to look at implications, at least not in all circumstances. E.g., "if water is wet, then zebras have stripes."
This implication is true whether or not water is actually wet. (I.e., it could be frozen or gaseous, as steam.) The state of water does not cause zebras to have stripes.

 

[Ibix]

It all boils down to sets. In both cases, $x^2 < 0$ or my Wimbledon entries, we are choosing from the empty set. That's what makes them logically similar.

Learning from your Wimbledon example (and assuming you're not Andy Murray), is the point that the OP's "if $x^{2}<0$, then $x=23$" is meant to be read as "every real $x$ that satisfies $x^2<0$ [i.e. none of them] also satisfies $x=23$"? And it's vacuously true because the first condition eliminates all options so the other is essentially irrelevant?

 

[PeroK]

Learning from your Wimbledon example (and assuming you're not Andy Murray), is the point that the OP's "if $x^{2}<0$, then $x=23$" is meant to be read as "every real $x$ that satisfies $x^2<0$ [i.e. none of them] also satisfies $x=23$"? And it's vacuously true because the first condition eliminates all options so the other is essentially irrelevant?

Yes. Another way to look at it is that for an implication of this type to fail, there must be a counterexample. We would need to find an $x$ with $x^2 < 0$, yet $x \ne 23$. Or, find a Wimbledon I played in that I didn't win.

In practical terms, you might look at an equation and determine whether the solutions must be rational. That only fails if you can find irrational solutions. You may even have proved a theorem:

If $x$ is a solution to equation $X$, then $x$ is rational.

That theorem stands even if you find that there are no solutions to equation $X$. Then, it's vacuously true.

 

[DaveC426913]

You didn't copy enough of what @phinds wrote. In the full quote he stated that 1) you don't live in California and 2) he doesn't live in Paris.

No, get it.

This makes both the hypotheses ("You live in California") and the conclusion ("I live in Paris") false, which makes the overall implication true.

Why?

The only way an implication can be considered to be false is if the hypothesis is true but the conclusion is false.

But why? (I mean other than because the discipline of formal logic says so).

 

[DaveC426913]

I don't think cause and effect are the right way to look at implications, at least not in all circumstances. E.g., "if water is wet, then zebras have stripes."
This implication is true whether or not water is actually wet. (I.e., it could be frozen or gaseous, as steam.) The state of water does not cause zebras to have stripes.

Right. But I don't see the difference between this example and yours.

 

[DaveC426913]

It all boils down to sets. In both cases, $x^2 < 0$ or my Wimbledon entries, we are choosing from the empty set. That's what makes them logically similar.

OK, this is can at least grasp. It's programming logic.

It's essentially saying all forms of false are equivalent. eg. (1+1=3) = (2+2=5).

Still, that that's not the same as "If (1+1=3) then (2+2=5)".

 

[Thecla]

3+5=9
subtract 3 from both sides
5=6
If 3+5=9 then 5=6
This conclusion makes sense

 

[PeroK]

Still, that that's not the same as "If (1+1=3) then (2+2=5)".

I'm probably getting out of my depth here. When you say "If (1 + 1 = 3)", then you can conclude nothing from that. Any logical steps depend on applying whatever rules of arithmentic we have already established. I suspect that I could get to 2 + 2 = 5.

But, with a statement like "if I live(d) in California", what can we do with that? What rules of habitation are we following? This is why that fits better into a syllogism. In other words, if that's our major premise, we need a minor premise. We cannot work from that single premise alone.

A typical syllogism would be:

If X lives in California. [Major premise.]
If all people living in California believe DJT is the new messiah. [Minor premise]
Then X believes DJT is the new messiah. [Conclusion]

That is a valid syllogism. But:

If X lives in California. [Major premise.]
Then Y lives in Paris. [Conclusion]

Is not a well-formed syllogism. And is not vacuously true.

That said, I'm not nor ever have been a member of the logician party.

 

[mcastillo356]

Athough that seems a pointless and absurd example, the key point is that it is valid to argue logically from a false premise. This is useful when you don't know whether your original premise is true or false - and, in fact, often you are trying to prove that it's false. If you start with a premise, argue logically and end up with something that you already know to be false (or that contradicts your original premise), then you have proved that the original premise is false. This is generally known as a proof by contradiction.

Note that in this case it is the entire statement that is vacuously true. Not accepting the concept of vacuously true statements undermines the concept of proof by contradiction.

Brilliant all the answers. @PeroK, thanks.

I don't understand what the problem or the point is.

I look for meaning, for common sense, in everything, and when I can't find it, I keep looking for it. As an example I put the thread on "Comedian", by Maurizio Cattelan. In this case, I gave up. Just because common sense tells to give up. But nonsense is useful in maths, as pointed out in the previous quote.

Yeah, I've started the thread the wrong way. The title is confusing: Nonsense can be truth in logic? Yes, it can, but in maths nonsense has got a great role. I am sure you agree, @martinbn.

Thanks once again, PF

Marcos

 

[Yuras]

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.)

 

[Mark44]

This makes both the hypotheses ("You live in California") and the conclusion ("I live in Paris") false, which makes the overall implication true.

Why?

See immediately below.

The only way an implication can be considered to be false is if the hypothesis is true but the conclusion is false.

By definition, any other combination of truth values for the hypothesis (the "if" part) and the conclusion (the "then" part) makes the implication true.

But why? (I mean other than because the discipline of formal logic says so).

By definition of a logical implication.

But:

If X lives in California. [Major premise.]
Then Y lives in Paris. [Conclusion]

Is not a well-formed syllogism.

We're not talking about syllogisms in this thread (major premise, minor premise, conclusion). We're talking about logical implications, which consist of a hypothesis and a conclusion. As already stated, an implication is defined to be true 1) if the conclusion is true or 2) if both hypothesis and conclusion are false.

Given values for X and Y, the truth value of the implication above can be determined.

 

[phinds]

We're not talking about syllogisms in this thread (major premise, minor premise, conclusion). We're talking about logical implications ...

Yes, that's my point exactly. I understand completely why some people find it weird and/or don't like but I don't get why folks can't accept well defined math logic. Your post #7 really should have ended the discussion.

 

[PeroK]

Given values for X and Y, the truth value of the implication above can be determined.

I'm still not sure. Something doesn't feel right. It feels like misapplying logic somehow. Like you are mixing up absolute truths with conditional truths somehow. It should depend on a test of whether X lives in California.

 

[PeroK]

Yes, that's my point exactly. I understand completely why some people find it weird and/or don't like but I don't get why folks can't accept well defined math logic. Your post #7 really should have ended the discussion.

I'm not convinced it's logically sound. See my last post.

I'm not saying I'm definitely right, but I'm not convinced.

 

[phinds]

I'm not convinced it's logically sound.

OK, fine. Explain to me and Mark where the truth table in post #7 goes wrong

 

[Mark44]

I'm still not sure. Something doesn't feel right. It feels like misapplying logic somehow. Like you are mixing up absolute truths with conditional truths somehow. It should depend on a test of whether X lives in California.

That's exactly why I said "given values for X and Y," better stated as given truth values for "X lives in California" and "Y lives in Paris."
The implication is true if "Y lives in Paris" is true (independent of where X lives) or if "X lives in California" is false. This is really pretty basic logic.

 

[PeroK]

OK, fine. Explain to me and Mark where the truth table in post #7 goes wrong

I feel like your not living in California is not a logical hypothesis. Not as I understand it.

 

[phinds]

I feel like your not living in California is not a logical hypothesis. Not as I understand it.

Well, then, you're just going to have to remain one of the folks who think that vacuous truths are weird (or, in your case, wrong). That won't change the math definitions.

The validity of the statements as logical or correct hypotheses is simply not at issue, despite your desire that it be. The only thing that matter is, is the first statement true or not. If it is not then it isn't even relevant what the second statement is, the overall statement is true whether or not common sense says that it is.

 

[Mark44]

I feel like your not living in California is not a logical hypothesis.

Both the hypothesis and conclusion are logical statements, that can be either true or false. The statement "you live in California" is true when you actually live there, and false if you don't. There's really nothing complicated about this.

the overall statement is true

What you're calling "overall statement" is what I'm referring to as the implication. I'm pretty sure we're on the same page here.

 

[Mark44]

From post #24:

What precisely is the logical construction here? I don't see it.

In case you're still not clear on this, I believe that you're confusing syllogism with implication. In a syllogism there are major and minor premises, followed by a conclusion. An implication has just a single premise (the hypothesis) followed by a conclusion.
For an implication, both the hypothesis/premise and conclusion are logical (i.e., Boolean) statements whose values are either true or false. The (corrected) table in post #7 lists the possible truth values for the two statements as well as the truth value for the implication.

Apologies if the previous posts in this thread cleared up your question from post #24.

 

[PeroK]

That's exactly why I said "given values for X and Y," better stated as given truth values for "X lives in California" and "Y lives in Paris."
The implication is true if "Y lives in Paris" is true (independent of where X lives) or if "X lives in California" is false. This is really pretty basic logic.

I looked on the Wikipedia page and they give an example very like this one: One example of such a statement is "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia".

So, I guess you must be right.

Perhaps I was thinking more in terms of mathematics, where things are clear cut. There could be a place in Spain called "Tokyo". There are the cities of Athens and Cairo in the US, for example. I still feel that additional assumption hovers over this whole question. That somehow it's not quite pure logic. It's logic mixed up with general knowledge.

When we say something like if $x^2 < 0$, then $x = 23$, then that is only vacuously true under strict conditions. $x$ cannot be a complex number. If that's understood, then we have something that is logically self-contained.

Whereas, something like "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia" doesn't feel logically self-contained.

 

[Mark44]

Perhaps I was thinking more in terms of mathematics, where things are clear cut.

When we say something like if $x^2 < 0$, then $x = 23$, then that is only vacuously true under strict conditions. $x$ cannot be a complex number.

You don't need any conditions here. For the statement $x^2 < 0$ to be true (and allowing both real and complex numbers), x must be pure imaginary. If a complex x has some nonzero real part, then its square will also be complex, so can't be compared with zero. Now, if x is pure imaginary, its square is real, but negative, so x can't be equal to 23.

OTOH, if x is real, then $x^2 \ge 0$, so $x^2 < 0$ can't be true in this case.

 

[PeroK]

You don't need any conditions here. For the statement $x^2 < 0$ to be true (and allowing both real and complex numbers), x must be pure imaginary. If a complex x has some nonzero real part, then its square will also be complex, so can't be compared with zero. Now, if x is pure imaginary, its square is real, but negative, so x can't be equal to 23.

OTOH, if x is real, then $x^2 \ge 0$, so $x^2 < 0$ can't be true in this case.

Either way, the implication is, by definition true. The hypothesis can't possibly be true, regardless of whether x is real or complex, so the truth value of the conclusion is irrelevant, making the implication itself true.

$i^2 = -1 < 0$ and $i \ne 23$

 

[Mark44]

$i^2 = -1 < 0$ and $i \ne 23$

You're correct and my logic had a flaw. If x is pure imaginary, then then the implication is false because the hypothesis is true (##i^2 < 0) while the conclusion is false (x = 23). Thank you for the correction. I edited my post but left what I wrote in what you quoted.

 

[PeroK]

Sorry to go on about this, but I think this is interesting. 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. To be logically valid it would need to be "If Tokyo is not in Spain and Tokyo is in Spain, then ...".

Logic itself is not supposed to depend on whether the statements themselves are true. It's supposed to be self-contained. As far as possible, mathematics is self-contained. This goes back to my original point that it's better to stick to mathematics when explaining this aspect of logic. That goes for my Wimbledon example as well. Although, I Imagine, that will always be true; and I tried to make it self-contained.

Whereas, @phinds example, is highly dependent on current facts. This is not to say it's wrong. But, it does miss the point, IMO, that these logical constructions are supposed to be self-contained.

 

[Paul Colby]

it all boils down to sets. In both cases, $x^2<0$ or my Wimbledon entries, we are choosing from the empty set. That's what makes them logically similar.

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