Nonsense can be truth in logic?

[mcastillo356]

TL;DR

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)".