B Nonsense can be truth in logic?

[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.

 

[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.