A spooky book

So I saw a book with total $100$ pages. First page of the book reads "The statement on page number 2 is true". Second page reads "The statement on page number $3$ is true" and so on. The $100$ th page of the book, however, reads "The statement on page number 1 is false!". Is the stament on page $10$ true or false?

#Paradox

Easy Math Editor

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Comments

As far as the structure of the paradox goes, pages 2 through 99 are irrelevant. Essentially, all of pages 1 through 99 are both literal and logical repetitions, so we in effect have a 2-page book, the first page of which states "the statement on the next page is true", while the last page states "the statement on the previous page is false". This is the two-statement version of the Liar paradox, on which an immense amount of thought has been expended, with no universally agreed-upon resolution.

So the logical fate of page 10 is the same as that of any of the other pages in the book - undetermined.

Brian Charlesworth - 6 years, 7 months ago

Yes. I posted it just for entertainment of those who don't know much about liar's paradox.

Snehal Shekatkar - 6 years, 7 months ago

I never get tired of thinking about this paradox; it's fun, frustrating and profound, all at the same time. :)

Brian Charlesworth - 6 years, 7 months ago

I am lying.

Did I lie?

Julian Poon - 6 years, 7 months ago

I always lie, so yes, you lied.

Brian Charlesworth - 6 years, 7 months ago

@Brian Charlesworth – :D May be we should start some serious discussion about this paradox on this page. What do you think?

Snehal Shekatkar - 6 years, 7 months ago

@Snehal Shekatkar – What do you think serious discussion can be, on this. Though I'd like to be a part.

Start On. I will follow, given I understand. I am not lying. :P

Arya Samanta - 6 years, 7 months ago

@Arya Samanta – :) Let's start discussion in new comment.

Snehal Shekatkar - 6 years, 7 months ago

@Brian Charlesworth – Lol. So since u always lie, were you lying about you lying? If thats the case, were you speaking the truth about you lying? And.... that means.... u were lying about you lying. Which means that you were... (and this goes on)

Julian Poon - 6 years, 7 months ago

Electron: All electrons are liars.

Is the electron lying or telling the truth?

John M. - 6 years, 7 months ago

Perhaps the electron is a positron in disguise. :)

If only electrons could actually talk; what stories they could tell.

Brian Charlesworth - 6 years, 7 months ago

Well, what I think is, the electron, instead of answering Yes or No, will answer Yo.

Yes + No = Yo

Cheers

Note: Or, we could always do this the hard way...

John M. - 6 years, 7 months ago

@John M. – Hahaha. Or perhaps the electron would answer "Yolo". :) I particularly like the one with the two cat eigenstates, divided by $\sqrt{2}$ . Thanks for the laughs. :D

Brian Charlesworth - 6 years, 7 months ago

This is somewhat different. This statement can be said to be false if their is at least one electron wo doesn't lie.

Snehal Shekatkar - 6 years, 7 months ago

Well now you're extrapolating. We're just going with the statement here - as we should. If we start adding extra conditions, we're changing the initial statement. It's just like math, you know?

And this is somewhat different - in a sense that I implied a quantum superposition of electrons as a quantum superstatement - that is, the electron tells a truth and a lie at the same time, or, rather, the combination of the two.

See Shcrodinger's Cat.

John M. - 6 years, 7 months ago

So what could be the possible resolutions of this paradox guys?

Snehal Shekatkar - 6 years, 7 months ago

With regard to the liar's paradox, I tend to take the view that statements which refer only to themselves and not to something "real" have no meaning and hence no truth value. They are like closed loops disconnected from anything of tangible significance. So without any truth value, any question regarding the trueness of a statement is invalidated, and hence no paradox existed in the first place. It only existed in our minds because we ascribed a truth value to it, a truth value to which it was never entitled. Likewise with the book paradox; no meaning, no truth value, no paradox.

Brian Charlesworth - 6 years, 7 months ago

Yeah and to add to that - I've read a Wiki article before on this - those statements are actually false. This is because they violate the Fundamental Law of Logic of Contradiction - if a statement contradicts itself, and a closed infinite loop is one such type, then it is simply illogical - false. And only because it's false doesn't mean that the opposite of what the statement is stating must be true - which is what seems intuitive. It simply means that the statement makes no logical sense and thus looking for a logical answer is pointless.

There - "paradox" resolved.

John M. - 6 years, 7 months ago

@John M. – True. :)

Brian Charlesworth - 6 years, 7 months ago

true.....

Ankan Biswas - 6 years, 7 months ago

My doubt...

What does the 99th page say?

Lets assume that Pages 1-99 are true and pages 1-98 state that the statement on the next page is true. i.e. page 98 says, '''the statement on page 99 is true''.

Now page 99 says, '''the statement on page 100 is false''. ----1

Now as per the question, page 100 says, '''the statement on page 1 is false''.
(from --1) the meaning of this is that '''the statement on page 1 is true''.

Which, according to me, ends the paradox.

Please state whether the above statements are true or false.

Ritu Roy - 6 years, 7 months ago

99th page says that statement on page 100 is true. This is quite obvious from the given description.

Snehal Shekatkar - 6 years, 7 months ago

If you read the book from page 1 to 100 considering all the written statements to be true, then obviously the statement on tenth page should be false.

Ninad Akolekar - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$