where's the flaw?

[kamil9876]

Suppose we want to prove that all Algebra  students in a class of students have the same grade. Denote this statement as . This is clearly true for . Assume it is true for where . Then for a class of n+1 students have the same grade. But all have the same grade. Hence all have the same grade. Hence true for and so by induction is true for all natural QED.

[zzdfa]

i got another induction paradox:

A man is sentenced to death on sunday. The judge tells him: 'You will be killed on either monday or tuesday, at 8am and you will not be able to logically deduce that you will be killed that day until 8am that day.' So the guy thinks,

I will be killed on monday, because if I don't get killed on monday then I'll be able to deduce that I'm to be killed on tuesday.

but the he thinks

I can't get killed on monday either because I just deduced that I will be killed on monday.

Therefore I can't be killed on either day.

On monday morning the guards shoot him at 8am, and he did not deduce this.

Where was the flaw in the prisoner's reasoning? ( i say induction paradox because his argument works for any finite number of days)

[kamil9876]

^ That one's a classic. My year 10 math teacher (sikkent btw) introduced us to this one except it had 7 days

He said, immediately after telling it so I didn't have a proper chance to figure this out myself but I think I agree, that it depends on what you mean by "know" (in his example the word "know" was used rather than 'logically deduced').

A man is sentenced to death on sunday. The judge tells him: 'You will be killed on either monday or tuesday, at 8am and you will not be able to logically deduce that you will be killed that day until 8am that day.'

Stopping at this list of premises, we get a set of statements that implies(by your next argument) "the guy cannot think at all()". Hence the introduciton of the next premise contradicts this. (possible resolution but maybe there are better ones, not sure at how cool this one is). (ie it's like saying is an even prime, and then suddenly introducing the premise )

[zzdfa]

yeh, i still haven't found a really convincing explanation. according to wikipedia its still an unsolved problem.

and I don't really understand your thing, you said

Stopping at this list of premises, we get a set of statements that implies(by your next argument) "the guy cannot think at all()". Hence the introduciton of the next premise contradicts this.

but you quoted the whole premises? I didn't introduce any new premises after that one. the prisoner looked at this premise:

A man is sentenced to death on sunday. The judge tells him: 'You will be killed on either monday or tuesday, at 8am and you will not be able to logically deduce that you will be killed that day until 8am that day.'

and deduced that he could not be killed on monday or tuesday. however, he was killed on monday. contradiction. if we have a contradiction then there must either be
1)a flaw in his logic
2)the judges statement is false  (this is how we prove things by contradiction remember, think of this q as one of those 0=1 proofs)

I think what you're trying to get at is that 2), the judges premise is false. but it's not!! it's clearly not false, the prisoner did end up getting killed on a day which he could not logically deduce!!
which leaves case 1).
so its just like a 0=1 proof, it starts off with a not false premise and then gets a contradictory conclusion, due to a flaw in the reasoning
except the flaw isnt obvious in this case

[kamil9876]

Didn't quote the whole premise, the man deducing reasonably is another one. The man is crucial here since if it was somebody on the outside performing that argument we wouldn't reach the conclusion that "he[man]" didn't deduce. i.e: if the problem was only what I quoted, one would deduce that the only possibility (to avoid contradiction) is that the man cannot reason. In other words the judge being true on "you will not be able to logically deduce" is like an implicit way of saying "you[man] are a stupid kent and cannot reason". But by introducing the next premise that he is not a stupid kent gives the contradiction. Just like n is an even prime leaves 2 as the only possibility but by introducing you get the paradox.

i.e: introduce not enough equations, no unique solution. Introduce just enough, unique solution. Introduce too many, contradicting solutions hence none.

That's what I meant.

[zzdfa]

Hmm i see. I'm not entirely convicned though. i don't think this is true:

"you will not be able to logically deduce" is like an implicit way of saying "you[man] are a stupid kent and cannot reason".

because in this case, the man was able to reason, yet he was still not able to deduce when he was to be killed (he thought he could, but he was wrong). i.e. it is possible for the man to be able to reason yet unable to deduce that he will be killed on the next day.

so  "you will not be able to logically deduce"  and "you are able to reason" are not mutually exclusive.

[kamil9876]

lol i was just trying to add some colour. Just treat the two as synonymous(or disregard it completelty, simply replace "cannot reason" with "cannot deduce in this particular instance").

Summary:

"You will not be able to deduce" implies "Man will not think", since otherwise we get a contradiction(as shown by the addition of the next premise that he does actually think").

Why doesn't wiki have a whole article on this paradox: " is an even prime, ".

Simply it's the same thing(something was added that caused contradiction) although this one was more clever and is open to many interpretations(I agree that mine is not the only one and there are probably better ones) since it's vague in it's description(informal english language(unlike the original post)) and there are various combinations of premises that one can throw out in order to get rid of contradiction.



Also: wiki calls it a philosophical problem, and philosophers are ussually harder to please even on the most trivial of shit(only way to keep the subject alive) that is meaningless(see Wittgenstein's argument on why philosophy is shit) like Zeno's paradox and thompson's lamp.