Discuss.
Discus.
All right! I am the paradox!
“Only the true paradox denies its own veracity.”
Would it be paradoxical if that statement turned out to be not true?
mcgoon said:
“Only the true paradox denies its own veracity.”Would it be paradoxical if that statement turned out to be not true?
Yes.
I mean no.
The difference between a proof by contradiction and a paradox is that a proof by contradiction gives (NOT A) IMPLY A, whereas a paradox gives (NOT A) EQUIVALENT A. In the former case A=TRUE is a solution and the only solution, whereas in the latter case A has no solution.
Only the true paradox denies its own veracity.
KJW said:
The difference between a proof by contradiction and a paradox is that a proof by contradiction gives (NOT A) IMPLY A, whereas a paradox gives (NOT A) EQUIVALENT A. In the former case A=TRUE is a solution and the only solution, whereas in the latter case A has no solution.
I’m not convinced any more that there is any such thing as a paradox, because I managed to resolve some 12 or so paradoxes involving infinity, showing that they weren’t paradoxes after all.
I’ll illustrate with some well-known paradoxes.
1. “This statement is false”. “The village barber shaves everyone who does not shave himself. Who shaves the barber?” “Consider the set of all sets that are not members of themselves. Is this set a member of itself?”
These three paradoxes are identical. In each case, what we have is a logical entity that is neither true nor false. I think that these can be handled using Belnap’s four-value relevance logic. Its possible values are true, false, both (true and false), and neither (true nor false).
2. Godel’s proof that any mathematics that contains the integers cannot be both complete and consistent.
Whenever any proposition is shown to be unprovable within the current framework of mathematics, it becomes an axiom. The best known example of this is the “axiom of choice”.:https://en.wikipedia.org/wiki/Axiom_of_choice
3. Is infinity odd or even?
The solution to this is found in Robinson’s hyperreal numbers. You are free to choose whether infinity is odd or even but, once chosen, that answer stays fixed for the remainder of the calculation.
4.
This is resolvable using infinitesimals.
mollwollfumble said:
Only the true paradox denies its own veracity.KJW said:
The difference between a proof by contradiction and a paradox is that a proof by contradiction gives (NOT A) IMPLY A, whereas a paradox gives (NOT A) EQUIVALENT A. In the former case A=TRUE is a solution and the only solution, whereas in the latter case A has no solution.
I’m not convinced any more that there is any such thing as a paradox, because I managed to resolve some 12 or so paradoxes involving infinity, showing that they weren’t paradoxes after all.
I’ll illustrate with some well-known paradoxes.
1. “This statement is false”. “The village barber shaves everyone who does not shave himself. Who shaves the barber?” “Consider the set of all sets that are not members of themselves. Is this set a member of itself?”
These three paradoxes are identical. In each case, what we have is a logical entity that is neither true nor false. I think that these can be handled using Belnap’s four-value relevance logic. Its possible values are true, false, both (true and false), and neither (true nor false).
2. Godel’s proof that any mathematics that contains the integers cannot be both complete and consistent.
Whenever any proposition is shown to be unprovable within the current framework of mathematics, it becomes an axiom. The best known example of this is the “axiom of choice”.:https://en.wikipedia.org/wiki/Axiom_of_choice
3. Is infinity odd or even?
The solution to this is found in Robinson’s hyperreal numbers. You are free to choose whether infinity is odd or even but, once chosen, that answer stays fixed for the remainder of the calculation.
4.
This is resolvable using infinitesimals.
1. I agree. There are no “true” paradoxes, only apparent paradoxes, that is apparent contradictions due to a logical error.
That makes the statement in the thread title simply False, rather than paradoxical.
But I don’t understand Belnap’s categories. Why are three categories not sufficient? (i.e. True, False, and Neither)
2. I haven’t yet worked out how this ties in with Godel. I’ll have another think about it.
3. Can’t we just say that odd and even are undefined for infinity?
4. Why do we need infinitesimals? a*0 = b*0 does not imply that a=b, so the step in red is just wrong, isn’t it?
By the way, I think the Village Barber one is different to the other two. It is not possible for the Village Barber to shave everybody (and only) those who do not shave themselves, so the statement is just false, not paradoxical.
The Rev Dodgson said:
By the way, I think the Village Barber one is different to the other two. It is not possible for the Village Barber to shave everybody (and only) those who do not shave themselves, so the statement is just false, not paradoxical.
If the barber does not shave himself then he does shave himself.
If the barber does shave himself then he does not shave himself.
there’s a softer more practical sense of paradox, of which whatever involves somewhat (apparently) contradictory requirements.
paradoxes are not necessarily or mostly an oblivion
mollwollfumble said:
The Rev Dodgson said:
By the way, I think the Village Barber one is different to the other two. It is not possible for the Village Barber to shave everybody (and only) those who do not shave themselves, so the statement is just false, not paradoxical.
If the barber does not shave himself then he does shave himself.
If the barber does shave himself then he does not shave himself.
But that doesn’t make it a paradox, it just makes it a statement that cannot be true.
Its just like saying If a >= b Then a > b
transition said:
there’s a softer more practical sense of paradox, of which whatever involves somewhat (apparently) contradictory requirements.paradoxes are not necessarily or mostly an oblivion
meaning a true paradox doesn’t necessarily result in an oblivion-of-contradiction.
transition said:
there’s a softer more practical sense of paradox, of which whatever involves somewhat (apparently) contradictory requirements.paradoxes are not necessarily or mostly an oblivion
I didn’t say there are no apparent paradoxes; just that there are no true paradoxes.
The Rev Dodgson said:
transition said:
there’s a softer more practical sense of paradox, of which whatever involves somewhat (apparently) contradictory requirements.paradoxes are not necessarily or mostly an oblivion
I didn’t say there are no apparent paradoxes; just that there are no true paradoxes.
okay, I had a mental disturbance after seeing the math, forgive me
transition said:
transition said:
there’s a softer more practical sense of paradox, of which whatever involves somewhat (apparently) contradictory requirements.paradoxes are not necessarily or mostly an oblivion
meaning a true paradox doesn’t necessarily result in an oblivion-of-contradiction.
What do you mean by a true paradox?
And what do you mean by an oblivion-of-contradiction?
>And what do you mean..”
I was waffling probably, only just finished my first coffee, not fully human.
More I was just saying paradoxical requirements feature commonly of thinking about many things.
maybe the barber has a beard.
transition said:
>And what do you mean..”I was waffling probably, only just finished my first coffee, not fully human.
More I was just saying paradoxical requirements feature commonly of thinking about many things.
OK, I can certainly agree with that :)
Or should that be :( ?
The Rev Dodgson said:
mollwollfumble said:
The Rev Dodgson said:
By the way, I think the Village Barber one is different to the other two. It is not possible for the Village Barber to shave everybody (and only) those who do not shave themselves, so the statement is just false, not paradoxical.
If the barber does not shave himself then he does shave himself.
If the barber does shave himself then he does not shave himself.
But that doesn’t make it a paradox, it just makes it a statement that cannot be true.
Which statement are you saying cannot be true? If it is the statement that the barber shaves those who do not shave themselves, then it cannot be true because it is a paradox. It isn’t not a paradox because is cannot be true.
>If it is the statement that the barber shaves those who do not shave themselves, then it cannot be true because it is a paradox.
I think it’s usually accepted that it cannot be true because such an individual can’t exist. So it’s not regarded as a “true paradox”, just a matter of stating a simple contradiction, like “a white black thing”.
mollwollfumble said:
2. Godel’s proof that any mathematics that contains the integers cannot be both complete and consistent.Whenever any proposition is shown to be unprovable within the current framework of mathematics, it becomes an axiom. The best known example of this is the “axiom of choice”.:https://en.wikipedia.org/wiki/Axiom_of_choice
Godels proof is not an example of something that is unprovable within the current framework of mathematics.
One example of something that is undecidable within the current framework of mathematics is the continuum hypothesis. But this can’t be axiomatised because either the set exists or it doesn’t exist.
KJW said:
Which statement are you saying cannot be true? If it is the statement that the barber shaves those who do not shave themselves, then it cannot be true because it is a paradox. It isn’t not a paradox because is cannot be true.
But it does.
A paradox is a statement that appears to state a logical possibility, but leads to contradictory statements which cannot both be true.
The statement that the barber shaves all, and only, those that do not shave themselves is clearly not a logical possibility, so it isn’t a paradox.
No more than “the black object is white” is a paradox.
By the way, the statement “Only the true paradox denies its own veracity” I’d put in the same category as the barber “paradox”.
There are many statements that deny their own veracity, that are not true paradoxes, so it is simply false.
Are paradoxes interesting?
dv said:
Are paradoxes interesting?
Some people think so.
The Rev Dodgson said:
dv said:
Are paradoxes interesting?
Some people think so.
And who am I to judge.
dv said:
The Rev Dodgson said:
dv said:
Are paradoxes interesting?
Some people think so.
And who am I to judge.
You are probably the best person in the Universe to judge what you find interesting.
The Rev Dodgson said:
transition said:
there’s a softer more practical sense of paradox, of which whatever involves somewhat (apparently) contradictory requirements.paradoxes are not necessarily or mostly an oblivion
I didn’t say there are no apparent paradoxes; just that there are no true paradoxes.
Of course their are true paradoxes, these men are both doctors and a pair
Cymek said:
The Rev Dodgson said:
transition said:
there’s a softer more practical sense of paradox, of which whatever involves somewhat (apparently) contradictory requirements.paradoxes are not necessarily or mostly an oblivion
I didn’t say there are no apparent paradoxes; just that there are no true paradoxes.
Of course their are true paradoxes, these men are both doctors and a pair
OK let me re-phrase.
There are no true strict logical paradoxes.
Yes, there are plenty of paradoxes in the sense of things that are true but make no sense.
But I’m not sure what is paradoxical about the two blokes you mentioned.
The Rev Dodgson said:
Cymek said:
The Rev Dodgson said:I didn’t say there are no apparent paradoxes; just that there are no true paradoxes.
Of course their are true paradoxes, these men are both doctors and a pair
OK let me re-phrase.
There are no true strict logical paradoxes.
Yes, there are plenty of paradoxes in the sense of things that are true but make no sense.
But I’m not sure what is paradoxical about the two blokes you mentioned.
a pairofdocs
Cymek said:
a pairofdocs
Groan.
:))
What about the saying “When we/I grow up we/I put away childish things”, which really should apply to religion as they are just childish stories with no proof but are taken as the truth and we wouldn’t accept this with anything else.
Also its from the bible which I was unaware until I just looked then which is even more funny as they don’t even get the irony in the statement
Reviewing this thread I see the second post had a circular discus on, and last has a pair o’ Docs.
We don’t seem to have made much progress.