hamad
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Post by hamad on Feb 11, 2018 17:43:29 GMT
Everyone knows the liars problem that says:
This statement is false.
where it's a self-contradictory statement or the other version that uses cards where it says
Front: The sentence on the other side of this card is TRUE.
Back: The sentence on the other side of this card is FALSE.
Trying to assign a truth value to either of them leads to a paradox.
If the first statement is true, then so is the second. But if the second statement is true, then the first statement is false. It follows that if the first statement is true, then the first statement is false.
If the first statement is false, then the second is false, too. But if the second statement is false, then the first statement is true. It follows that if the first statement is false, then the first statement is true.
How would you think of it in terms of mathematics?
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Post by Elizabeth on Feb 12, 2018 3:14:22 GMT
You're looking for a formula where one side equals produces a true answer while the other side does not? Hmm
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Mocha
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Post by Mocha on Feb 12, 2018 3:28:38 GMT
It's already pretty mathematical as-is. Alternatively, going the route Elizabeth suggested, you could say:
1=0
Where 1 is 'The other side is 1 (=boolean true)'
And 0 is 'The other side is 0 (=boolean false)'
The difference here is we can now evaluate the entire expression as false, since 0!=1.
A more mathematical equivalent to that kind of paradox is an indeterminate form.
For example, 0^0 is indeterminate. The limit of x^0 as x approaches 0 is 1, however, the limit of 0^x as x approaches 0 is 0. This makes 0^0 indeterminate - its value will depend on the construction of the limit. Other indeterminate forms include x/0, infinity - infinity, infinity * 0, etc.
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hamad
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Post by hamad on Feb 12, 2018 11:06:49 GMT
It's already pretty mathematical as-is. Alternatively, going the route Elizabeth suggested, you could say: 1=0 Where 1 is 'The other side is 1 (=boolean true)' And 0 is 'The other side is 0 (=boolean false)' The difference here is we can now evaluate the entire expression as false, since 0!=1. A more mathematical equivalent to that kind of paradox is an indeterminate form. For example, 0^0 is indeterminate. The limit of x^0 as x approaches 0 is 1, however, the limit of 0^x as x approaches 0 is 0. This makes 0^0 indeterminate - its value will depend on the construction of the limit. Other indeterminate forms include x/0, infinity - infinity, infinity * 0, etc. I don't think it's as simple as that. Let's say one side is B and another side is A A <=>~B B <=> A If B is true then A is true then ~B is true which is a contradiction Alternatively, if B is false it means A is false then ~B is false so B is true. Again, I don't understand what you mean by the entire expression is false, because if something is false then its opposite is true. What is the true opposite for that statement? |
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Mocha
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Post by Mocha on Feb 12, 2018 14:03:24 GMT
It's already pretty mathematical as-is. Alternatively, going the route Elizabeth suggested, you could say: 1=0 Where 1 is 'The other side is 1 (=boolean true)' And 0 is 'The other side is 0 (=boolean false)' The difference here is we can now evaluate the entire expression as false, since 0!=1. A more mathematical equivalent to that kind of paradox is an indeterminate form. For example, 0^0 is indeterminate. The limit of x^0 as x approaches 0 is 1, however, the limit of 0^x as x approaches 0 is 0. This makes 0^0 indeterminate - its value will depend on the construction of the limit. Other indeterminate forms include x/0, infinity - infinity, infinity * 0, etc. I don't think it's as simple as that. Let's say one side is B and another side is A A <=>~B B <=> A If B is true then A is true then ~B is true which is a contradiction Alternatively, if B is false it means A is false then ~B is false so B is true. Again, I don't understand what you mean by the entire expression is false, because if something is false then its opposite is true. What is the true opposite for that statement? The true opposite is 0!=1 >Again, I don't understand what you mean by the entire expression is false, Not the original question, but my terrible analogy. Also, I meant to say equality. >What is the true opposite for that statement? 0!=1. |
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hamad
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Post by hamad on Feb 12, 2018 17:23:06 GMT
0=1
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Deleted
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Post by Deleted on Feb 12, 2018 17:42:25 GMT
-1=1, here's how:
1= sqrt(1) = sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) = i X i = sqr(i) = -1
sqr - square
sqrt - square root
i - imaginary unit of complex number
Shrug
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hamad
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Post by hamad on Feb 12, 2018 17:53:45 GMT
-1=1, here's how: 1= sqrt(1) = sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) = i X i = sqr(i) = -1 sqr - square sqrt - square root i - imaginary unit of complex number  sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) , sqrt(a*b)=sqrt(a)*sqrt(b), unfortunately that's only true when argument is positive :P |
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Deleted
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Post by Deleted on Feb 12, 2018 17:56:58 GMT
-1=1, here's how: 1= sqrt(1) = sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) = i X i = sqr(i) = -1 sqr - square sqrt - square root i - imaginary unit of complex number sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) , sqrt(a*b)=sqrt(a)*sqrt(b), unfortunately that's only true when argument is positive :P I'm actually surprised there is another obsevant math person around here. Most people would stumble and take that equation for granted  |
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hamad
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Post by hamad on Feb 12, 2018 19:15:06 GMT
sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) , sqrt(a*b)=sqrt(a)*sqrt(b), unfortunately that's only true when argument is positive :P I'm actually surprised there is another obsevant math person around here. Most people would stumble and take that equation for granted  Well, being an electrical engineer kind of plays into that  |
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Mocha
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Post by Mocha on Feb 12, 2018 20:01:21 GMT
-1=1, here's how: 1= sqrt(1) = sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) = i X i = sqr(i) = -1 sqr - square sqrt - square root i - imaginary unit of complex number sqrt((-1)X(-1))= sqrt(-1) X sqrt(-1) , sqrt(a*b)=sqrt(a)*sqrt(b), unfortunately that's only true when argument is positive :P Dammit, beat me to it! xD |
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Deleted
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Post by Deleted on Feb 12, 2018 20:08:55 GMT
I'm actually surprised there is another obsevant math person around here. Most people would stumble and take that equation for granted Well, being an electrical engineer kind of plays into that Makes no difference, as I saw people with engineering background who failed to spot the error |
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Post by archlogician on Apr 20, 2018 0:22:45 GMT
From the perspective of classical logic it seems that we must reject this situation as describing an inconsistent theory and leave it at that. However, this is not terribly satisfying. By accepting the principle ex falso quodlibet, stating that from contradiction anything follows, classical logic is rendered completely unable to say anything interesting about inconsistent theories. However, given that we reason in the face of conflicting information, and this reasoning appears to be meaningfully capable of being judged rational or irrational, logicians have developed systems of logic rejecting ex falso quodlibet which allow for the analysis of these sorts of paradoxes. Such systems are known as paraconsistent logics. I highly recommend the article Paraconsistent Logic on the Stanford Encyclopedia of Philosophy for more on this approach. Paraconsistent logic is by no means a trivial subject, as it allows for exploration of a variety of self-contradictory mathematical objects. The book Inconsistent Mathematics by C. E. Mortensen covers this possibility in detail. |
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Post by Deleted on Apr 26, 2018 19:09:53 GMT
From the perspective of classical logic it seems that we must reject this situation as describing an inconsistent theory and leave it at that. However, this is not terribly satisfying. By accepting the principle ex falso quodlibet, stating that from contradiction anything follows, classical logic is rendered completely unable to say anything interesting about inconsistent theories. However, given that we reason in the face of conflicting information, and this reasoning appears to be meaningfully capable of being judged rational or irrational, logicians have developed systems of logic rejecting ex falso quodlibet which allow for the analysis of these sorts of paradoxes. Such systems are known as paraconsistent logics. I highly recommend the article Paraconsistent Logic on the Stanford Encyclopedia of Philosophy for more on this approach. Paraconsistent logic is by no means a trivial subject, as it allows for exploration of a variety of self-contradictory mathematical objects. The book Inconsistent Mathematics by C. E. Mortensen covers this possibility in detail. And also the stuff of logic that bases on not extensional, but intentional aspects. I don't know exactly, but it seems the theorems of Godel can be proved in paraconsistent logic too? |
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