- #1
AcidRainLiTE
- 90
- 2
I just started reading a book ("The Number Systems of Analysis" - by Little, Teo, and Brunt) which attempts to resolve the paradox resulting from the statement "this statement is false" by claiming that it is neither true nor false. However, I am a little confused because I don't think that solves the problem. Here's why:
(1) Let p = this statement is false.
(2) According to the book, p is not true and p is not false. Let's suppose the book is correct (I will show that we will get a contradiction).
(3) Let q = p is not false.
So, q is true (since, as stated in (2), we are assuming the book is correct).
(4) Let r = q is false.
(5) But, since we know that q is true, r is false.
(6) Now, it seems to me that r is equivalent to p. If this is true (which I will show in a minute), then we have a problem, because we said r is false and p is not false.
Here is why I think r is equivalent to p:
r = q is false (By definition from (4))
= "p is not false" is false (By substituting for q)
So r says that it is false that "p is not false". In other words, r says that it would be a lie to say "p is not false." Thus, r says that p is false. We thus have,
(7) r = p is false.
= "this statement is false" is false (by substituting for p)
But it turns out that p is equivalent to that statement. You can see this by substituting as follows:
p = this statement is false
= "this statement is false" is false." (by substituting p for 'this statement')
This is exactly what we got for r in (7).
So, we have that r = p.
But from (5), r is false and from (2), p is not false. Since r and p are equivalent, we have a contradiction.
I hope that made sense. I tried to state it clearly, but it is inevitably convoluted.
(1) Let p = this statement is false.
(2) According to the book, p is not true and p is not false. Let's suppose the book is correct (I will show that we will get a contradiction).
(3) Let q = p is not false.
So, q is true (since, as stated in (2), we are assuming the book is correct).
(4) Let r = q is false.
(5) But, since we know that q is true, r is false.
(6) Now, it seems to me that r is equivalent to p. If this is true (which I will show in a minute), then we have a problem, because we said r is false and p is not false.
Here is why I think r is equivalent to p:
r = q is false (By definition from (4))
= "p is not false" is false (By substituting for q)
So r says that it is false that "p is not false". In other words, r says that it would be a lie to say "p is not false." Thus, r says that p is false. We thus have,
(7) r = p is false.
= "this statement is false" is false (by substituting for p)
But it turns out that p is equivalent to that statement. You can see this by substituting as follows:
p = this statement is false
= "this statement is false" is false." (by substituting p for 'this statement')
This is exactly what we got for r in (7).
So, we have that r = p.
But from (5), r is false and from (2), p is not false. Since r and p are equivalent, we have a contradiction.
I hope that made sense. I tried to state it clearly, but it is inevitably convoluted.