Relevantly, because it’s structurally identical to Warrigal’s sample sentence, so whatever definition Warrigal is using (a perfectly standard one, it seems to me) must apply to both.
It’s structurally identical to a sample sentence that Warrigal used in describing a different approach, not the one he/she is taking.
If it manifestly has a discontinuity, you should be able to say where it is.
(In fact, it does not have a discontinuity. For not(S) = 1-S, it is completely linear: it is the iterated system f(S) = 1-f(S), having a fixed point at .5. )
Okay, this is getting silly. Warrigal says “The sentence ‘this sentence’s truth value is less than 0.5’ has a sharp jump in truth value at 0.5, but the sentence ‘this sentence’s truth value is significantly less than 0.5’ does not [and we will ban the first form]”. In the same way, my sentence “This sentence’s truth value is less than epsilon” has a discontinuity at epsilon. Both sentences make discontinuous claims about their truth values.
What is the “different approach” that you claim this sentence is in reference to?
(Incidentally, I agree with you that my sentence has a fixed point at 0.5 under Warrigal’s system. That’s why my original comment was criticizing the presentation and not necessarily the content of the theory.)
The sentence we were discussing was “This statement has truth value 0”. I assumed that when you said it was structurally identical to Warrigal’s sample sentence, you were referring to this passage:
“This sentence is false” is not a valid sentence there, because it refers to itself, but no ordinal number is less than itself.
That sentence refers to the traditional ways around Russell’s paradox.
You seem to say discontinuity when you mean a noncontinuous first derivative.
Why does it have a discontinuity?
Folks, you shouldn’t vote down legitimate questions.
Relevantly, because it’s structurally identical to Warrigal’s sample sentence, so whatever definition Warrigal is using (a perfectly standard one, it seems to me) must apply to both.
It’s structurally identical to a sample sentence that Warrigal used in describing a different approach, not the one he/she is taking.
If it manifestly has a discontinuity, you should be able to say where it is.
(In fact, it does not have a discontinuity. For not(S) = 1-S, it is completely linear: it is the iterated system f(S) = 1-f(S), having a fixed point at .5. )
Okay, this is getting silly. Warrigal says “The sentence ‘this sentence’s truth value is less than 0.5’ has a sharp jump in truth value at 0.5, but the sentence ‘this sentence’s truth value is significantly less than 0.5’ does not [and we will ban the first form]”. In the same way, my sentence “This sentence’s truth value is less than epsilon” has a discontinuity at epsilon. Both sentences make discontinuous claims about their truth values.
What is the “different approach” that you claim this sentence is in reference to?
(Incidentally, I agree with you that my sentence has a fixed point at 0.5 under Warrigal’s system. That’s why my original comment was criticizing the presentation and not necessarily the content of the theory.)
The sentence we were discussing was “This statement has truth value 0”. I assumed that when you said it was structurally identical to Warrigal’s sample sentence, you were referring to this passage:
That sentence refers to the traditional ways around Russell’s paradox.
You seem to say discontinuity when you mean a noncontinuous first derivative.