A simple explanation of a flaw that makes no reference to Lob’s Theorem, meta-anythings, or anything complicated. Of course, spoilers.
“Let ◻Z stand for the proposition “Z is provable”. Löb’s Theorem shows that, whenever we have ((◻C)->C), we can prove C.”
This statement is the source of the problem. For ease of typing, I’m going to use C’ = We can prove C. What you have here is (C’->C)->C’. Using Material Implication we replace all structures of the from A->B with ~A or B (~ = negation). This gives:
~(~C’ or C) or C`
Using DeMorgan’s laws we have ~(A or B) = ~A and ~B, yielding:
(C’ and ~C) or C`
Thus the statement (C’ → C) → C’ evaluates to true ONLY when C’ is true. You then proceed to try and apply it where C’ is false. In other words, you have a false premise. Either you can in fact prove that 2=1, or it is not in fact the case that (C’ → C) → C’ when C is “2=1”.
PS: I didn’t actually need to read Lob’s Theorem or even know what it was about to find this flaw. I suspect the passage quoted is in fact not the result of Lob’s Theorem. You can probably dig into Lob’s Theorem to pinpoint why it is not the result, but meh.
If you “have ((◻C)->C)”, that is an assertion/assumption that ◻((◻C)->C). By Loeb’s theorem, It implies that ◻C. This is different from what you wrote, which claims that ((◻C)->C) implies ◻C.
Wow. I’ve never run into a text using “we have” as assuming something’s provability, rather than assuming its truth.
So the application of the deduction theorem is just plain wrong then? If what you actually get via Lob’s theorem is ◻((◻C)->C) ->◻C, then the deduction theorem does not give the claimed ((◻C)->C)->C, but instead gives ◻((◻C)->C)->C, from which the next inference does not follow.
I don’t think I’ve ever used a text that didn’t. “We have” is “we have as a theorem/premise”. In most cases this is an unimportant distinction to make, so you could be forgiven for not noticing, if no one ever mentioned why they were using a weird syntactic construction like that rather than plain English.
And yes, rereading the argument that does seem to be where it falls down. Though tbh, you should probably have checked your own assumptions before assuming that the question was wrong as stated.
A simple explanation of a flaw that makes no reference to Lob’s Theorem, meta-anythings, or anything complicated. Of course, spoilers.
“Let ◻Z stand for the proposition “Z is provable”. Löb’s Theorem shows that, whenever we have ((◻C)->C), we can prove C.”
This statement is the source of the problem. For ease of typing, I’m going to use C’ = We can prove C. What you have here is (C’->C)->C’. Using Material Implication we replace all structures of the from A->B with ~A or B (~ = negation). This gives:
~(~C’ or C) or C`
Using DeMorgan’s laws we have ~(A or B) = ~A and ~B, yielding:
(C’ and ~C) or C`
Thus the statement (C’ → C) → C’ evaluates to true ONLY when C’ is true. You then proceed to try and apply it where C’ is false. In other words, you have a false premise. Either you can in fact prove that 2=1, or it is not in fact the case that (C’ → C) → C’ when C is “2=1”.
PS: I didn’t actually need to read Lob’s Theorem or even know what it was about to find this flaw. I suspect the passage quoted is in fact not the result of Lob’s Theorem. You can probably dig into Lob’s Theorem to pinpoint why it is not the result, but meh.
You are wrong, and I will explain why.
If you “have ((◻C)->C)”, that is an assertion/assumption that ◻((◻C)->C). By Loeb’s theorem, It implies that ◻C. This is different from what you wrote, which claims that ((◻C)->C) implies ◻C.
Wow. I’ve never run into a text using “we have” as assuming something’s provability, rather than assuming its truth.
So the application of the deduction theorem is just plain wrong then? If what you actually get via Lob’s theorem is ◻((◻C)->C) ->◻C, then the deduction theorem does not give the claimed ((◻C)->C)->C, but instead gives ◻((◻C)->C)->C, from which the next inference does not follow.
I don’t think I’ve ever used a text that didn’t. “We have” is “we have as a theorem/premise”. In most cases this is an unimportant distinction to make, so you could be forgiven for not noticing, if no one ever mentioned why they were using a weird syntactic construction like that rather than plain English.
And yes, rereading the argument that does seem to be where it falls down. Though tbh, you should probably have checked your own assumptions before assuming that the question was wrong as stated.