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.
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.