I guess the result you’re referring to is that there is no TM whose output converges to 1 if its input is a true statement (according to standard semantics) in second order PA and 0 if it’s false. But don’t we need stronger results to conclude “we can’t do that either”? For example, take a human mathematician and consider his or her beliefs about “the standard numbers” as time goes to infinity (given unlimited computing power). Can you show that those beliefs can’t come closer to describing the standard numbers than some class of other mathematical structures?
Yes, you’re right. I was being too severe. We can’t “capture the meaning” probabilistically any better than we can with classical truth values if we assume that “capture the meaning” is best unpacked as “compute the correct value” or “converge to the correct value”. (We can converge to the right value in more cases than we can simply arrive at the right value, of course. Moving from crisp logic to probabilities doesn’t actually play a role in this improvement, although it seems better for other reasons to define the convergence with shifting probabilities rather than suddenly-switching nonmonotonic logic.)
The main question is: what other notion of “capture the meaning” is relevant here? What other properties are important, once accuracy is accounted for as best is possible? Or should we just settle for as much accuracy as we can get, and then forget about the rest of what “capture the meaning” seems to entail?
We know we can’t do that either, although we can do slightly better than we can with monotonic logic (see my response).
I guess the result you’re referring to is that there is no TM whose output converges to 1 if its input is a true statement (according to standard semantics) in second order PA and 0 if it’s false. But don’t we need stronger results to conclude “we can’t do that either”? For example, take a human mathematician and consider his or her beliefs about “the standard numbers” as time goes to infinity (given unlimited computing power). Can you show that those beliefs can’t come closer to describing the standard numbers than some class of other mathematical structures?
Yes, you’re right. I was being too severe. We can’t “capture the meaning” probabilistically any better than we can with classical truth values if we assume that “capture the meaning” is best unpacked as “compute the correct value” or “converge to the correct value”. (We can converge to the right value in more cases than we can simply arrive at the right value, of course. Moving from crisp logic to probabilities doesn’t actually play a role in this improvement, although it seems better for other reasons to define the convergence with shifting probabilities rather than suddenly-switching nonmonotonic logic.)
The main question is: what other notion of “capture the meaning” is relevant here? What other properties are important, once accuracy is accounted for as best is possible? Or should we just settle for as much accuracy as we can get, and then forget about the rest of what “capture the meaning” seems to entail?