‘From A follows A’ can be in two meta-stable situations; If A is false and if A is true. But typically circular logic is used to prove that A is true and ignores the other situation.
which I agree with. This can be contrasted with “From A does not follow not A”, which I believe entails A (as a false statement implies everything?).
When trying to prove a logical statement B from A, we generally have a sense of how much B resembles A, which we could interpret as a form of distance. Both A and not A resemble A very much.
I’ll formalise this mathematically in the following sense. The set of logical statements form a metric space where the metric is (lack of) resemblance. The metric is normalised so that A and not A lie in the unit ball centred at A, denoted by U. When reading a proof from premise A, we move from one point to the next, describing a walk.
Therefore, we might think of circular reasoning as an excursion (that is to say, a walk that returns to where it started) of logical statements. A proof that A implies not A is then an almost-excursion; it returns to U.
If there is reason to believe that (1) excursions and almost-excursions are comparably likely in some sense not defined (2) lots of excursions are recorded but not any almost-excursions, then this seems to lend evidence to the proposition that A does not imply not A i.e. A is true.
The issue is obviously that it’s not clear that (1) is true. However, intuitively, it does seem at least more likely to be true if conditioned on the walk becoming very distant from A before its return to U.
Avtur Chin writes:
which I agree with. This can be contrasted with “From A does not follow not A”, which I believe entails A (as a false statement implies everything?).
When trying to prove a logical statement B from A, we generally have a sense of how much B resembles A, which we could interpret as a form of distance. Both A and not A resemble A very much.
I’ll formalise this mathematically in the following sense. The set of logical statements form a metric space where the metric is (lack of) resemblance. The metric is normalised so that A and not A lie in the unit ball centred at A, denoted by U. When reading a proof from premise A, we move from one point to the next, describing a walk.
Therefore, we might think of circular reasoning as an excursion (that is to say, a walk that returns to where it started) of logical statements. A proof that A implies not A is then an almost-excursion; it returns to U.
If there is reason to believe that (1) excursions and almost-excursions are comparably likely in some sense not defined (2) lots of excursions are recorded but not any almost-excursions, then this seems to lend evidence to the proposition that A does not imply not A i.e. A is true.
The issue is obviously that it’s not clear that (1) is true. However, intuitively, it does seem at least more likely to be true if conditioned on the walk becoming very distant from A before its return to U.