Generally, I don’t think you should reject any possibility with 100% certainty unless you can demonstrate it’s contradictory, or inconsistent with current evidence. You can assign it low probability based on intuition and things like Occam’s razor, but not 0. Furthermore, we do have consistent models of the universe with infinities that are consistent with current evidence, so it really should be treated like a live possibility on this basis. Lots of domain experts (probably most physicists) think infinities are possible, too.
I’d go even further, since all evidence is in principle questionable, so maybe nothing gets ruled out with certainty other than logical impossibilities. However, any or every specific logical inference you’ve made could be mistaken, too, so you can’t really know with certainty that you’ve demonstrated that something is logically impossible. People make invalid inferences all the time, and in the worst case, you could be under a delusion constantly convincing you of invalid inferences, although this seems unlikely.
This is probably very controversial, but maybe even the rules of classical logic are questionable. However, I’d take the rules of logic as “working assumptions”, because I’m totally clueless/confused about what to do without them. Similarly if you doubt your specific logical inferences too much even for very basic ones when extremely careful and explicit. Similarly if you doubt your own experiences too much.
So, I would take the rules of logic for granted and treat them as if they’re true with certainty, and that’s basically it. Then I’d assign very high weight to my own experiences, simple logical inferences and basic standard probability theory (maybe 100% to probability theory, but I think there are competitors for dealing with uncertainty). I pretty strongly endorse Cromwell’s rule:
https://en.m.wikipedia.org/wiki/Cromwell’s_rule
Generally, I don’t think you should reject any possibility with 100% certainty unless you can demonstrate it’s contradictory, or inconsistent with current evidence.
I think this is a great principle for almost everyting, but does not apply to axioms. These cannot be falsified, and therefore are neither supported nor rejected by evidence. I do not even know what to think about the concept of probability without axiomatically accepting/rejecting some stuff. As far as I can tell, when we talk about probability of X, we are really talking about probability of X if a certain set of axioms (e.g. rules of logic) are true.
However, any or every specific logical inference you’ve made could be mistaken, too, so you can’t really know with certainty that you’ve demonstrated that something is logically impossible.
I agree. However, this does not seem to apply to axioms, whose acceptance does not depend on inference (otherwise they would be theorems).
This is probably very controversial, but maybe even the rules of classical logic are questionable. However, I’d take the rules of logic as “working assumptions”, because I’m totally clueless/confused about what to do without them.
Ok, I guess my rejection of infinities can also be interpreted as a working assumption. I understand there are tools to deal with infinities, but these also rely on their own definitions/axioms, so they would also require working assumptions (while adding complexity).
I pretty strongly endorse Cromwell’s rule
Me too. It is just that I consider infinities to be at the same level as logical impossibilities (e.g. A>B and A<B being true simultaneously). There will never be evidence for or against them, so I can just set my prior for their existence to 0.
Hi Michael,
What else can I use besides my strongest intuitions (which I guess can be called motivated reasoning) to know which axioms to accept/reject?
Generally, I don’t think you should reject any possibility with 100% certainty unless you can demonstrate it’s contradictory, or inconsistent with current evidence. You can assign it low probability based on intuition and things like Occam’s razor, but not 0. Furthermore, we do have consistent models of the universe with infinities that are consistent with current evidence, so it really should be treated like a live possibility on this basis. Lots of domain experts (probably most physicists) think infinities are possible, too.
I’d go even further, since all evidence is in principle questionable, so maybe nothing gets ruled out with certainty other than logical impossibilities. However, any or every specific logical inference you’ve made could be mistaken, too, so you can’t really know with certainty that you’ve demonstrated that something is logically impossible. People make invalid inferences all the time, and in the worst case, you could be under a delusion constantly convincing you of invalid inferences, although this seems unlikely.
This is probably very controversial, but maybe even the rules of classical logic are questionable. However, I’d take the rules of logic as “working assumptions”, because I’m totally clueless/confused about what to do without them. Similarly if you doubt your specific logical inferences too much even for very basic ones when extremely careful and explicit. Similarly if you doubt your own experiences too much.
So, I would take the rules of logic for granted and treat them as if they’re true with certainty, and that’s basically it. Then I’d assign very high weight to my own experiences, simple logical inferences and basic standard probability theory (maybe 100% to probability theory, but I think there are competitors for dealing with uncertainty). I pretty strongly endorse Cromwell’s rule: https://en.m.wikipedia.org/wiki/Cromwell’s_rule
I think this is a great principle for almost everyting, but does not apply to axioms. These cannot be falsified, and therefore are neither supported nor rejected by evidence. I do not even know what to think about the concept of probability without axiomatically accepting/rejecting some stuff. As far as I can tell, when we talk about probability of X, we are really talking about probability of X if a certain set of axioms (e.g. rules of logic) are true.
I agree. However, this does not seem to apply to axioms, whose acceptance does not depend on inference (otherwise they would be theorems).
Ok, I guess my rejection of infinities can also be interpreted as a working assumption. I understand there are tools to deal with infinities, but these also rely on their own definitions/axioms, so they would also require working assumptions (while adding complexity).
Me too. It is just that I consider infinities to be at the same level as logical impossibilities (e.g. A>B and A<B being true simultaneously). There will never be evidence for or against them, so I can just set my prior for their existence to 0.