Hah, I’ll let Decartes go (or condition him on a workable concept of existence—but that’s more of a spitball than the hardball I was going for).
But in answer to your non-contradiction question… I think I’d be epistemically entitled to just sneer and walk away. For one reason, again, if we’re in any conventional (i.e. not paraconsistent) logic, admitting any contradiction entails that I can prove any proposition to be true. And, giggle giggle, that includes the proposition “the law of non-contradiction is true.” (Isn’t logic a beautiful thing?) So if this mathematician thinks s/he can argue me into accepting the negation of the law of non-contradiction, and takes the further step of asserting any statement whatsoever to which it purportedly applies (i.e. some P, for which P&~P, such as the whiteness of snow), then lo and behold, I get the law of non-contradiction right back.
I suppose if we wanted to split hairs, we could say that one can deny the law of non-contradiction without further asserting an actual statement to which that denial applies—i.e. ~(~(P&~P)) doesn’t have to entail the existence of a statement P which is both true and false ((∃p)Np, where N stands for “is true and not true?” Abusing notation? Never!) But then what would be the point of denying the law?
(That being said, what I’d actually do is stop long enough to listen to the argument—but I don’t think that commits me to changing my zero probability. I’d listen to the argument solely in order to refute it.)
As for the very tiny credence in the negation of the law of non-contradiction (let’s just call it NNC), I wonder what the point would be, if it wouldn’t have any effect on any reasoning process EXCEPT that it would create weird glitches that you’d have to discard? It’s as if you deliberately loosened one of the spark plugs in your engine.
There are, apparently, certain Eastern philosophies that permit and even celebrate logical contradiction. To what extent this is metaphorical I couldn’t say, but I recently spoke to an adherent who quite firmly believed that a given statement could be both true and false. After some initial bewilderment, I verified that she wasn’t talking about statements that contained both true and false claims, or were informal and thus true or false under different interpretations, but actually meant what she’d originally seemed to mean.
I didn’t at first know how to argue such a basic axiom—it seemed like trying to talk a rock into consciousness—but on reflection, I became increasingly uncertain what her assertion would even mean. Does she, when she thinks “Hmm, this is both true and false” actually take any action different than I would? Does belief in NNC wrongly constrain some sensory anticipation? As Paul notes, need the law of non-contradiction hold when not making any actual assertions?
All this is to say that the matter which at first seemed very simple became confusing along a number of axes, and though Paul might call any one of these complaints “splitting hairs” (as would I), he would probably claim this with far less certainty than his original 100% confidence in NNC’s falsehood: That is, he might be more open-minded about a community of mathematicians explaining why actually some particular complaint isn’t splitting hairs at all and is highly important for some non-obvious reasons and due to some fundamental assumptions being confused it would be misleading to call NNC ‘false’.
But more simply, I think Paul may have failed to imagine how he would actually feel in the actual situation of a community of mathematicians telling him that he was wrong. Even more simply, I think we can extrapolate a broader mistake of people who are presented with the argument against infinite certainty replying with a particular thing they’re certain about, and claiming that they’re even more certain about their thing than the last person to try a similar argument. Maybe the correct general response to this is to just restate Eliezer’s reasoning about any 100% probability simply being in the reference class of other 100% probabilities, less than 100% of which are correct.
(Note: This comment is not really directed at Paul himself, seeing as he’s long gone, but at anyone who shares the sentiments he expresses in the above comment)
I think I’d be epistemically entitled to just sneer and walk away.
Note that there is almost certainly at least one person out there who is insane, drugged up, or otherwise cognitively impaired, who believes that the Law of Non-Contradiction is in fact false, is completely and intuitively convinced of this “fact”, and who would sneer at any mathematician who tried to convince him/her otherwise, before walking away. Do you in fact assign 100% probability to the hypothesis that you are not that drugged-up person?
Hah, I’ll let Decartes go (or condition him on a workable concept of existence—but that’s more of a spitball than the hardball I was going for).
But in answer to your non-contradiction question… I think I’d be epistemically entitled to just sneer and walk away. For one reason, again, if we’re in any conventional (i.e. not paraconsistent) logic, admitting any contradiction entails that I can prove any proposition to be true. And, giggle giggle, that includes the proposition “the law of non-contradiction is true.” (Isn’t logic a beautiful thing?) So if this mathematician thinks s/he can argue me into accepting the negation of the law of non-contradiction, and takes the further step of asserting any statement whatsoever to which it purportedly applies (i.e. some P, for which P&~P, such as the whiteness of snow), then lo and behold, I get the law of non-contradiction right back.
I suppose if we wanted to split hairs, we could say that one can deny the law of non-contradiction without further asserting an actual statement to which that denial applies—i.e. ~(~(P&~P)) doesn’t have to entail the existence of a statement P which is both true and false ((∃p)Np, where N stands for “is true and not true?” Abusing notation? Never!) But then what would be the point of denying the law?
(That being said, what I’d actually do is stop long enough to listen to the argument—but I don’t think that commits me to changing my zero probability. I’d listen to the argument solely in order to refute it.)
As for the very tiny credence in the negation of the law of non-contradiction (let’s just call it NNC), I wonder what the point would be, if it wouldn’t have any effect on any reasoning process EXCEPT that it would create weird glitches that you’d have to discard? It’s as if you deliberately loosened one of the spark plugs in your engine.
There are, apparently, certain Eastern philosophies that permit and even celebrate logical contradiction. To what extent this is metaphorical I couldn’t say, but I recently spoke to an adherent who quite firmly believed that a given statement could be both true and false. After some initial bewilderment, I verified that she wasn’t talking about statements that contained both true and false claims, or were informal and thus true or false under different interpretations, but actually meant what she’d originally seemed to mean.
I didn’t at first know how to argue such a basic axiom—it seemed like trying to talk a rock into consciousness—but on reflection, I became increasingly uncertain what her assertion would even mean. Does she, when she thinks “Hmm, this is both true and false” actually take any action different than I would? Does belief in NNC wrongly constrain some sensory anticipation? As Paul notes, need the law of non-contradiction hold when not making any actual assertions?
All this is to say that the matter which at first seemed very simple became confusing along a number of axes, and though Paul might call any one of these complaints “splitting hairs” (as would I), he would probably claim this with far less certainty than his original 100% confidence in NNC’s falsehood: That is, he might be more open-minded about a community of mathematicians explaining why actually some particular complaint isn’t splitting hairs at all and is highly important for some non-obvious reasons and due to some fundamental assumptions being confused it would be misleading to call NNC ‘false’.
But more simply, I think Paul may have failed to imagine how he would actually feel in the actual situation of a community of mathematicians telling him that he was wrong. Even more simply, I think we can extrapolate a broader mistake of people who are presented with the argument against infinite certainty replying with a particular thing they’re certain about, and claiming that they’re even more certain about their thing than the last person to try a similar argument. Maybe the correct general response to this is to just restate Eliezer’s reasoning about any 100% probability simply being in the reference class of other 100% probabilities, less than 100% of which are correct.
That would be Jain logic.
(Note: This comment is not really directed at Paul himself, seeing as he’s long gone, but at anyone who shares the sentiments he expresses in the above comment)
Note that there is almost certainly at least one person out there who is insane, drugged up, or otherwise cognitively impaired, who believes that the Law of Non-Contradiction is in fact false, is completely and intuitively convinced of this “fact”, and who would sneer at any mathematician who tried to convince him/her otherwise, before walking away. Do you in fact assign 100% probability to the hypothesis that you are not that drugged-up person?