2023

To any logician put there: Does not the Grue problem incorporate a contradiction (Green, then not green)? Thus would a deduction for Gruemess not be vulnerable to the Principle of Explosion where one can infer any conclusion from the green-not green contradiction? I'm no logician, just asking.

Oh you already said it. Nvm

Hmm, I like this idea. But aren't there continuous differentiable functions that would sort of look like this function. It would be low for a long time and then jump up quickly (though not instantly).

@@Oskar1000 It is that it has a singularity, a point at which it is not differentiable that I am underscoring because that is what grue appears to be for me. If you have an example of something 'gruelike' where it is continuously differentiable, that would be interesting. Do you have one in mind?

@@pmyou2 Well sine-waves are continously differentiable right. And sums of any number of sine-waves of different magnitudes would also be continously differentiable. And it's a known fact that you can combine sine-waves to produce all sorts of lines. This technique is used in compression of music (and images). So we could form something that would look like a line that's at 0 for a long time and then (almost) instantaneously jumps up to 1. It would take an insane amount of sine-waves but it should work.

No idea. Sounds like "Heeeeere's Johnny". Somebody must have been watching The Shining.

@@KaneB I thought the exact same thing!

Problem is, same thing applies to deduction. Just look at Curry's paradox.

@@DarrenMcStravick : Very few people take self-referential sentences seriously. Any sentence that begins "If this sentence is true..." is obvious silliness, so if that is what you have to do to twist deduction into "proving" what you want, then deduction is on pretty solid footing.

@Ansatz66 Curry's paradox isn't the only problem, you also have the problem of deduction, the logo-centric predicament, and the uninformativeness/triviality of logical truths. Also, how exactly does one not "take self-referential sentences seriously"? Do you have a reason for not permitting them into logical arguments?

@@DarrenMcStravick : The problem of deduction is just a case of what can happen when people fail to communicate due to having different interpretations for certain words. That problem is not really about deduction so much as it is a fundamental problem for all human communication, regardless of what people are trying to talk about. The logo-centric predicament is just a misunderstanding of the purpose of axioms. Axioms are not supposed to be claims to be justified; they are rules of discourse that are accepted to help facilitate communication, not because they represent real truths about the world. If X is an axiom, that means we treat X as being true in our discussion regardless of whether it is really true, just so that we have a common foundation upon which we can build. You do not need to believe a thing in order to assume it for the sake of discussion, and you do not need to justify a belief that you do not hold. Deductive truths may seem trivial and uninformative because you use them so pervasively and effortlessly. People mostly use deduction without even giving it a single conscious thought, but if we were somehow forbidden from using deduction then all communication would quickly fall apart. It seems trivial when we take it for granted, but we would miss it terribly if we did not have it. "How exactly does one not 'take self-referential sentences seriously'? Do you have a reason for not permitting them into logical arguments?" Not all sequences of words are meaningful, even if each word in the sequence is a well-accepted English word with clear meaning in isolation. For example, "Apples kite koala rustic amber." We might ask why this sequence of words has no meaning, but there is no good reason. It just so happens that the English speaking world has not collectively devised a meaning for it. Maybe someday in some slang dialect that sequence will become a meaningful sentence, but not today. That is why we exclude self-referential sentences from logical arguments, because they are usually meaningless.