4 ms·

This post is basically a paraphrase of a well-known example, which has its own Wikipedia page https://en.m.wikipedia.org/wiki/All_horses_are_the_same_color http

This post is basically a paraphrase of a well-known example, which has its own Wikipedia page https://en.m.wikipedia.org/wiki/All_horses_are_the_same_color https://en.m.wikipedia.org/wiki/All_horses_are_the_same_colo... I think the fallacy is explained better there tbh. I also think it’s confusing to say one isn’t the base case. When you apply the inductive step there you hit N=2, and indeed the step fails because there’s no overlap.

tux3 5y ago
This is indeed much clearer than the OP's version. In this one, the transitivity is obvious, because the article clearly explains that you may take out any horse from n + 1 and note that the remaining n have the same color, then do that observation again with another one, and use the horses that never left the group as a transitive link. I agree this is much more worth looking at.
wruza 5y ago
(wiki) Thus in any group of horses, all horses must be the same color. Yes, iff n horses have the same color. I fail to see any paradox here (and in subj tfa too) and why induction is of any importance to it. We worked it all out of a restricting assumption which wasn’t contradicted, but that’s it. It doesn’t tell us anything about all other cases, e.g. when n horses do not have the same color. Can someone please explain what I’m missing here?
- dsizzle 5y ago
  The inductive step allows you to stipulate what seems like a strong assumption, in this case that all horses in sets of size N have the same color. I think you're reacting to that seeming like it's the problem, because at first that seems like where the "cheating" is occuring: you can obviously have a set of horses that aren't all the same color. But actually that's part the power of the inductive step: you do have the freedom to choose whatever assumptions you want. The question is whether it's true for N+1, for all N. In this case, it's not: the inductive step fails. But there's nothing flawed with the starting premise per se (aside from it being intuitively obvious that it will never work). The interest here is that it seems you can get further than you should be able to, and it's a little tricky to figure out where the logic falls apart.
- Dylan16807 5y ago
  What you're missing is that "n horses have the same color" is true for n=1. So therefore we should be able to invoke the induction to bigger and bigger numbers. If the induction actually was valid for 1->2, we'd have a hell of a problem.
  - wruza 5y ago
    Now I think I understand the subtlety, thanks! The induction part is important here because we start from n=1 (where the assumption is not as obviously questionable as for n in general, because it’s coincidentally/degeneratively true in relation to our reasoning) and then the “base error” allows us to generalize into a bigger mistake.
- lambdatronics 5y ago
  Yeah, it's phrased weird. In the inductive step, what you want to do is prove that "if all sets of horses with size n have the same color, then all sets of horses with size n+1 have the same color." In order to prove that, you start with the assumption that all sets of horses with size n have the same color. They're not actually assuming it's true for all n right away, they're just assuming that they've gotten it for some n.

tux3 5y ago

This is indeed much clearer than the OP's version. In this one, the transitivity is obvious, because the article clearly explains that you may take out any horse from n + 1 and note that the remaining n have the same color, then do that observation again with another one, and use the horses that never left the group as a transitive link. I agree this is much more worth looking at.

wruza 5y ago

(wiki) Thus in any group of horses, all horses must be the same color. Yes, iff n horses have the same color. I fail to see any paradox here (and in subj tfa too) and why induction is of any importance to it. We worked it all out of a restricting assumption which wasn’t contradicted, but that’s it. It doesn’t tell us anything about all other cases, e.g. when n horses do not have the same color. Can someone please explain what I’m missing here?

dsizzle 5y ago
The inductive step allows you to stipulate what seems like a strong assumption, in this case that all horses in sets of size N have the same color. I think you're reacting to that seeming like it's the problem, because at first that seems like where the "cheating" is occuring: you can obviously have a set of horses that aren't all the same color. But actually that's part the power of the inductive step: you do have the freedom to choose whatever assumptions you want. The question is whether it's true for N+1, for all N. In this case, it's not: the inductive step fails. But there's nothing flawed with the starting premise per se (aside from it being intuitively obvious that it will never work). The interest here is that it seems you can get further than you should be able to, and it's a little tricky to figure out where the logic falls apart.
Dylan16807 5y ago
What you're missing is that "n horses have the same color" is true for n=1. So therefore we should be able to invoke the induction to bigger and bigger numbers. If the induction actually was valid for 1->2, we'd have a hell of a problem.
- wruza 5y ago
  Now I think I understand the subtlety, thanks! The induction part is important here because we start from n=1 (where the assumption is not as obviously questionable as for n in general, because it’s coincidentally/degeneratively true in relation to our reasoning) and then the “base error” allows us to generalize into a bigger mistake.
lambdatronics 5y ago
Yeah, it's phrased weird. In the inductive step, what you want to do is prove that "if all sets of horses with size n have the same color, then all sets of horses with size n+1 have the same color." In order to prove that, you start with the assumption that all sets of horses with size n have the same color. They're not actually assuming it's true for all n right away, they're just assuming that they've gotten it for some n.