My question involves the word "same" apropos to identity vs. comparison, especially to the base case of a particular induction proof. I was trying to find the flaw in the induction for: "P = All horses are the same color. Base Case: P(1) = One horse is the same color as itself. Induction: We have n+1 horses. Take any one away, and the rest will be the same (because of P(n)). Since it didn't matter which we took away, all horses must be the same." I posit that the flaw in this proof isn't simply the lack of "sameness" overlap in the P(2) instance, but in the choice of base case and use of the word "same." I say that there needs to be a comparison (i.e., 2 or more unique objects) in the base case to use the word "same" as it is in "P".
If I say Horse A is the same color as Horse A, and you say Horse A is the same color as Horse B, are we really using the word "same" in the same sense? If not, doesn't it follow that it's better not to use them interchangeably in an induction proof such as the one above? If so, am I just naively over analyzing something that's actually very obvious and elementary?