I am no expert in formal logic, so please forgive me if this question sounds stupid. It's about a common pattern used in many mathematical proofs.(adsbygoogle = window.adsbygoogle || []).push({});

For me it' "obvious" or "trivial" - but I can't prove it.

For a friend of mine it's far from obvious or even wrong - but I don't get his point and I am quite sure that he does not really understand mathematical methods at all ;-)

Let's make a simple example: suppose there's a natural number n and a statement A(n) like "a natural number n is always either even or odd". In many cases there's a proof which does not use a specific property or value of n, so the proof is valid for all n and we conclude that "∀n∈N : A(n)". Now my friend is saying that such a proof is never valid for all numbers n, but only for one single but unspecified number (which I think is nonsense); so he is saying that the "∀n" is wrong (which is think is hogwash).

The common pattern in such proofs is the following: we have a statement A(n) and a proof P which does not use a specific property or value of n. So we could say

"P → ∃n∈N : A(n)" ∧ "P does not use a specific property or value of n" → "∀n∈N : A(n)"

Again: for me it sounds strange; if I have a proof which works for all n then I immediately conclude that "∀n" is right.

Anyway - let me ask the question whether my conclusion is really obvious. Or if one needs a proof for the above mentioned pattern, and how it looks like.

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# A question about a common pattern in mathematical proofs

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