I have recently read a lot of proofs that like to say "But..." right before the punchline. I feel that the word "But..." should be used if what follows is contradictory in some way, as in proofs by contradiction. However, many proofs that don't use contradiction simply use "But..." to introduce the punchline (the example that inspired this post is at the end of this proof of the Alternating Series Test. Is this considered acceptable form? It's so common but sounds strange to me.
I think that "but" can also be used to exemplify or to provide a unintuitive result (consequence) or more evidence that was not immediately obvious. For example:
claim a evidence a some more evidence.... but.. since we know that from evidence a... then we have theorem b!
claim a evidence a claim b ..but notice that... x = y because of evidence a and claim a as assumed so... claim b is true!
I also think that such usage is acceptable and is not necessarily required to be an introduction to a contradiction.
A common use seems to be
We have situation X.
We also have additional feature Y, that causes the situation X to assume a much simpler or different-looking form Z.
Z is different enough from X to call attention to the contrast between them with words like "but", "however", "actually", or "in fact".
The ability to draw such contrasts is useful even when Z is a special case of X and not in contradiction to anything.
Generally, if the initial idea of the proof changes as the narrative develops, maybe to handle some complication or special cases, some words like "but" or "however" may appear when indicating that.
In English, the word "but" can function two different ways:
- As "not this, but that", it signifies an exclusive or.
- As "not only this, but also that", it signifies a conjunction.
The word "but" by itself is often an abbreviated version of one of those two. In the usage you ask about, the latter is meant.