I'm not a mathematician, but I think normally, when you say "X but Y" you mean:
$$X \wedge Y \wedge P(Y|X)<P(\neg Y | X)$$
As in, X and Y is true, but the probability of Y is low given X.
This works with the examples too:
- Alice was proud but poor - Most poor people are not proud, and Alice is both proud and poor
- My brother went but I did not - Most of the time I go where my brother goes, however this time I did not
In these cases, stating the probability is often an important part of the statement. In "My brother went but I did not" stating that I usually go with my brother is an important part of what the author is trying to communicate.
In some cases, Y is not special because of X, but because of something else implied by X, like even stating X itself. Consider the case:
- Thus we can conclude Y, but this is obvious. - "I am telling you Y. This means there is a high chance Y is important. However, Y is not important since it is obvious"
Now we get into high-level meta reasoning where we have to include the probability of the author saying X when computing
There is another special case when X is subjunctive:
- I would have saved her, but I could not - "If I could have saved her I would"
In this case you can replace "but" with "if not" with the same meaning.