Is "$X$, but $Y$" logically equivalent to "$X$ and $Y$"? While reading about Mathematical Logic in a book, I found the following,


  Conjunction. If $X$ is a statement and $Y$ is a statement, the
  statement "$X$ and $Y$" is true if $X$ and $Y$ are both true, and is false
  otherwise. ...
  
  ...Interestingly, the statement
   "$X$, but $Y$" is logically the same statement as "$X$ and $Y$", but
  they have different connotations (both statements affirm that $X$
  and $Y$ are both true, but the first version suggests that $X$ and
  $Y$ are in contrast to each other, while the second version suggests
  that $X$ and $Y$ support each other). Again, logic is about truth,
  not about connotations or suggestions.


It is the second part which I can't understand. Precisely my questions are,


How does the statement "$X$, but $Y$" is interpreted?
How does both statements affirm that $X$
and $Y$ are both true? 
What is meant by saying that, "$X$, but $Y$" suggests that $X$ and
$Y$ are in contrast to each other, and "$X$ and $Y$" support each other?
"Her judgement is firm, but fair" is both firm and fair with the connotation that judgement is not often both. &nbsp; The qualities are held in contrast.

"This ice-cream is soft and creamy" is both soft and creamy, with the connotation that this is better than ice-cream being either alone. &nbsp; The qualities reinforce each other.
Logically X and Y is the same as X but Y. There is no difference between them. There is no Objective difference. But psychologically, they influence your Subjective conclusion.

Let me bring some examples of X but Y:



"Fermat's claim was discovered after his death in the margin of a book, but he provided no proof"

X= "Fermat's claim was discovered after his death in the margin of a book"

Y= "He provided no proof"

Here, after hearing X, you expect Fermat to provide the proof too.

Z= "The proof is provided"

While, such idea is rejected by Y. 

X satisfies Z but Y contradicts Z.



"There is an elementary proof for this theorem but it involves millions of steps"

X="There is an elementary proof for this theorem"

Y="This proof involves millions of step"

Here after hearing X, you expect Z

Z="There is a basic proof for this theorem"

But, role of Y is to reject such idea. Again X satisfies Z and even make some people to expect Z and Y contradicts Z. Y comes to clarify it.



"Her judgement is firm, but fair"

X="Her judgement is firm"

Y="Her judgement is fair"

Z="She might have biased judgement"

X satisfies Z and Y rejects Z