# 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?

Suppose I introduce you to one of my friends, and tell you the following:

My friend here is French, but he speaks fluent Japanese.

If you later find out my friend is Dutch, you know I was lying. Likewise if you ask my friend an easy question in fluent Japanese and he isn't able to answer, you will know I was lying.

But if you have a conversation in Japanese with my friend, and in the course of that you find out that he is certainly French, then you will know I told the truth.

Those are exactly the kinds of outcomes that can happen if I told you

My friend here is French and he speaks fluent Japanese.

Logic only cares what makes statements are true or false, so it considers the two statements equivalent. Logic doesn't care about any feelings (such as surprise or amusement) that the statements may cause you to have.

"Her judgement is firm, but fair" is both firm and fair with the connotation that judgement is not often both.   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.   The qualities reinforce each other.

The are comments on this question in Frege if I am right.

The conjunction " but " has a pragmatic import; it can be translated as " I say $$A$$. Do not infer from this I do not say $$B$$. I also say $$B$$". The function of " but" is to prevent a mistake.

If one admits that "$$A$$ but $$B$$" is equivalent to "$$A \& \sim(A\to \sim B)$$", one can see it is extensionnally equivalent to

$$A\ \& \sim ( \sim A \vee \sim B)$$

$$A\ \&\ (A\ \&\sim\sim B)$$

$$A\ \& ( A\ \&\ B )$$

$$(A\ \&\ A) \& B$$

$$A\ \& B$$

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

• How does this answer my second and third question?
– user170039
Commented Feb 9, 2015 at 5:39
• Yes, logically X but Y means X and Y both are true. Commented Feb 9, 2015 at 5:40
• "$X$ but $Y$" doesn't require a third condition $Z$. Commented Feb 9, 2015 at 5:53
• @Théophile: So can "$X$, but $Y$ is true" be elaborated as "$X$ is true but $Y$ is also true"?
– user170039
Commented Feb 9, 2015 at 6:02
• Yes. Let me make it clear this way. Logically, always replace but with and. Commented Feb 9, 2015 at 6:10