The logical and psychological differences between the conjunctions "and" and "but"

The following question comes from Logic for Mathematicians by J. B. Rosser on page 17.

Exercise II.1.1 Write two short essays (not more than five sentences apiece) concerning the use of “and” and “but” as conjunctions between complete statements, telling:

(a) The logical difference between “and” and “but”.

(b) The psychological difference between “and” and “but”.

I am not too sure what to write, but I have wrote the following:

(a) Usually the statement to the right of the conjunction “but” is the negation of some statement; for instance, consider the conjunction “I like coffee but I do not like caffeine.” If we let `A` be the statement “I like coffee” and `B` be the statement “I like caffeine”, then the aforementioned conjunction is analogous to `A&~B`. So, one might conclude that the conjunction “but” is a special form of the conjunction “and” in which the statement to the right of the ampersand is negated. However, this a poor conclusion because we might have a conjunction as follows “I like snow but I also like warm weather”; if we let `A` be the statement “I like snow” and `B` be the statement “I like warm weather”, then the preceding conjunction is quite analogous to `A&B` which disagrees with our previous conclusion.

(b) Let `A` and `B`be some statements in English. Psychologically, in a sentence of the form “`A` but `B`” we might expect `B` to be contrary to `A`. For example, a student might say to his/her teacher “I stayed up all night working on my homework, but my dog ate it for breakfast”. Part `A` suggests that the student has homework to turn into the teacher, while `B` suggest the student does not have homework to turn in. In other word, psychologically we might expect the conjunction “but” to be succeeded by an excuse/explanation for the falsehood of the statement preceding the conjunction “but”. Additionally, the conjunction “but” is often followed by a statement which refers back to the statement preceding the conjunction. For instance, “I like dogs, but I like cats more” in which case “I like dogs” is a suffices to be a statement, but “I like cats more” is an ambiguous statement.

I am not satisfied with either of my responses, I would like a more definitive response. Thanks.

• Grice would say that 'but' conventionally implicates contrast, according to his theory of implicatures. You may wish to look at the relevant Stanford Encyclopedia of Philosophy article. Commented Dec 19, 2013 at 20:52

You raise a number of good observations about some common usages, but be careful about suggesting that any of them is a usual or typical usage. They strike me as common, but not more than that.

Logically, used as a connective between two sentences, "but" is simply a conjunction. It does not imply anything about the relationship of the meanings of the sentences — not even anything about negation, as you eventually conclude in part (a) — other than that they are both true.

Psychologically, or one might say pragmatically, "but" always implies contrast or exception.

• So, if we were to draw out a truth table we would find that `A&B` is only true when both `A` and `B` are true. However, if * represents the logical connective "but" then `A*B` would be everywhere true? Is this a precise meaning of "but"? Commented Dec 19, 2013 at 19:01
• They have the same truth table. "A&B" and "A*B" (or A-but-B?) are both true when both A and B are true, but false otherwise. Example: "You can have the car-keys, but it's too snowy to drive" means it's true you can have the car keys and it's also true that it's too snowy to drive. Commented Dec 19, 2013 at 20:02
• I did mean A-but-B. So A-and-B is logically equivalent to A-but-B? So the response to part (a) should be that there is no difference? Commented Dec 19, 2013 at 20:10
• Yes. Sorry if I was unclear about that! Commented Dec 19, 2013 at 21:11
• @ChrisopherE No problem, I see that it is implicit after a reading your answer more carefully. Thanks for your help. Commented Dec 19, 2013 at 21:22