# What is the logic of the coordinating conjunction BUT?

What is the logical difference between "and" and "but"?

Is it really possible to oppose "and" and "but" in this sense?

Is it really possible to reduce the logic of "but" to the logic of "and"?

So my question is:

What is the logic of "but"?

Did any academic author ever offered a logical account of the coordinating conjunction "but" distinct from that of "and"?

• The propositional logic of "but" is the logic of "and". The difference is intensional, "but" indicates contrast, as in "both A and B are red" vs "A is red but B is green". For a formalization, see Meyer-Hoek, A modal contrastive logic: The logic of ‘but’:"We use a simple modal logic, which is an extension of the well-known S5 logic, and base the contrastive operators proposed by Francez in [2] on the basic modalities... We thus obtain a logic for contrastive operators that is more in accord with the tradition of intensional logic..." Commented Sep 13, 2023 at 8:54
• But (sic!) see Lloyd Humberstone, The Connectives (MIT Press, 2011), page 674: "Mill [1843], Book I, Chapter IV, §3, gives the ‘contrast’ formulation of what but involves, saying that “Caesar is dead but Brutus is alive” is tantamount to the fourfold assertion: (i) Caesar is dead; (ii) Brutus is alive; (iii) It is desirable that (i) and (ii) should be thought of together; (iv) between (i) and (ii) there exists a contrast." Commented Sep 13, 2023 at 8:57
• @MauroALLEGRANZA All your conditions (i)-(iv) are symmetric with respect to swapping the two statements. According to that, "Caesar is dead but Brutus is alive" would mean the same as "Brutus is alive but Caesar is dead." But there is clearly a difference between the two sentences. So your conditions (i)-(iv) are missing something. The missing condition is that in "A but B," B is stated as more relevant the point being made, than A. Commented Sep 13, 2023 at 15:53
• If you are satisfied with one of the received answers, please accept it. Commented Sep 14, 2023 at 6:51
• @MauroALLEGRANZA "please accept it" Sorry, I can't reasonably do that. Commented Sep 14, 2023 at 16:31

If you had no idea of what the word "but" means and would just identify it as a conjunction, given that it rests between 2 propositions and look at the resulting truth table then "A BUT B" would be true if A and B are both true and false otherwise. Which is functionally equivalent to "A AND B".

When it comes to how it is used in natural language it seems to be used to (preemptively) contradict a stated or more often unstated assumption.

So take the example from the original question:

“I like coffee but I do not like caffeine.

if you instead said "I like coffee" then people might assume. "They like coffee? It's quite bitter and most people just like it because of the stimulating effect. They probably like it because of the stimulating effect".

“I like snow but I also like warm weather”

Again if you just said "I like snow", then people would assume. "They like snow so they probably like winter, coldness, snowball fights [other snow related things] or because snow is quite extreme might dislike [not snow related things]".

“I stayed up all night working on my homework, but my dog ate it for breakfast”

"They stayed up all night working on their homework? So they probably have made some progress and are ready to turn it in"

“I like dogs, but I like cats more”

"They like dogs? So it's probably their favorite animal."

"Caesar is dead but Brutus is alive"

"Ceasar is dead? Then he was probably ambushed and his fellow Brutus, who was with him, is probably also dead.

Where A on it's own is rather something like: A -> ◊B, so from A follows it is possible that B.

So BUT could be an operator applied to A -> ◊B that transforms it into an AND.

So BUT(A->◊B) -> A AND/BUT not B

or conversely A BUT not B as opposed to A AND not B also implies that there is a connection of A -> ◊B and that it's not just a list of required features.

• "A but B" can interpreted in terms of what supports or detracts from a point P that we're trying to make. We could say "A but B" if: (1) A and B are both true. (2) A at least slightly detracts from P (provides partial evidence reducing the support for P, however slim). (3) B supports P. I was going to make another answer saying this and showing how it holds for each of your examples, but I realized I would only be repeating what you said. Commented Sep 13, 2023 at 18:49
• @haxor789 This is interesting, but is it your personal two-cents or is this lifted without reference from some academic authors? And in this case which ones? Given the approximation, it seems it is just your personal opinion. Commented Sep 14, 2023 at 5:31
• @Speakpigeon Unfortunately it's just an opinion Commented Sep 14, 2023 at 6:44

To my rexkoning, it boils down to matching natural language connectives to logical connectives. In classical logic there are only 4 connectives (AND, OR, IMPLICATION, BIIMPLICATION). To which of these 4 connectives should we match BUT to? AND seems the least unreasonable choice.

In addition, as Conifold mentions, BUT is a contrasting conmective and BUT = AND + INFERENCE (that the conjuncts are somwhow incompatible). In other words, you may not use BUT unless you have an argument that demonstrates the logical inconsistency between the conjuncts. So if I say, Angela is good BUT Shannon is bad, I mean Angela is good AND Shannon is bad AND This state of affairs is somehow odd

In conclusion if BUT = !& then ...
p !& q = (p & q) & (q = !p) [worst-case scenario, p !& q = p & !p (contradiction)

• "classical logic" A non-starter. I asked What is the logic of "but"?. So-called "classical logic" is neither classical nor logic. 2. "logical inconsistency" There is no logical inconsistency between "Angela is good" and "Shannon is bad". Commented Sep 14, 2023 at 16:28
• On point. Poor example, but I'm sure you can come up with a better instance. Commented Sep 15, 2023 at 1:21

When you go from natural language to formal language, you lose nuance and gain precision. In natural language, "but" and "and" have distinctively different usages, you wouldn't typically substitute one for another.

In terms of the mapping they cast on the more common formal logical languages, however, both reduce to AND. That means that the portion of the meaning of "but" that readily translates into logical form is "AND."

It's possible to create more elaborated formal languages that capture more of the meaning and nuance of natural language, but there isn't always much to gain from it. "But" is more a way of highlighting a particular fact about the world than a change in the content of that fact.

• "lose nuance and gain precision" So you lose exactitude. Something which is precise but can't show the time is not a clock. 2. "mapping they cast" They certainly don't. Rather, formal languages postulate their own interpretations. 3. "logical form is "AND."" I don't think we know this is true. This is what is done, mostly, but not necessarily what should be done. 4. "there isn't always much to gain from it" I don't think you know this. 5. "change in the content of that fact" A true sentence does not change the content of the fact of which it is true. Commented Sep 14, 2023 at 16:19