# Metamathematics of buts

Something I learned (probably in middle school) that always bothered me is that the truth value of "and" and "but" are basically the same. If you were going to assign a truth-functional interpretation of "but" in first-order logic, it would be the same as "and".

There's been a explosion of logical systems that are alternatives to first-order logic, such as fuzzy logic. Is there a logical system that can distinguish "and" and "but"?

• It seems to me that the common usage of "but" is equivalent to "and" plus the expectation that the listener should be at least a little bit surprised. This expectation of surprise doesn't seem formalizable to me . . . but who knows? Jan 19, 2021 at 14:14
• There are modal logics that formalize notions of belief, and there are temporal logics that formalize the possibility of truth values changing over time. A suitable combination of these should be able to formalize "X but Y" as something like "X and Y and at some time in the past it was believed that at no time in the future (X and Y)." But (!) before attempting any formalization, we should try to agree on the intended meaning(s) of "but" in natural language. For example, is it a commutative operation on statements? Jan 19, 2021 at 14:51
• No, it shouldn't be closed just because it doesn't look like an exam question. Several settings have already been proposed and maybe in five year's time someone will stumble on this question and give a good formal answer. Jan 19, 2021 at 19:45
• I think Aristotle wrote a bit about the logic of buts in his Posterior Analytic. (Sorry ...) Jan 20, 2021 at 6:42
• To those saying this is out of scope: the MSC (Mathematics Subject Classification) 03B65 is for logic of natural languages. We have had a class for it for 40 years. Jan 20, 2021 at 17:48

Interpreting “$$X \text{ but } Y$$“ as $$X \wedge Y \wedge \diamond(X\wedge\neg Y)$$ is a reasonable starting point. (“X and Y and it would be possible to have X and not Y”.)

This works for the basic examples I found in online dictionaries:

• “He was poor but proud”
• “She’s 83 but she still goes swimming every day”
• “My brother went but I did not”
• “He stumbled but did not fall”
• “She fell but wasn’t hurt”

This correctly identifies that “he is a bachelor but unmarried” is not an appropriate use of “but”.

And this also shows the difference between such examples as:

• “That comment was harsh but fair.” (It was harsh and fair, while some comments are harsh and unfair.)
• “That comment was fair but harsh.” (It was fair and harsh, while some comments are fair and compassionate.)
• How about, “the girl was young but blonde”? Counterexample? Jan 20, 2021 at 7:54
• I'm saying that "young but blonde" sounds wrong. I would say it's because being blonde doesn't violate expectations after learning about being young. Yet it's possible to be young and not blonde. Jan 20, 2021 at 8:14
• $\diamond$ is the standard symbol for possibility in modal logic: en.wikipedia.org/wiki/Modal_logic#Axiomatic_systems
– user44143
Jan 20, 2021 at 10:22
• "It's cloudy but it's raining" sounds wrong, while "it's cloudy but it's not raining" sounds right. Since it's possible for it to be either raining or not when it's cloudy, I don't think your suggested interpretation gets at the difference between these two examples. Jan 20, 2021 at 14:11
• @AlexKruckman, that’s a good example. Maybe the conclusion is that $A\wedge B$ is the best interpretation of “A but B” in ordinary propositional logic; the above (or perhaps $A\wedge B \wedge \neg\square (A\to B)$) is the best interpretation in propositional modal logic; and there probably other logics that are even more faithful to the ordinary usage.
– user44143
Jan 20, 2021 at 15:26

In a paper entitled "Contrastive Logic" (Logic Journal of the IGPL 3 (1995), 725–744), Nissim Francez introduced something he called bilogics, which are logics intepreted over a pair of structures instead of a single structure, in order to study words such as but and already. The idea in the case of but is that one must simultaneously consider two states of affairs, namely the actual state of affairs and the "expected" state of affairs. A later paper by J.-J. Ch. Meyer and W. van der Hoek, A modal contrastive logic: The logic of ‘but’ (Ann. Math. Artif. Intell. 17 (1996), 291–313) showed how more or less the same idea could be captured using an extension of the well-known modal logic S5, which provides a framework for analyzing possible worlds.

There is a small literature on related topics that you can find by searching for "contrastive reasoning."

• This is great. I was just thinking about Matt F.'s answer, and wondering if you could sharpen it up by developing a model logic of "expected", when I saw your answer pointing to papers that develop exactly that idea. Jan 21, 2021 at 7:44
• Jan 21, 2021 at 13:09
• Incidentally, the comment by LSpice that looking at the literature on the unexpected hanging paradox might be relevant was correct. The paper by J.-J. Ch. Meyer and W. van der Hoek does discuss that paradox. In fact, searching for papers on the paradox is how I first came across their paper many years ago. Jan 22, 2021 at 0:49
• The Francez and Meyer/van der Hoek papers are exactly the sources Humberstone gives re: mathematical treatments of "but" (The Connectives, page 676) in contrast with more linguistic or philosophical sources. Humberstone's tome is my go-to for references of this kind, so I suspect that this indicates a meaningful sparsity of the literature. Jan 28, 2021 at 1:36

(not enough reputation to comment)

I would understand "but" to introduce some unexpected consequences (against implicit assumptions) of the truth value of a claim, or that there are some other elements that effect the truth value of the claim at some point. For example

• It is windy outside, but laundry will not dry faster because it will rain soon.
• Lisa is not able to join us, but we can discuss with her over video call.

I dont't see "but" to be equal to "and" in first-order logic. It is more on the structure of a sentence and hidden assumptions that could be translated to "but" in textual presentation. I would guess (perhaps ignorantly) that no logic can define "but" because it refers to something we would not expect and hence unknown. (this is something that might carry some cultural differences, too)

• Ok, now I upvoted so you have enough rep to comment. Good answer! Jan 20, 2021 at 17:24
• I feel like there is a strong argument for "but" and "and" being equivalent in first-order logic, but not in higher order logic: In language, the difference is that "X but Y" implies that there is either a) noteworthy probability of $X \land \neg Y$ or b) contextual relevance to $X \nRightarrow Y$ which isn't communicated with "and". That is: There is significance to $X \nRightarrow Y$. The argument is that "X but Y" implies some extra semantics over "X and Y", but it doesn't add anything that is representable in first-order logic. Jan 21, 2021 at 0:25

I'm not a mathematician, but I think normally, when you say "X but Y" you mean:

$$X \wedge Y \wedge P(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 X(Y|X).

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.

• I was going to suggest "$X$ and $Y$ and $P(Y|X)$ is small". (I think your version is equivalent to $P(Y|X) < .5$.) Jan 21, 2021 at 11:34
• End of a proof: "We thus reduced our statement to $Y$. But the latter is obvious." Jan 21, 2021 at 12:39
• Or, say: "Ramanujan would be able to solve that, but he is dead". Jan 21, 2021 at 12:47
• Or, even: "I wish I would not have done that, but you cannot alter the past". In this example, $P(Y|X)=1$ (in fact, $P(Y)=1$). Jan 21, 2021 at 12:54
• Will Sawin made an earlier comment along similar lines. Jan 21, 2021 at 13:44

In Lojban, a constructed language meant to embody logical thinking, the distinction between "and" and "but" is made with two different sorts of words which have different grammar. Conjunctions are made by simply uttering multiple propositions in a row, but each proposition can be tagged with a non-logical modifier which annotates it relative to prior propositions.

As explained in Complete Lojban Language, the discursive particle {ku'i} tags a proposition as contrary to the preceding proposition. This gives a way to annotate "but", but without changing what is logically asserted. Similarly, other particles give ways to translate "similarly" or "in parallel".

While Lojban does not directly correspond to a second-order formal logic (yet), there are tools like tersmu which can extract logical sentences, and these tools discard discursive annotations.

This answer isn't worth accepting, but it was too long for a comment.