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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"?

Will Sawin
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arsmath
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    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? – Will Brian Jan 19 '21 at 14:14
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    Maybe the place to look for a formalisation of surprise would be in the literature on the unexpected hanging paradox? – LSpice Jan 19 '21 at 14:43
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    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? – Andreas Blass Jan 19 '21 at 14:51
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    I’m voting to close this question because it’s is too philosophical. – Monroe Eskew Jan 19 '21 at 14:52
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    In some kind of probabilistic logic, one could do "$X$ and $Y$ and $P(X \textrm{ and }Y) < P(X) P(Y)$" where the probabilities are based on prior information... – Will Sawin Jan 19 '21 at 15:15
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    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. – Paul Taylor Jan 19 '21 at 19:45
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    Isn't there a difference at the level of well-formed sentences, rather than truth values? - I supported him and his brother supported him. - *I supported him but his brother supported him. – anemone Jan 19 '21 at 22:27
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    I think Aristotle wrote a bit about the logic of buts in his Posterior Analytic. (Sorry ...) – Noah Schweber Jan 20 '21 at 06:42
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    @PaulTaylor Whether or not it looks like an exam question is irrelevant. What is relevant that this is a question on the semantics of natural language and its philosophical interpretation, with hardly any mathematical content, and as such it is off topic for this site. It might be appropriate for https://philosophy.stackexchange.com . If and when someone figures out a formal system that describes such a logic, then questions about the mathematical properties of such a system may be on topic here. – Emil Jeřábek Jan 20 '21 at 07:37
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    The arguments to close don't make any sense to me. Either there is a mathematical system that distinguishes "and" from "but", or there isn't. This a question about mathematics. I don't care about the philosophical question, and I wouldn't be equipped to understand the philosophical answer. – arsmath Jan 20 '21 at 08:56
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    Whether there is a mathematical system that distinguishes “and” from “but” is (1) not a question of mathematics, but of semantics of natural language, and (2) subjective, so it’s certainly not “either there is or there isn’t”. If you think otherwise, here is a test case. I give you the system that only has constants 0 and 1 (no other connectives, propositions, or what not), where 0 represents “and”, and 1 represents “but”. Does it answer the question? If yes, how is it not trivial? If not, why not? Give me a purely mathematical reason that does not involve any semantics of natural language. – Emil Jeřábek Jan 20 '21 at 10:24
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    So you are saying that the meaning of mathematical concepts (which must be expressed in natural language) is off-topic for MathOverflow? I definitely don't agree with that. – arsmath Jan 20 '21 at 11:02
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    You are not asking about the meaning of a mathematical concept. You are asking about the meaning of a non-mathematical construct in order to model it by a mathematical structure. – Emil Jeřábek Jan 20 '21 at 11:07
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    I'm asking if there is a mathematical concept with a natural language meaning, which is just the reverse question of what is the natural language meaning of a mathematical concept. – arsmath Jan 20 '21 at 11:08
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    Actually, https://philosophy.stackexchange.com is not the best fit. I didn’t realize we have a dedicated site for linguistics (https://linguistics.stackexchange.com), and indeed, there are various questions on formal semantics there. This is your best chance to get a sensible answer from an expert that actually knows what they are doing, rather than the feeble amateurish attempts we’ve seen here. – Emil Jeřábek Jan 20 '21 at 12:13
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    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. – Andrés E. Caicedo Jan 20 '21 at 17:48
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    "All", "some", "probably", "almost certainly", "necessarily", "possibly", "always" and "eventually" are natural language, but they have been given formal mathematical meanings. (I mean the plural for each of them.) Maybe "but" could have a formal mathematical meaning too. – Paul Taylor Jan 20 '21 at 17:48
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    @MonroeEskew As Andres and Paul are suggesting, it's hard to rule out that mathematics might have some bearing on the matter, and much in logic that was once considered as "belonging to" philosophy has been mathematicized. The question is not a beginner question and merits consideration. – Todd Trimble Jan 20 '21 at 20:22
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    @EmilJeřábek Reading the first paragraph of the question, I had the same reaction your comments show: it sounded like a question of linguistics, not mathematics. But the second paragraph makes clear that it is a mathematical question — it’s asking if there are mathematical logics that can express this linguistic distinction. – Peter LeFanu Lumsdaine Jan 21 '21 at 00:04
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    wow, there is so much in the world that I fortunately or unfortunately have no idea about. Thanks for posting the question, answers, and all the comments. – TheVillageIdiot Jan 21 '21 at 05:54
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    It should be mentioned here that the word "but" is often used mathematically in a kind of strange way -- e.g. "$x = y$; but $y = z$; therefore $x = z$". This makes the most sense if one is arguing by contradiction, and arriving at the contradiction in the argument -- the "but" signals the strangeness of the arriving contradiction. But sometimes the usage creeps into arguments which are not by contradiction. It can give the writing a sense that the author is battling to prove to you something that you don't want to believe. – Tim Campion Jan 21 '21 at 15:09
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    A clearer understand of 'and' would help as a starting point. Mere conjunction doesn't suffice. I propose that dependent type theory fares better in Sec 2.3 of my recent book, https://global.oup.com/academic/product/modal-homotopy-type-theory-9780198853404. – David Corfield Feb 01 '21 at 09:36

5 Answers5

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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.)
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    How about, “the girl was young but blonde”? Counterexample? – Monroe Eskew Jan 20 '21 at 07:54
  • Or how about simply "you would think it is true but it is not"? – მამუკა ჯიბლაძე Jan 20 '21 at 07:58
  • @MonroeEskew, I think this analysis works well enough with that example. It would be unusual or awkward to say “the girl was young and blonde even though so many young girls have darker hair”, but if that’s what you’re saying then “young but blonde” is an appropriate summary. –  Jan 20 '21 at 08:00
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    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. – Monroe Eskew Jan 20 '21 at 08:14
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    @MattF. What is the $\diamond$ meaning? – Turbo Jan 20 '21 at 09:51
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    $\diamond$ is the standard symbol for possibility in modal logic: https://en.wikipedia.org/wiki/Modal_logic#Axiomatic_systems –  Jan 20 '21 at 10:22
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    @MonroeEskew "He was old but black-haired"? or "..young but grey" – David Roberts Jan 20 '21 at 11:03
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    "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. – Alex Kruckman Jan 20 '21 at 14:11
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    The addition of the third term is a great improvement over standard 'and' but for the reasons pointed out above the 'it would be possible' symbol is a bit too weak. In an ideal world I guess we would replace it by a symbol that says 'but conditioned on $X$, not $Y$ it is more probable than $Y$'. If only such a symbol existed... – Vincent Jan 20 '21 at 15:05
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    @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. –  Jan 20 '21 at 15:26
  • So, one example that confuses me involves the use of "not only" and "also". From an example at https://dictionary.cambridge.org/us/dictionary/english/but: She's not only a painter, but also a writer. Does this fit into the pattern still? I suspect it might but I can't quite cram it in there (although again, it could be the awkwardness of representing this in English throwing me off). – Jason C Jan 21 '21 at 00:03
  • Another confusing example, to me, is: Exclusive-or means that one or the other is true, but not both. Can this fit the given logical pattern? – Jason C Jan 21 '21 at 00:12
  • In thinking, maybe it is more accurate to say $\diamond(X \nRightarrow Y)$ rather than $\diamond(X\wedge\neg Y)$ ? That is, "there is a possibility that X does not imply Y", which is different than the current form in that it does not force consideration of $X \wedge \neg Y$ as a valid logical state. – Jason C Jan 21 '21 at 00:29
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    @JasonC, to your last three comments: 2) I think "but also" is best analyzed as a unit within the "not only" / "but also" pair, so explaining "but also" would be separate from this explanation of "but". 3) On my proposal, "We'll serve you coffee or tea, but not both" is roughly "We'll serve you coffee or tea, and we won't serve you both, even though some other places might serve you both". 4) The "material conditional" that is common in math differs so much from the conditionals of ordinary English that I found it easier not to bring conditionals into this at all, but you could try that too. –  Jan 21 '21 at 01:41
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    In fact "unexpectedness of $Y$ when $X$" can be, at least over S4, be expressed by something like $X\to\diamond(X\land\neg Y)$, which, again over S4, is implied by your "$X$ but $Y$". So maybe when not over S4, this has to be modified somehow? – მამუკა ჯიბლაძე Jan 21 '21 at 07:05
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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."

Timothy Chow
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  • 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. – arsmath Jan 21 '21 at 07:44
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    Nissim Francez's contrastive logic. – user5402 Jan 21 '21 at 13:09
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    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. – Timothy Chow Jan 22 '21 at 00:49
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    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. – Noah Schweber Jan 28 '21 at 01:36
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(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.
  • Adam has a car, but he is not able to join us because he does not have a driving license.
  • 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)

Jii
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    Ok, now I upvoted so you have enough rep to comment. Good answer! – KingLogic Jan 20 '21 at 17:24
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    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. – Jason C Jan 21 '21 at 00:25
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I'm not a mathematician, but I think normally, when you say "X but Y" you mean:

$$X \wedge Y \wedge P(Y|X)<P(\neg Y | X)$$

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.

Manfred Weis
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mousetail
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    I was going to suggest "$X$ and $Y$ and $P(Y|X)$ is small". (I think your version is equivalent to $P(Y|X) < .5$.) – Nik Weaver Jan 21 '21 at 11:34
  • End of a proof: "We thus reduced our statement to $Y$. But the latter is obvious." – მამუკა ჯიბლაძე Jan 21 '21 at 12:39
  • @NikWeaver Yes, that is the same thing. I wanted to avoid using constants but it doesn't make a difference. – mousetail Jan 21 '21 at 12:41
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    Or, say: "Ramanujan would be able to solve that, but he is dead". – მამუკა ჯიბლაძე Jan 21 '21 at 12:47
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    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, $P(Y)=1$). – მამუკა ჯიბლაძე Jan 21 '21 at 12:54
  • @მამუკაჯიბლაძე Perhaps I could add a corollary that if X is subjunctive, X but Y could mean $\neg Y \rightarrow X$ – mousetail Jan 21 '21 at 12:58
  • Well, it could be also something like $Y\land\Box(Y\to X)$ but I am not sure... – მამუკა ჯიბლაძე Jan 21 '21 at 13:18
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    Will Sawin made an earlier comment along similar lines. – Timothy Chow Jan 21 '21 at 13:44
  • Well, ordinary English words are often used in multiple inconsistent ways. I guess the point is that "$P(Y|X)$ is small" captures the intuition that $Y$ is not expected once $X$ is known. – Nik Weaver Jan 21 '21 at 13:46
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    @TimothyChow thanks for pointing that out, it is indeed similar. I think I like "$P(Y|X)$ is small" better than "$P(X \wedge Y) < P(X)P(Y)$" because the latter condition is symmetric in $X$ and $Y$, while "but" usually implies that the second clause is the unexpected one, it seems to me. – Nik Weaver Jan 21 '21 at 13:50
  • I agree that the linguistic aspects interfere and probably confuse me as a non-native speaker, but I still think that one important feature of "but" is that $Y$ witnesses inevitability of something crucial that somehow diminishes $X$, no? – მამუკა ჯიბლაძე Jan 21 '21 at 14:19
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    What about "I usually get chocolate, but this time I want strawberry? I don't sense any "inevitability" in this $Y$ ... – Nik Weaver Jan 21 '21 at 16:36
  • Another example would be sentence in your comment: "I agree ... but I still think ..." --- this $Y$ also seems to me not to have any flavor of inevitability. – Nik Weaver Jan 21 '21 at 16:38
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    Anyway it is fun to debate these questions. – Nik Weaver Jan 21 '21 at 16:39
  • Yes it's definitely fun. Inevitability was unfortunate term I must admit. But I really meant something very close in meaning to inevitability. Like, the fact that I want strawberry sort of inevitably undermines (this time) my usual habit of getting chocolate. Or, the fact that [I still think ...] sort of inevitably spoils my [agree]ment. – მამუკა ჯიბლაძე Jan 21 '21 at 17:59
  • @NikWeaver but I also tend more and more to agree with Emil that one needs professional linguist rather than a mathematician to argue about these things... – მამუკა ჯიბლაძე Jan 21 '21 at 18:08
  • Oh, I see what you mean. But still ... it was raining, but I decided to go out anyway. This Y doesn't seem to "undermine" X. – Nik Weaver Jan 21 '21 at 19:53
  • Counterexample: "Coin flipping may result in tails ten times in a row, but that would be a rare occurrence". I'd say the probability of "coin flips resulting in tails ten times in a row" being a "rare occurrence" is not smaller than the probability of it *not being a rare occurrence". – Will Jan 22 '21 at 03:48
  • In other words, constructions of the form "something could have happened, but it didn't!" do not indicate a probability, but rather juxtapose the narration of the speaker. I think there is an implicit expectation by the audience that any scenario described by the speaker relates to some reality (given the statistics of more usual contexts), so the subsequent "but" then "surprises" the audience (on some level) that this scenario did not happen. The probability here does not relate to the facts described, but rather to how they are described: a modality in a higher order of logic it seems. – Will Jan 22 '21 at 04:06
  • @NikWeaver Well, in a sense it does: raining in this context means a weather not suitable for going out. – მამუკა ჯიბლაძე Jan 22 '21 at 06:19
  • @მამუკაჯიბლაძე The relevant profession depends on what question one is asking. If one is asking for a taxonomy of different shades of meaning of a specific word in a specific language, then that is the job of a lexicographer. Such a study would be unlikely to be published in a linguistics journal unless the topic were broadened to multiple languages, or maybe multiple "contrastive words." But if the goal is to isolate a particular subset of meanings and capture them with reasonable fidelity using a mathematical model, then I'd look to either mathematics, logic, or the philosophy of language. – Timothy Chow Jan 22 '21 at 13:53
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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.

Corbin
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