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Many times in class, we are asked to answer, "What is 2+2?" or "What is the derivative of the function x?". It would not be the intended answer to write "2+2" or "The derivative of the function x". But, why not? A tautology is technically a correct answer. Is there some formal definition of an "expression", that can distinguish between two different "expressions" that refer to the same "object"? I would like to know if other philosophers have thought of this problem, and which if any references apply.

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    What you are looking for is the idea behind the technical term ground term of rewrite systems. Alternatively normal form in the lambda calculus. No it's not trivial and this is not the best SE for it. Math, CS or theoretical CS would be more helpful Nov 10 '20 at 15:54
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    A tautology is not a correct answer even technically. What is correct is determined by the rules of linguistic practice, class in this case, and they prescribe, in particular, that the answers must be informative and relevant, see Grice's communication maxims. Tautologies are not. The difference between two different expressions with the same referent is described by what Frege called sense, see sense and reference
    – Conifold
    Nov 10 '20 at 18:48
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    "It would not be the intended answer..." and "...technically a correct answer". It seems you're already aware there is a difference between these two. Are you asking why educators aren't more precise with their questions? Or are you asking what those intentions are? In either case, the question might fit better on Academia.SE or perhaps MathEducators.SE. Nov 10 '20 at 18:59
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    Now I show you a circle and ask: what is the ratio of the perimeter and the diameter? The answer is Pi. Now I ask: what is Pi? The answer is: it is the ratio of the perimeter and diameter of any circle. That's circular. You could also say that Pi is approximately 3.14, but it's not exactly 3.14, so there really is no better answer than the circular "Pi is the ratio of the perimeter and the diameter, and the ratio of the perimeter and the diameter is Pi."
    – Stef
    Nov 11 '20 at 12:47
  • 3
    Related question: Is mathematics one big tautology? Nov 11 '20 at 17:23

12 Answers 12

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The unstated assumption is that the person asking the question is asking for an answer that is in the simplest form. "What is 2+2?" could better be expressed as "What natural number is equal to 2+2?". The answer 4 is simpler than the description 2+2. "What is the derivative of the function x?" means, "What function, expressed in its simplest form, is identical to the derivative of x?"

In some cases, it might not be entirely obvious what the simplest form is. For example, the square root of i is a constant and looks simple enough, but it is possible to write it in different ways. It would be more precise to ask, "How can the square root of i be expressed without using any powers of i?"

In non-mathematical cases, arguably there is a distinct difference between identifying an individual by using a definite description, as contrasted with using their name or by indicating them in some direct way. To ask, "Who is the president of France?" is obviously a request for a name. To ask, "Who broke this vase?" is obviously a request for a name or some finger-pointing. We usually prefer to identify people using names rather than definite descriptions, though there is a theory of names which has it that names are a kind of definite description in disguise.

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    As an informal response in the context of informal math it's ok. Formally there are many ahems. Eg consider replacing in the question above "2+2" by "10 ** 10 **10" where ** denotes "raised to". More prosaically writing (ie "simplifying") Avogadro's number without scientific notation! A systematic response would need to use (something like) rewrite-systems or lambda-calculus. Nov 10 '20 at 9:52
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    Another issue is that a question "what is X, expressed in the simplest form" might well be X, but that would hardly be a tautology. For example, the simplest-form answer to "what is nine divided by six" might be "one and one half", but the simplest-form answer to "what is five divided by six" would be "five sixths". That X divided by six is X sixths may be a tautology, but the fact that the simplest form description of the answer is X sixths wouldn't be.
    – supercat
    Nov 11 '20 at 0:06
  • Hypothesis: This answer is false. Well is it? Prove: I might as well ask what is 4. The assumption is that 4 is the simplest term. But it is not. It is invariant over several representations, 2+2, or e.g. Roman IIII which is arguably more simple in the sense that it is homiconic, but it is not simpler to type. Corallary: There is no unique measure for simplicity in life. The simpelest answer would be nothing, inarguably, but that is not satisfying. Conclusion: The answer might be right, but it is resting on further unspoken assumptions, which is essentially what OP was asking about.
    – vectory
    Nov 13 '20 at 21:31
  • It is by the way incomprehensible why this answer was accepted while OP was clearly asking about reference to prior research (which should in effect exclude own research, for lack of a better word). The added insult "...is obviously..." is obviously not up to standards either. Although, intuition seems to be a bare necessity for communication, I think a more intuitive answer is possible.
    – vectory
    Nov 13 '20 at 21:34
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The teacher's goal when asking is not merely to obtain a correct answer (spoiler alert, because they already know the answer), but for the students to demonstrate knowledge, in order to make sure the lesson is assimilated. By merely parroting the question, the student demonstrates no knowledge.

Also, "the derivative of f is the derivative of f" is correct, but not very useful. If the teacher were to follow with a small application project that depends on derivation, like setting the strength of a toy catapult to hit a target with a ball, the student won't be able to do it. For this kind of project, they could succeed by trial and error, but imagine a plane engineers who, instead of computing the amount of fuel for a plane to reach its destination, would tell you "listen, just jump in, we'll see what happen."

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    The slope of the graph of f at x=42 is the slope of the graph of f at x=42 :^)
    – user
    Nov 10 '20 at 14:51
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    @user well then the teacher can ask is it larger than 35 or smaller. (correct answer: larger or smaller or equal :-) )
    – lalala
    Nov 10 '20 at 19:22
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    Sort of a variation on this old meme... Nov 10 '20 at 20:08
  • @lalala yes and/or no - aladeen youtube.com/watch?v=NYJ2w82WifU
    – emory
    Nov 10 '20 at 21:57
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    So in other words, while the answer is not logically wrong, it is pragmatically wrong.
    – Philip Klöcking
    Nov 11 '20 at 13:06
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Short Answer

When you use talk about "expressions" and "objects" to which they refer, you are in the domain of semiotics, linguistics, and the philosophy of language. Semioticians talk about symbol, reference, and referent. Analytical philosophers tend to talk in terms of sense and reference. In computer science, there are variables that refer to memory addresses that refer to values.

But the question of why tautologies are not acceptable answers has much more to do with implicature. Tautologies while logically correct are pragmatically wrong! This is related to Wittgenstein's observation that meanings of words are contextually upon the language-games they are played in.

Long Answer

Gottlob Frege in his Über Sinn und Bedeutung (On Sense and Reference) led the charge to address how to deal with semantics, or the meaning of words, and is considered the father of analytical philosophy, which is very driven since the linguistic turn to understand how words contribute to philosophical ideas. In the twentieth century, many philosophers like Bertrand Russell, Saul Kripke, John Searle, and Ludwig Wittgenstein put their brilliant minds to these sorts of questions. I'll try to give you short version.

When one asks the question, 'What is 2+2?', the answer is not determined solely by logical function. There are motivations involved in questions and answers, and those constitute the rules of the 'language-game'. Human beings are considered agents and as such manifest intentionality. What constitutes a correct answer is not solely logical or even grammatical (think of rhetorical questions, for instance), but rather are questions of implicature. From the article:

An implicature is something the speaker suggests or implies with an utterance, even though it is not literally expressed. Implicatures can aid in communicating more efficiently than by explicitly saying everything we want to communicate.1 This phenomenon is part of pragmatics, a subdiscipline of linguistics. The philosopher H. P. Grice coined the term in 1975. Grice distinguished conversational implicatures, which arise because speakers are expected to respect general rules of conversation, and conventional ones, which are tied to certain words such as "but" or "therefore".2 Take for example the following exchange:

So, when one asks, 'What is 2+2'? There are several possible correct answers all of which rely on context.

A1. '4.' (The goal when asking a young child during an arithmetic lesson.)
A2. '2+2, Duh!' (The goal when responding to a math teacher to be a smart aleck.)
A3. 'A binary functional expression of the summing operation.' (The goal when trying to show how arithmetic can be expressed with a predicate calclus.)
A4. 'An idiom among philosophers to express the nature of analytical truth. (The goal when trying to express an understanding of the analytical and synthetic divide.
A5. 'Okay, I get it. It's supposed to be easy, but you don't have to be a jerk and use sarcasm!' (When responding to someone who is trying to insult you when you make a mistake.)

Note that EVERY aforementioned answer is logically correct, but the real question of which is the appropriate response has nothing to do with grammar or logic, but rather the implicature. There are unspoken rules when communicating, the most famous being likely Grice's Maxims:

In social science generally and linguistics specifically, the cooperative principle describes how people achieve effective conversational communication in common social situations—that is, how listeners and speakers act cooperatively and mutually accept one another to be understood in a particular way. As phrased by Paul Grice, who introduced it in his pragmatic theory,

Make your contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged.1:45

So, why cannot one respond with a tautology like '2+2' when asked 'What is 2+2?' Well, one certainly can, so it's not a matter of whether one can, but rather whether of one should!

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    The Grice reference reminds me of a joke about teaching English as a foreign language: if a teenage boy asks a teenage girl "Have you seen the latest Star Wars Movie yet?" the correct translation is often something like "Do you want to spend a couple of hours with me in the dark?"
    – alephzero
    Nov 10 '20 at 21:52
  • @alephzero lol Point well made.
    – J D
    Nov 11 '20 at 2:14
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Language is about communication, not formal logic.

Most of the time, a question is asked to gain information. Usually, the information desired is about the subject of the question (e.g. “When's the train gonna get here?” is requesting information about the train), though sometimes the asker might want indirect information (e.g. whether the askee knows the answer to a question), or to accomplish something (e.g. the Socratic method of teaching). This situation has aspects of all three, though mainly the last two.

A trivial tautology communicates no information (about the question's subject, anyway), so is not (usually) a useful answer to a question. In the sense that it doesn't help with what the question asker intended answers to be useful for, the answer is wrong.

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It is not wrong to state that P is P, but it is implicitly assumed so, according to Aristotle's law of identity.

So, in case of having such type of question, "What is P?", an answer of the form "P is P" is not necessary. One can assume that being it not the expected answer, in most cases the answer will probably be wrong.

Remark that it is not the proposition ("P is P") that is wrong: it is the answer that is wrong. The error could be considered to come from a red herring/avoiding the issue fallacy (the same as answering "bananas are yellow": it is true that bananas might be yellow, but that's not the expected answer).

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    Aristotle's LoI is the red-herring here 😆 : Just replace "2+2" in q by "2+3" and in answer by "3+2" LoI stops applying but I guess his teacher remains dissatisfied! As @Bumble said it's a q about simplification. And as I've said to bumble and to OP simplification is not a simple matter Nov 10 '20 at 16:54
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This answer is based primarily on pragmatics, a field in the intersection of linguistics and philosophy of language.

You are right that the answer "2+2" to the question "what does 2+2 equal" is correct. The issue is that such a tautological answer does not help the other speaker, and so violates our basic assumptions about how a conversation should function.

Essentially, unless we have evidence to the contrary, we assume that our conversation partners are cooperating with us in conveying information. This is called the Cooperative principle and is summed up in Grice's Maxims:

  1. Maxim of Quality: Try to make your contribution one that is true. Do not say what you believe is false. Do not say that for which you lack adequate evidence.
  2. Maxim of Quantity: Make your contribution as informative as is required (for the current purposes of the exchange). Do not make your contribution more informative than is required.
  3. Maxim of Relevance: Be relevant.
  4. Maxim of Manner: Be perspicuous (despite being self-violating, this is the usual framing). Avoid obscurity of expression. Avoid ambiguity. Be brief (avoid unnecessary prolixity). Be orderly.

A tautological answer gives less information than is required (failing the maxim of quantity), and is not relevant (failing the maxim of relevance). The question was asked expecting a good faith cooperative answer, and instead we found out that our conversation partner was not communicating cooperatively. This is not the answer we wanted or expected, hence the listener feels put out.

Non-cooperative communication does of course occur in the wild. Many jokes are based on a certain amount of non-cooperativity (showing that it isn't even always viewed as a social faux pas), as of course is lots of deliberately manipulative communication (i.e. propaganda).

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  • Just a nitpick on your introductory sentence: There literally is a branch called philosophy of language and many philosophers have contributed to theories of communication, like C.S. Peirce, Cassirer, Wittgenstein, Ayers, Ryle, Habermas. Ironically, Grice was a philosopher as well, so the answer is based in philosophy which has been adapted in linguistic theory ;)
    – Philip Klöcking
    Nov 12 '20 at 10:34
  • ah ok, I'd usually heard Grice described as a Pragmatist within linguistics, rather than a Philosopher of Language, but that could easily be due to my coming at it primarily from that direction. I've reworded that sentence
    – Tristan
    Nov 12 '20 at 10:37
  • It's a common trope that philosophers are claimed to be a "social/political scientist", "economist", "mathematician/logician", "biologist", "linguist", etc. because they provided influential contributions to these fields. It does not help philosophy's academic and social standing, as you can imagine. Did you know that Adam Smith wrote his major works giving birth to modern economics as moral philosophy, for example? Grice, in particular, taught philosophy at Oxford and UC Berkeley and made major contributions to the study of ethics, metaphysics, and artificial intelligence as well.
    – Philip Klöcking
    Nov 12 '20 at 11:07
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    I'm upvoting, but we encourage users to use references like Wikipedia, Stanford Encyclopedia of Philosophy, the Internet Encyclopedia of Philosophy, and PhilPapers to strengthen the quality of responses.
    – J D
    Nov 12 '20 at 16:24
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    @Tristan: I replaced the link with the corresponding entry in the Stanford Encyclopedia. It starts with the Cooperative Principle and the maxims and continues with some more technical implications of them (pun intended - the main entry is "implicature"). The contributions there are written and kept up to date by professional philosophers working in the corresponding field and a pretty good go-to source.
    – Philip Klöcking
    Nov 12 '20 at 20:53
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The answers this question is asking about give linguistic identity, when the questions ask to resolve the denotation (or linguistic reference). See Frege's "Sense and Reference" and Russell's "On Denoting". To use Russell's example, if I ask "What is the Morning Star?", the question is about the referred-to object: the planet Venus. Clearly the question-asker already knows the linguistic utterance "the Morning Star" — the question-asker used that very utterance in forming his question — so returning that utterance as an answer cannot fulfill the 'what is...' which seeks a resolution. "The Morning Star" would only be a correct answer to the question "What is the other name for the planet Venus", because then we have reversed the process: asked about the alternate linguistic utterance that refers to or denotes the planet.

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  • Arguably, 2+2 = 4 is also a tautology.

Let's admit that (a) the succcessor of n is n+1 (b) 2 is the successor of 1 (c) 3 is the successor of 2 (d) 4 is the successor of 3 (e) addition is associative

We therefore get

2+2 = 4

is equivalent to 2+(1+1) = 4

is equivalent to (2+1) + 1 = 4

is equivalent to 3+1 = 4

is equivalent to 4=4.

If 4=4 is a tautotolgy ( or at least a vacuously true statement) and if 2+2 = 4 is equivalent to 4=4, then, 2+2 = 4 is also a tautology.

  • However, though equivalent objectively, not alll tautologies are equally informative subjectively. A young child may not know that 2+2 and 4 have the same denotation; he may not know that, though conceptually different, " the sum of 2 and 2 " and " the successor of 3 " refer to the same object.

In the same way, supposing f is a the function such that f(x)=2x. One may understand the expression ' the derivative of g(x) = x² ", but may not know at the same time that " f " and " the derivative of g " refer to the same object ( for there is a conceptual / intensional distinction betweeen f and g). So being able to say that " f = g' " shows one possesses some knowledge that a person only capable of saying " f = f " does not have.

  • In the same way , the inspector that is capable to say " the murderer is John Doe" is much more capable than the inspector that is only capable to say " the murderer is the murderer".

  • Conclusion : teachers try to make sure we know that 2 expressions have the same referent/denotation , in spite of the fact these expressions do not have the same intension ( = conceptual content)

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In linguistics, the study of meaning is often broken down into semantics (the intrinsic value of an utterance) and pragmatics (how people actually use language). "2+2 = 2+2" is a semantically correct statement, but generally not a pragmatically useful one.

In particular, when someone asks a question, they generally have a purpose behind it. That purpose might be to gain new knowledge ("okay, can you show me an example? if you use these new axioms to define addition, what's 2+2?"), or to test whether a student has learned the material ("quick, tell me, what's 2+2?"), or for another reason completely separate from arithmetic ("the Party says that 2+2=5, so tell me Winston, what's 2+2?").

But in all of these situations, "2+2 = 2+2" doesn't satisfy that purpose. It doesn't provide the questioner with new information, it doesn't demonstrate that a student has learned how to do arithmetic, and it doesn't demonstrate obedience to the Party.

And since it doesn't satisfy the purpose, it's not a useful answer to the question, even if it's technically a correct one. Philosophers generally concern themselves with semantics more than pragmatics, but language is generally used for a purpose, and in day-to-day life, fulfilling the speaker's purpose matters just as much as the literal truth values of what is said.

Consider also: "Can you pass the salt?" "Yes, I am capable of passing the salt." In most situations, the asker isn't just curious whether they're capable of it (which most people are), but actually wants the salt. So expressing capability, while semantically correct, is not pragmatically useful.

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  • I'm upvoting, but we encourage users to use references like Wikipedia, Stanford Encyclopedia of Philosophy, the Internet Encyclopedia of Philosophy, and PhilPapers to strengthen the quality of responses.
    – J D
    Nov 12 '20 at 16:25
  • I'm upvoting, because I encourage users to make references to George Orwell's 1984 and its IngSoz Party's math curriculum. The underlying more philosophical reference to the different messages in a single utterance is also appreciated, since I have asked about it on linguistics.SE and another answer here concedes that it' s noteworthy; anyway I think you missed the question by a bit with regards to the pragmatics of rethorically repeating a question.
    – vectory
    Nov 13 '20 at 23:32
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A tautology is unlikely to be a correct answer in this case, because it’s not on the answer sheet and you want to pass the test/quiz/worksheet.

You could of course write “four”, but that isn’t the answer the teacher is looking for and so will likely get points taken off, if not outright marked incorrect. 2+2 is 100% incorrect.

As with everything else, context is king. The question is asked for a reason, the answer either serves that reason or not, and the one asking the question will judge the answer as sufficient or not.

2+2 = You’re missing the vig, and I’m gonna break your legs.

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The answer is incorrect as it observes an absence of variation between the original assertion and that which follows it. It is the variation, which allows for a contrast through difference, which reflects itself under the "correct" answer given one phenomenon is expressed under a new form. This new form allows for a distinction between the original assertion and the new assertion, that of the answer, which allows for definition.

Definition occurs through contrast, contrast through difference, thus definition occurs through difference. All answers are defined through their difference to the original assertion with this definition being that of the answer itself. To answer a question is to give proof, to give proof is to give definition of a state of being.

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Compare your stated question to if the teacher had instead asked "What are some other expressions that equal 2+2?".

While "2 + 2 = 2 + 2" is a True statement, it is not the "correct" answer because there is an unspoken implied extension to statements of this kind in education: "What is 2+2 in the commonly given most simplified form?"

Through your use of the term "technically correct", you suggest awareness of this. There are infinite expressions you can enter on the right side of the "=" to produce a True equation. But the correct answer is typically unique.

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    Nov 12 '20 at 16:26

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