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For example, one may say that 'happy' is a synonym of 'joyous.'

However, one may also say that '5' is a synonym of '2+3'. Weird, I know, but bear with me.

In the former case, the two are not exactly the same. The words, while difficult to define, have slightly different connotations. However, the latter are exactly the same. In an ontology, one might say that the latter two identifiers (I mean '5' and '2+3') are referring to literally the same object, whereas the former two identifiers ('happy' and 'joyful') are referring to related, but slightly different, things.

So can we only say that 'x and y are synonyms' if x and y are referring to exactly the same things? Or can they be synonyms if they are merely referring to similar, but slightly different things?

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    There are no exact synonyms. – Mitch Jun 21 '18 at 3:07
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    While in mathematics 5=2+3, it's not true to say that 5 is exactly the same as 2+3 conceptually. – Jason Bassford Jun 21 '18 at 3:39
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    As with synonymous words having different connotations in different contexts, the same can apply with mathematics. Your answer to 2+3 may differ when operating with different bases, for example. – Miral Jun 21 '18 at 6:30
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The received wisdom in linguistics is that there are no exact synonyms; normally there is some difference between any two proposed synonyms (usually in syntax, presuppositions, idiom participation, social level, or speech-group identification), such that one will work, and the other won't.

If you ever do have a situation in which there is absolutely no difference between two words, what happens in general is that one of them disappears (or survives with a different meaning), and the other one inherits the wordhood. After all, why go to the trouble of memorizing two words that have no difference at all when one will do?

The closest to syntactic and semantic synonymy that I've ever encountered is the two verbs seem and appear, which come from very different sources, but have invaded the same word space and can be substituted one for another in any number of situations without changing meaning or producing ungrammaticality. This includes some fairly sophisticated syntax, as well as some epistemologically interesting semantics.

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    Yes. And still, somebody can "appear in a room" but not "seem in a room." So, it's really only certain senses that are synonymous—not the words as a whole. – Jason Bassford Jun 21 '18 at 3:40
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    You say that if there are exact synonyms, "what happens in general is that one of them disappears". I don't find this very convincing on three grounds: (1) The mental lexicon is huge, so one more word is not a problem. (2) How can we reliably generalize on a phenomenon if it is so rare? (3) Investigating diachronic semantics is notoriously difficult because all you have is the written record. So, can you support this claim with examples of exact synonyms where one of them disappeared? Or even better, do you have references that convincingly show that this is what really happens? – Schmuddi Jun 21 '18 at 8:01
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    "Suddenly he seemed appearingly from nowhere." – Kris Jun 21 '18 at 12:46
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    @Schmuddi it's a form of blocking, a pretty basic linguistic principle. You can read about it in an introductory linguistics text, but it's been long enough, i can't cite a definitive study. – De Novo Jun 22 '18 at 17:03
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    @DanHall: Blocking is only a related phenomenon: the term is used to refer to situations where the morphology of a language allows a particular form, but that form is not used because there is already an established synonymous form in the lexicon. The textbook example is the potential word stealer, which uses the productive suffix -er, but which is blocked by the existing word thief. Another example would be *gooder, which is blocked by better. This is quite different from the case described in this answer where one of two existing exact synonyms is claimed to disappear. – Schmuddi Jun 22 '18 at 21:23
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Are synonyms exact? This is an excellent question, and it gets to the concept of how meaning is constructed in language.

With your example from arithmetic, the thing the symbol 5 refers to is fixed. Mathematicians have a series of absolute rules for how you can use it and what it is equal to.

Language, and in particular, the relationship between a thing in a language (e.g., the word happy or phrase a happy occasion) and the meaning of that thing, is not fixed. Many linguists and philosophers would say the meaning of the set of symbols or sounds "happy" is constructed based on how it is used.

You can think of an individual's understanding of a word -- you may look up a word in the dictionary and see a string of other words that describe a meaning, and you may also encounter that word in text and speech. All of these things help you understand what you and others believe the word to mean.

You can also refer to the way dictionaries generate their definitions. Merriam-Webster describes this process in an answer to the faq "how does a word get into the dictionary". The dictionary meanings themselves are based on the way the words are used.

If meaning comes from usage, since two synonyms will occur in different texts, with different contexts, they may have similar or very similar meanings, but they can't have exactly the same meaning.

You may make an argument that in a specific utterance you could replace one word for another without a change in meaning, that the thing either of those words referred to would be the same. However, once that utterance occurs, only one of the possible words was used, and both meanings are changed either by being selected or not.

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I apologize for not being a native English speaker. Please correct me if needed.

You can't say "5 synonym of 2+3", with some mathematical value in mind, if you didn't define synonym correctly. Mathematics can't be applied to blur objects/concepts, because it is the science of rigourous and coherent definition and manipulation of quantities, structures and transformations. There is 2 concepts related to your blurry "synonym" concept which are commonly used in mathematics: equality and equivalence. Equality is about value and equivalence about semantic. For example 5x3 is not the same thing as 3x5, even if both contains the same number of objects! If you are just interested in the number (the value of the result) then both equals to 15 of course. But the structure of the result is not the same : one is 3 collections of five objects and the other 5 collections of three objects. While 5=2+3=4+3 they may not be equivalent (commonly denoted by ≡) depending on which equivalent relation you define (note that = is an equivalence relation).

Natural languages are blurry, ambiguous, non coherent, etc. This is what makes them so powerful (and fun!). Words may have several definition, different words may have the same acception, etc. Synonymy is about similarity (a synonym?). In general synonymy is not an exact correspondance, in fact I have no example of two words for the same exact thing, there is always small differences.

Natural language are inherently of that blurry nature because to speak easily, conveniently, concisely you need to use "shortcuts" and "semantical approximation". Mathematical language is the converse when you say something in the mathematical language it hasn't different meanings, one mathematical phrase is about one exact object, this makes that language hard to handle and not useful in everyday life. Scientist/engineers always makes a lot of efforts to specify/precise their intentions because just describing it in a natural language is too ambiguous and lead to mistakes. Standards are such things, if you read one, you will see that it always starts with a (tentative) precise definition of the terms used in it! Specification languages are also of the same nature: useful to describe something precisely.

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    Never apologise for not being an English native! That was hardly your fault - even though you were there at the time! An English native speaker, or native English speaker. Now that's different ! – Tim Jun 21 '18 at 9:30
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There is a well-known paper by Nelson Goodman, where he argued that words can never have exactly the same meaning.

The philosopher of language John Searle criticizes this argument in he following way:

[Goodman] discussing synonymy once offered an analysis which has the consequence that no two words can be exactly synonymous. Since, for example, the expression "eye doctor that is not an oculist" can be described as an eye doctor description but not as an oculist description, he argues that this shows there is something in the " secondary extension" of "eye doctor" which is not in that of "oculist". And since a similar point can be made about any pair of words, he argues that no two different words can ever have "quite the same meaning". But now let us reflect on what exactly is proven by such an argument. Is it not quite clear that what it shows is that such facts about secondary extensions have simply no bearing on whether two terms are synonymous? The starting point for any search for a criterion of synonymy is (and must be) such facts as that "oculist" means eye doctor. Any extensional criterion for a concept like synonymy would first have to be checked to make sure that it gave the right results, otherwise the choice of the criterion would be arbitrary and unjustified. The proposed criterion does not give the right results, nor is there any a priori reason why it should, and we must therefore abandon it.

The claim that "oculist" means eye doctor is not a claim that has to satisfy any criteria which philosophers might propose for synonymy, but rather any proposed criterion for synonymy has to be consistent with such facts as that "eye doctor" is synonymous with "oculist". Nor does the maneuver with the notion of exactness offer any help; for, as Wittgenstein pointed out, exactness is relative to some purpose; and relative to the purposes for which we employ synonyms, "oculist" is exactly synonymous with "eye doctor". For example, my child, who knows the meaning of "eye doctor" but not of "oculist", asks me, "What does oculist mean?" I tell him, ""Oculist" means eye doctor." Have I not told him exactly what he wanted to know?

So according to Searle, two ordinary words can indeed be "exactly" synonymous.

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