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Let's look at some examples:

— Would you like some ice cream?
— No.

— Are you happy?
— Yes.

According to Wiktionary “yes” is a particle:

Particle
yes
Used to show agreement or acceptance...

“No” and “Yes” are sentences, they start with a capital letter and end with a period.

However, a particle cannot be either a subject or a predicate. Thus, we have the sentences without a subject and a predicate. What is weird.

So, I want to understand, what parts of speech and sentence constituents are "yes" and "no" in such cases.

According to etymonline.com:

Old English gise, gese "so be it!," probably from gea, ge "so" (see yea) + si "be it!," from Proto-Germanic *sijai-, from PIE *si-, optative stem of root *es- "to be."...

It turns out, that "yes" etymologically means "so to be".

Could it be that "yes" and "no" are not particles, but something like verbs which express being or nonbeing of agreement? Could it be that in this way they are actually predicates, and subjects are the essences of questions, which are not repeated in the answers but are implied?

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    I'm afraid your definition of "sentence" is incorrect. Punctuation and capitalization are irrelevant. Yes and No are not sentences. They are Utterances, something said by someone. Sentences are a special class of utterance, which has syntax and parts of speech; Yes and No, not being sentences, don't have syntax, and therefore don't need to be categorized for POS. If you want to label them, "discourse particle" is a good term. – John Lawler Jun 1 at 14:11
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    Responses such as 'Yes,' 'On the table', 'Just a minute', 'If you must' are often called sentence substitutes in that they convey (in context) a complete idea but don't have the subject + finite verb structure that satisfies the syntactic definition of a sentence. One-worders such as 'Yes,' 'No,' 'Obviously,' 'Almost,' 'Why?' and ('OKAY') often have obvious expansions 'I obviously didn't get the job of Professor in charge of teaching Defence against the Dark Arts' and in these cases POS identification doesn't ... – Edwin Ashworth Jun 1 at 14:14
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    achieve much (ex-sentence-adverb?). 'Particles' is sometimes used for 'all the rest' by those who think the 'adverb' dustbin is over-full, if a label is really desired. But one needs to be aware that the infinitive marker to is very different from the multi-word verb particle to in Heave to and discourse particles like Hello .... – Edwin Ashworth Jun 1 at 14:14
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They are adverbs as answers to questions according to the Cambridge Grammar of the English Language p.570, Merriam-Webster and the American Heritage Dictionary. This seems pretty unanimous and seeing as the apparent lack of any subject or predicate can be explained by the following, there's no reason to cast them as verb-like.

The role they play in clause structure is that of a polarity adjunct, CaGEL p847:

Yes and no answers

Yes and no serve as markers of positive and negative polarity in answers to questions. They may stand alone, or combine with a clause that expresses the answer more explicitly:

7ia. A: Is this car yours? B: Yes (it is). / No (it isn't)

7ib. A: Isn't this car yours? B: Yes (it is). / No (it isn't)

7iia. A: He has gone, hasn't he? B: Yes (he has). / No (he hasn't).

7iib. A: He hasn't gone has he? B: Yes (he has). / No (he hasn't).

The choice between yes and no depends simply on the polarity of the answer – not, for example, on agreement vs disagreement with what may be suggested by the question. Polar questions, especially negative ones, may be biased, indicating the questioner’s predisposition to think that one or other answer is the right one, but that has no bearing on the choice between yes and no. In [ib], for example, the appropriate response is yes if the car is B’s and no if it isn’t, irrespective of what A appears to expect is the case.

Similarly with answers to tag questions, as in [ii]. ∗Yes it isn’t and ∗No it is are thus ungrammatical as single clauses. In Yes it is and No it isn’t, the yes and no can be regarded as a special type of adjunct, a polarity adjunct, which agrees in polarity with the clause – a further case of polarity concord in English. The adjunct can also be placed at the end of the clause, with prosodic detachment: It is, yes and It isn’t, no. A response to [7i] with the form No, it’s Kim’s would not of course violate the polarity concord rule, because here we have not a single clause but a sequence of two, just as we do in No it’s not mine, it’s Kim’s.

One respect in which the agreement vs disagreement factor is relevant concerns the choice between single-word and expanded responses. Suppose you ask Didn’t you post the letter after all, then?, indicating that you think I didn’t. If in fact I did post it, I would normally say Yes I did, not just Yes.

In short, the rest of the answer has simply been elided because it can be retrieved from context leaving behind the polarity adjunct only.

This is the same logic as:

A: When did you have breakfast? B: Early this morning.

where B's answer consists only of an adjunct of temporal location.

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    'This seems pretty unanimous'? ... 'Yes and No, not being sentences, don't have syntax, and therefore don't need to be categorized for POS. If you want to label them, "discourse particle" is a good term.' (I believe John Lawler was an associate of McCawley's.) – Edwin Ashworth Jul 1 at 11:30
  • I'd appreciate some reasoning along with those summary pronouncements. On discourse markers, the Oxford Dictionary of English Grammar p124 points out "The term is somewhat specialized, and not easily defined, but the label generally includes words that are not part of the syntactic structure of a clause or sentence, e.g. ah, oh, well, you see, I mean". I hardly see how yes and no fit considering the analysis provided in CaGEL. – DW256 Jul 2 at 9:03
  • The comment is John Lawler's. I agree with his not classifying 'Yes' etc as adverbs. I can't point to myself as an authority, but JL certainly is, making 'pretty unanimous' unsound. // Defining 'clause' and 'sentence' in the traditional way, yes (as in "Yes!") is certainly a'word[] that are not part of the syntactic structure of a clause or sentence. 'eg' means that a partial set-list follows. – Edwin Ashworth Jul 2 at 13:54
  • Here are other unanimity-disprovers: Gabriele Scheler / Kerstin Fischer: ResearchGate: The Many Functions of Discourse Particles: A Computational Model of Pragmatic Interpretation. (... 1. 'Yes' ...) Cherry-picking favoured analyses is anathema on ELU, which tries to give a balanced treatment in debated areas. – Edwin Ashworth Jul 2 at 14:00
  • It certainly would be nice to have a fuller explanation from John Lawler, yet he has not provided one, so we're left in the dark. In the mean time, we might benefit from a fully fleshed-out analysis like the one in CaGEL. – DW256 Jul 2 at 14:27
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YES AND NO ARE PROFORMS. The utterances, "Yes, I did." and "No, I didn't." do substantiate it.

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    Like 'do' in "You like strawberries? So do I." Is this helpful? If you want to use this conceptualisation, you need pro-sentences. – Edwin Ashworth Jul 1 at 15:31

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