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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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    There’s nothing weird about utterances with no subject or predicate – they’re everywhere. Whether you choose to interpret them as sentences (S) depends on your beliefs. ‘Yes’ and ‘no’ aren’t the only such utterances either, of course – lots of particles can function similarly, and verbless utterances consisting only of noun phrases are perfectly common as replies as well (“How many people live in London?” – “Somewhere between 8 and 12 million all in all”). – Janus Bahs Jacquet Jun 1 at 8:50
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    “No” and “Yes” are sentences, they start with a capital letter and end with a period. This is a statement about the etiquette of that part of writing called punctuation. It has absolutely nothing to do with any part of grammatical theory or analysis. – Colin Fine Jun 1 at 11:38
  • @Colin Fine Cambridge dictionary: Sentence — a group of words, usually containing a verb, that expresses a thought in the form of a statement, question, instruction, or exclamation and starts with a capital letter when written. "Yes" expresses a thought in the form of a statement, for example, «I agree», and starts with a capital letter. So this is a sentence. – Eagle Jun 1 at 11:53
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    @Eagle That definition does not tally with what you wrote either – starting with a capital letter and ending with a period is a by-product of sentence-ness, not a defining property. A sentence doesn’t stop being a sentence if it doesn’t start with a capital letter or end with a period; that just happens to be how sentences are generally represented in writing. You can say that “yes” and “no” are used as sentences here, and because they are, they are capitalised and end with periods – but you can’t use the capital and period as evidence that they are sentences. – Janus Bahs Jacquet Jun 1 at 12:45
  • @Janus Bahs Jacquet Even if we don't talk about punctuation, it's still a sentence because it expresses a thought in the form of a statement. An oral or written sentence is an expressed thought that is completed. Yes meets these criteria. – Eagle Jun 1 at 13:00
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The fact that the English word yes was a contracted verb form some centuries ago does not affect its status in present day English: Part of Speech classification is essentially synchronic and all tests for certain kinds of part of speech are synchronic judgements of grammaticality. Particle is kind of "none of the parts of speech mentioned and defined before", and the answer particles yes and no are kind of canonical examples for words that are neither verbs, nouns, adjectives, adverbs, determiners, prepositions, conjunctions, interjections, nor cardinal numbers.

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