There are some sounds called the "R-colored diphthong" in English, such as [or] sound in "court" or the [ir] sound in "clear".
My question is simple: are these R-colored diphthongs regarded as phonemes, respectively for native speakers?
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How can you tell?
I think the right thing to do is ask a bunch of Americans who haven't thought much about phonetics whether the vowel in beard is the same as the vowel in bead or as the vowel in bid. If they instantly answer bead (or bid), then you know that for them /ir/ is an allophone of /i/ (or /ɪ/). If they hesitate and can't decide instantly, you can conclude that beard has its own phoneme.
Has anybody actually tried this experiment? Not as far as I know. What answer would you get if you did? My guess is that some Americans think of the vowel in /ir/ as an allophone of /i/, some as an allophone of /ɪ/, and some think of /ir/ as its own phoneme.
For me, as far as I can tell, the vowel of /ir/ is an allophone of /ɪ/, but /ɔr/ is definitely its own phoneme. I pronounce it /or/ and don't perceive it as either /ɔ/ or /oʊ/.
In my experience, rhotic vowels generally aren't regarded as separate vowel phonemes. But I am not a linguist so you shouldn't take my word for it. Hopefully someone else will post a better answer soon.
As a native speaker of a rhotic variety of English, I do think a somewhat convincing argument can be made that rhotic vowels are phonemes. For example, I have the cot-caught merger, so for me the phoneme /ɔ/ doesn't occur anywhere except before /r/. That's a pretty odd distribution. Now, I also have the horse-hoarse merger, so we could just analyze my [ɔr] as consisting phonemically of tautosyllabic /oʊr/ (assuming we're going with something like John Wells's theory of English syllabification where a word like "glory" gets divided as "glor.y," the stress drawing the intervocalic consonant into the coda of the preceding syllable). But I think it's reasonable enough to analyze it as a diphthong instead. It behaves fairly similarly in some ways to /ɔɪ/ (it also has [ɔ] as the first element, and it can result in some cases from historical sequences of /ɔ/ + a consonant (for example, "drawer" /drɔ˞/ and "lawyer" /lɔɪɚ/).