or is it simply an unstressed allophone of unstressed lax vowels? I'm curious because I've heard some people claim that [ə] is not a phoneme and it is just a reduced allophone of all the unstressed vowels in English.
First, "allophone" is the wrong term to use for many-to-one mappings between phonemes and surface realizations. Allophonic rules are not neutralizing, so final devoicing in German (for example) is not an allophonic rule. Aspiration in Engilsh is, because only /t/ becomes [tʰ].
Second, just to be clear, we say that the voiceless alveolar stop in English is the phoneme /t/ and not the phoneme /tʰ/ because the rule deriving [tʰ] from /t/ is simpler than the converse rule deriving [t] from /tʰ/. Or, so it has been thought – but it turns out that if you state the rule in terms of stress-foot position ("deaspirate foot-medially") you not only can state deaspiration as easily as you can state aspiration – plus, you can also explain why /h/ deletes in words like "prohibition". So the analogous question is, is it easier to derive [ʌ] from /ə/, or is it easier to derive [ə] from /ʌ/? There is no difference in simplicity: you get schwa when unstressed and caret when stressed. So it seems to be a coin toss.
But third: lots of (unstressed) vowels become schwa in English, as one can see from alternation like monotone ~ monotony [ˈmɔnətoʊn, məˈnɔtəni] though in some dialects, my [ɔ] is [a]; telegraph ! telegraphy [ˈtɛləgræf, təlɛgrəfi]. There is a many-to-one mapping between all sorts of vowels and schwa: so this is not an allophonic relation. Not all rule-governed relations in phonology are allophonic.
The conclusion therefore would be that every instance of schwa in English can be derived from some other vowels when unstressed. Sometimes you can tell that it comes from /ɪ/ or /ɛ/ given stress alternations as in telephone ~ telephonic. Sometimes you cannot tell: the first and last syllables of vanilla, first syllables of collapse, giraffe. The decision to say that the first vowel of vanilla must be underlyingly some vowel other than schwa is based on an unsupported claim that English underlyingly has no instances of schwa. That vowel could be ɪ, or ʌ, or ɛ, or æ...
The motivation for purging schwa from underlying forms is the idea that there is an intolerable acquisition burden to learning 12 phonemes compared to 11 phonemes. However, the rationale is countered by the heavier burden on acquisition that comes from having to make an arbitrary decision regarding the underlying form of many words. Whereas, if the default assumption is that the underlying form is the same as the surface form unless there is good evidence saying otherwise (and "saves a phoneme" is not good enough evidence), then schwa can be a phoneme of English, meaning that it is present in underlying forms.
It is not an allophone for sure. An allophone must be associated with a single phoneme; [ə] isn't, it's associated with multiple phonemes.
You might want to consider it as a phoneme or not, but that depends on a bunch of factors, like:
- Are phonemes "real"? They might be as well just a convenient abstraction¹. If phonemes are not real, it's up to the author to decide to include or exclude /ə/ as its own phoneme.
- If however phonemes are real, how do speakers interpret [ə]? Do they associate it with other vowels, or do they treat it as a unit of its own?²
Personally speaking I do not think that speakers keep track of "phonemes", but of articulations and how they contrast with each other. As such, I wouldn't analyse [ə] as belonging to its own phoneme; it's simply easier to analyse it as a fully unstressed realisation of other vowels.
And, even when you don't know which vowel that [ə] is "supposed" to be associated with, for me it's just for convenience, so might as well assign it a random value like /ɪ/ or /ʌ/ depending on the dialect in question³. The net result will be the same, call it a chicken or a duck, in that position it's simply a "bird".
- See Is the very concept of the phoneme disputed? for further discussion.
- Harder to analyse than it looks like. Not only different people will give you different answers, but the spelling itself will interfere on the data quite a bit.
- /ɪ/ for NZE, /ʌ/ for GA.