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Do they perceive it as an allophone of the vowel that is reduce to a schwa? so for example if /i/ is reduced to schwa would it still be perceived as /i/ by a native speaker of that language? or would it be perceived as a different vowel sound which /i/ can sometimes turn into?

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This is an interesting question, but not one that has been answered in a systematic, controlled fashion. The main issue is that there is no way to directly determine how a person "perceives a sound as". Sometimes you can make a reasonable approximation that suggests how they perceive certain sounds. For example, Makonde has a rule changing "unstressed" mid vowels to [a], which is also an underlying vowel vowel of the language, and it is easy to construct three-way ambiguous words that establish that speakers cannot tell the difference between [a] from underlying /a/, unstressed /o/, and unstressed /e/. However, this is not an allophonic rule, it is a full phonological neutralizing rule (whereas aspiration in English is allophonic because there is no underlying contrast between aspirated and unaspirated stops). Many vowel reductions have this characteristic: and English is an example, because unstressed lax vowels /ɪ ʊ ɛ ɔ æ a/ reduce to some schwa-like vowel (conventionally ʌ, which is a phoneme of English) and then are allophonically reduced to [ə] in words like [əˈtʰɔmɪk]. Russian and Catalan vowel reductions similarly merge two or more distinct vowel phonemes into one, so these are not allophonic rules.

The rule of English that gives you [ʌ] and [ə] in complementary distribution is an example of the type that you ask for. We can't meaningfully ask what speakers perceive unstressed /ʌ/ "as", but we can say that speakers do not (normally) perceive that they are phonetically different, indeed this is a point that is very difficult to convince a skeptical audience of students of.

Apart from the problem is finding an actually allophonic rule, there is the procedural problem that most experimental subjects are being tested on a language with a well-established writing system, and writing systems do not usually record allophonic differences – thus you have to devise your test to distinguish actual perception from orthographic reconstruction. Ojibwa theoretically presents a test case since reduction is non-neutralizing, however finding speakers who don't know how words are spelled is probably impossible.

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Russian is a language with massive vowel reduction, but the morphological principle is essential in Russian orthography, every morpheme is to be written in its full form irrespective of whether the vowel is actually full (stressed) or reduced (unstressed). In unstressed syllables /o/ is reduced to [ə] and [ɐ], and /e/ to [ɪ]. When children learn to write, first they write them “as it is heard”, namely as a and и (i) correspondingly, that is, they associate them with the sounds closest to what they are reduced to, and such spelling is wrong and unacceptable, very childish. Later at school they are taught to find out the original phoneme, the main allophone, by finding such forms of the word in which that syllable gets stressed and the full form of the morpheme is revealed.

For example, “leg” is [nɐ.ˈɡa] and “river” is [rʲɪˈka], so an untaught child would tend to write them as нага (naga) and рика (rika) which is considered wrong, spelling mistakes, since in the plural the forms are ['no.ɡʲi] and ['rʲe.kʲi] which means the root vowel is actually /o/, not /a/ in the first word and /e/, not /i/ in the second one. This is the first Russian spelling rule and cognitive method a child usually learns.

What I mean is that without knowing how to conduct even a simple morphological analysis like I showed above, without knowing exactly which vowel was reduced, without being taught at least the basic spelling rules one would have not a single motive to think anything at all about such things as vowels, allophones, reduction and schwa. An illiterate person is strongly unlikely to invent the idea of phonemes and allophones without being taught at least something about it. We can see illiterate people everywhere around, not yet spoiled by dogmas, those are children, and since they tend to write “as I hear”, the answer to your question is that it is natural for a person to perceive the sounds as they are without all of that schwa and allophone stuff. But if one has already learned at least something about the phonology of one's language, why would one then disagree with that theory? Maybe only if that person created a better phonological theory about their language, but that is a different story.

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