Is it really impossible for two different languages to have the exact same set of phonemes?
I read that somewhere and wanted to know if it was true.
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An example of two languages with exactly the same phonemes is Zezuru and Karanga, which are Shona languages. The same can be said about Skagit, Snohomish and Puyallup. These can be called "dialects", or separate languages, depending on your criteria: they pass the test of mutual intelligibility, but there are various non-inventory differences. It does depend on how you define "the same" in terms of a phonemic system. Usually people mean "same segments with the same phonetic values", which is what allows inventories to be compared across languages. In substance-free theories, features for classifying sounds (i.e. the phonemes) are not phonetically based, so the idea that two sounds could be "the same" versus "different" is meaningless. So it would also depend on what your definition of "phoneme" is, and what the basis for comparison across languages is.
North Pare and South Pare have the same phonemes, except that the voiceless lingual fricative in NP is pronounced [θ] and in SP is is pronounced [s]. In terms of abstract analysis, they are not different and there is no reason in either language to say "This phoneme is [+strident] (or [-strindent]) in this language". If you define phoneme properties in abstract relational and functional terms, they are the same; if you define them in terms of pronunciation, they are different.
It depends what you mean by "the exact same".
Phonemes tend to be named with IPA characters, but they're really just algebraic symbols, representing mental objects that we can't poke and prod at directly. That's why you see symbols like
/ḱ/ (PIE) and
/ḥ/ (Semitic) and
/ƀ/ (Germanic) and
/ъ/ (Slavic) that aren't anywhere on the IPA chart. Alice Kober, who laid the groundwork for deciphering Linear B, famously used numbers for all the phonemes (like
/2/) so as not to bias anything.
The upshot is that, while languages may have phonemes with the same symbol, they may not be pronounced at all alike. PIE and its daughter Proto-Germanic both have a
/d/, for example, but it's pretty clear that the two sounded nothing alike. Or for a living example, English
/r/ and Brazilian Portuguese
/r/ tend to sound completely different: in English it's an alveolar approximant, while in Brazilian Portuguese it's sometimes a glottal fricative!
If you want two languages that have to have the same phoneme names/symbols, well, those symbols are arbitrary. All you have to do is pick languages with the same number of phonemes and rename them to match. Nothing's stopping you from using
/&/ as phoneme names, and non-IPA symbols can be neutral about how exactly the phonemes were realized.
On the other hand, if you want two languages that have exactly the same allophones and realizations of every phoneme, that's going to be impossible. Take two speakers of English from different areas, and I guarantee their phonetic realizations will be different in some way.
So when linguists say "English and Japanese both have
/t/", they're making a compromise, saying that English and Japanese both have phonemes that are "close enough" in their realization: English
/t/ can't be
[ts], and Japanese
/t/ can't be
[ʔ], but they're vaguely similar enough to be called "the same".