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I relied on the Logic of English (LoE) phonograms to give myself a better understanding of English pronunciation since the spelling gives me a hard time (even as native speaker), but I noticed that the phonograms do not match any accent/dialect I am aware of. LoE appears to be using a variant of the American Heritage Dictionary (AHD) pronunciation key, but represents /u/ and /ʊ/ with <ö> and <ü>.

LoE seems to be intended for speakers of General American (GenAm), not Received Pronunciation (RP), yet it distinguishes /ɑ/ (<ä>) and /ɒ/ (<ŏ>) but not /ɔ/ (represented in AHD as <ô>). As far as I know, the father-bother merger (/ɑ/=/ɒ/) and cot-caught merger (/ɒ/=/ɔ/) never led to that outcome.

LoE matches neither RP nor GenAm. RP distinguishes /ɔ/ but LoE does not, while LoE distinguishes /ɒ/ but GenAm does not. What does it match?

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    I would guess this matches the COT/CAUGHT merged pronunciation (generally Western American English), but they've chosen to use ä and ŏ as alternate representations for the same phoneme. If they really were treating the /ɒ/ vowel as separate from the /ɑ/ vowel, they would also need to list ŏ under "a" (as in swan).
    – Peter Shor
    Commented Jun 14, 2018 at 19:05
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    Most American English speakers don't distinguish [ɑ] from [ɒ] phonemically. They both occur as /a/, and in the West they're merged with /ɔ/. Native Californians tend to say father, bother, author with the same stressed vowel in the first syllable.
    – John Lawler
    Commented Mar 24, 2023 at 17:38

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I don't think they're distinguishing /ɑ/ from /ɒ/.

They use different symbols for taught (ä) and thought (ŏ), which rhyme in all dialects of English I know of (including the Boston one). So I think they're just confused.

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