apodictic vs. apodeictic

Solution 1:

Well, style manuals don't need a reason beyond simple convention. Some people just like consistency, and to achieve that, you have to prescribe some standard and proscribe the alternatives, even if they're legitimate variants in general. And as Edwin Ashworth points out in a comment, apodictic seems to be the more commonly used spelling, which can be considered a reason to prefer it.

There's nothing etymologically wrong with either apodictic or apodeictic. They just use different conventions for transliterating Greek ει.

Josh61's quote from the Online Etymology Dictionary is informative:

apodictic:

• "clearly demonstrated," 1650s, from Latin apodicticus, from Greek apodeiktikos

The transliteration with i dates back to how the Romans romanized Greek words.

From Wikipedia: Romanization of Greek

Traditional English renderings of Greek names originated from Roman systems established in antiquity. [...] ⟨ει⟩ and ⟨ου⟩ were simplified to ⟨i⟩ (more rarely—corresponding to an earlier pronunciation—⟨e⟩) and ⟨u⟩.

This talks about names, but the same applies to words that originated in Greek but came into English via a Latin intermediary: consider irony, from Latin ironia, from Greek εἰρωνεία (eironeia).

The transliteration with ei is more letter-by-letter and occurs more frequently with names or with terms that were coined in English directly from Greek roots.