IPA: /ɪˈlɒdʒɪkli/
KK: /ɪˈlɑːdʒɪkli/
In a way that does not make sense or is not based on clear reasoning.
He argued illogically, making it hard for anyone to follow his point.
Illogically → It is formed from "il-" (meaning not) and "logica" (from Greek "logikē", meaning reason or logic) and the suffix "-ly" (meaning in a manner). The word "illogically" means in a manner that is not logical or reasonable.
Think of 'not' ('il-') combined with 'reason' ('logica') to remember that 'illogically' means acting in a way that lacks reason.