'antirationally' originates from Modern English, specifically from the adjective 'antirational' plus the adverbial suffix '-ly'; the prefix 'anti-' comes from Greek 'anti-' meaning 'against', and 'rational' derives from Latin 'rationalis' (from 'ratio') meaning 'reason'

'rational' changed from Latin 'rationalis' into Old French and then Middle English 'rational'; the formative prefix 'anti-' (Greek) was later attached in Modern English to create 'antirational', and adding '-ly' produced the adverb 'antirationally'

Initially elements meant 'against' (anti-) and 'of reason' (rational); combined and suffixed with '-ly' the modern word has come to mean 'in a manner opposed to reason' or 'opposing rationality'