'antirationality' is a modern English formation combining the prefix 'anti-' (from Greek 'anti-' meaning 'against') with 'rationality' (from Latin-derived 'rationalitas', via French/Medieval Latin), where the whole means 'against rationality'.

'rationality' originates from Latin 'rationalitas' (from 'ratio' meaning 'reason' or 'calculation'); the productive English prefix 'anti-' (from Greek 'anti-') was attached in modern English to form compound nouns like 'antirationality'.

The components originally meant 'against' (anti-) and 'reason' (ratio); combined in modern coinage they convey the specific sense 'opposition to or rejection of rational thought', a neologistic shift to name the attitude or condition.