"Absence of logic; the quality or fact of not being influenced or determined by logic or rationality. Usually in neutral rather than pejorative sense: compare illogic", 1930s; earliest use found in Journal of Philosophy. From a- + logic.
"That does not follow logic; contrary to logic; illogical", Mid 17th cent.; earliest use found in Thomas Blount (1618–1679), antiquary and lexicographer. From (i) Middle French, French alogique.