The Collaborative International Dictionary of English v.0.48:

irrationality \ir*ra`tion*al"i*ty\, n.
   The quality or state of being irrational; absence of
   rationality. "Brutish irrationaliity." --South.
   [1913 Webster]

WordNet (r) 3.0 (2006):

irrationality
    n 1: the state of being irrational; lacking powers of
         understanding [syn: irrationality, unreason]

Moby Thesaurus II by Grady Ward, 1.0:

98 Moby Thesaurus words for "irrationality":
   Urdummheit, aberration, abnormality, alienation, antilogy,
   brain damage, brainlessness, brainsickness, callowness,
   childishness, clouded mind, craziness, daftness, dementedness,
   dementia, derangement, disorientation, distraction, folie,
   foolishness, furor, ignorance, illogic, illogicality,
   illogicalness, immaturity, imprudence, inadvisability, inattention,
   incapacity, inconclusiveness, incongruity, inconsideration,
   inconsistency, indiscreetness, indiscretion, ineptitude,
   inexpedience, injudiciousness, insaneness, insanity, invalidity,
   lackbrainedness, lackwittedness, loss of mind, loss of reason,
   low IQ, lunacy, madness, mania, mental deficiency,
   mental derangement, mental disease, mental disorder,
   mental disturbance, mental illness, mental instability,
   mental sickness, mental weakness, mind overthrown, mindlessness,
   mindsickness, oddness, pixilation, pompousness, possession,
   primal stupidity, puerility, queerness, rabidness, reasonlessness,
   recklessness, senselessness, shattered mind, sick mind, sickness,
   slackmindedness, slackwittedness, strangeness, stuffiness,
   thoughtlessness, unbalance, unbalanced mind, unintellectuality,
   unintelligence, unreason, unreasonableness, unsaneness,
   unscientificness, unsensibleness, unsound mind, unsoundness,
   unsoundness of mind, untenableness, unthoughtfulness, unwisdom,
   unwiseness, witlessness