The Collaborative International Dictionary of English v.0.48:

Illogical \Il*log"ic*al\, a.
   Ignorant or negligent of the rules of logic or correct
   reasoning; as, an illogical disputant; contrary of the rules
   of logic or sound reasoning; as, an illogical inference. --
   Il*log"ic*al*ly, adv. -- Il*log"ic*al*ness, n.
   [1913 Webster]

WordNet (r) 3.0 (2006):

illogicalness
    n 1: invalid or incorrect reasoning [syn: illogicality,
         illogicalness, illogic, inconsequence] [ant:
         logicality, logicalness]