Source : Webster's Revised Unabridged Dictionary (1913)

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.

Source : WordNet®

illogicalness
     n : invalid or incorrect reasoning [syn: {illogicality}, {illogic}]
         [ant: {logicality}, {logicality}]