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®

illogically
     adv : in an illogical manner; "she acted illogically under the
           pressure" [ant: {logically}]