Source : Webster's Revised Unabridged Dictionary (1913)

Paradoxical \Par`a*dox"ic*al\, a.
   1. Of the nature of a paradox.

   2. Inclined to paradoxes, or to tenets or notions contrary to
      received opinions. --Southey. -- {Par`a*dox"ic*al*ly},
      adv. -- {Par`a*dox"ic*al*ness}, n.

Source : WordNet®

paradoxically
     adv : in a paradoxical manner; "paradoxically, ice ages seem to
           occur when the sun gets hotter"