The Collaborative International Dictionary
Paradoxicalness
Paradoxical \Par`a*dox"ic*al\ (p[a^]r`[.a]*d[o^]ks"[i^]*kal), a.
Of the nature of a paradox.
Inclined to paradoxes, or to tenets or notions contrary to received opinions.
--Southey. [1913 Webster] -- Par`a*dox"ic*al*ly, adv. -- Par`a*dox"ic*al*ness, n.
Wiktionary
paradoxicalness
n. The state or quality of being paradoxical.