The Collaborative International Dictionary of English v.0.48:

Inconsequence \In*con"se*quence\, n. [L. inconsequentia: cf. F.
   incons['e]quence.]
   The quality or state of being inconsequent; lack of just or
   logical inference or argument; inconclusiveness. --Bp.
   Stillingfleet.
   [1913 Webster]

         Strange, that you should not see the inconsequence of
         your own reasoning!                      --Bp. Hurd.
   [1913 Webster]

WordNet (r) 3.0 (2006):

inconsequence
    n 1: having no important effects or influence [ant:
         consequence, import, moment]
    2: invalid or incorrect reasoning [syn: illogicality,
       illogicalness, illogic, inconsequence] [ant:
       logicality, logicalness]