The Collaborative International Dictionary of English v.0.48:

Inconsequent \In*con"se*quent\, a. [L. inconsequens: cf. F.
   incons['e]quent. See In- not, and Consequent.]
   Not following from the premises; not regularly inferred;
   invalid; not characterized by logical method; illogical;
   arbitrary; inconsistent; of no consequence.
   [1913 Webster]

         Loose and inconsequent conjectures.      --Sir T.
                                                  Browne.
   [1913 Webster]

WordNet (r) 3.0 (2006):

inconsequent
    adj 1: lacking worth or importance; "his work seems trivial and
           inconsequential"; "the quite inconsequent fellow was
           managed like a puppet" [syn: inconsequent,
           inconsequential]

Moby Thesaurus II by Grady Ward, 1.0:

75 Moby Thesaurus words for "inconsequent":
   abnormal, absonant, absurd, adrift, anomalous, beside the mark,
   beside the point, beside the question, contradictory,
   contrary to reason, disproportionate, extraneous, extrinsic,
   fallacious, faulty, flawed, illogical, immaterial, impertinent,
   inadmissible, inapplicable, inapposite, inappropriate, inauthentic,
   incidental, incoherent, incommensurable, incommensurate,
   incompatible, inconclusive, incongruous, inconsequential,
   inconsiderable, inconsistent, inconsonant, invalid, irrational,
   irreconcilable, irrelative, irrelevant, loose, measly, niggling,
   nihil ad rem, nonessential, nonscientific, not at issue,
   not following, off the subject, out of proportion, out-of-the-way,
   oxymoronic, paltry, paradoxical, paralogical, parenthetical,
   peanut, pettifogging, picayune, picayunish, reasonless,
   self-annulling, self-contradictory, self-refuting, senseless,
   small, trifling, trivial, unauthentic, unconnected, unessential,
   unphilosophical, unreasonable, unscientific, without reason