The Collaborative International Dictionary of English v.0.48:

Syllogize \Syl"lo*gize\, v. i. [imp. & p. p. Syllogized; p.
   pr. & vb. n. Syllogizing.] [Gr. ?: cf. F. syllogiser.]
   To reason by means of syllogisms.
   [1913 Webster]

         Men have endeavored . . . to teach boys to syllogize,
         or frame arguments and refute them, without any real
         inward knowledge of the question.        --I. Watts.
   [1913 Webster]

WordNet (r) 3.0 (2006):

syllogize
    v 1: reason by syllogisms [syn: syllogize, syllogise]