Syllogistic Logic with Complements

This paper continues the development of complete fragments of natural language by adding a noun-level complement operator to the basic syllogistic language of All X are Y and Some X are Y : in the fragment of this paper we can also say, for example, Some non-X are Y . We axiomatize this logic of this fragment in two ways: in a direct fashion, allowing the inference of an arbitrary sentence from a contradiction, and alternatively using a rule of reductio ad absurdum. The fact that the direct system is as strong as the indirect calls on a proof-theoretic examination. The completeness work overall uses representation of orthoposets.