Logics for Two Fragments beyond the Syllogistic Boundary

This paper is a contribution to natural logic, the study of logical systems for linguistic reasoning. We construct a system with the following properties: its syntax is closer to that of a natural language than is first-order logic; it can faithfully represent simple sentences with standard quantifiers, subject relative clauses (a recursive construct), and negation on nouns and verbs. We also give a proof system which is complete and has the finite model property. We go further by adding comparative adjective phrases, assuming interpretations by transitive relations. This last system has all the previously-mentioned properties as well. The paper was written for theoretical computer scientists and logicians interested in areas such as decidability questions for fragments of first-order logic, modal logic, and natural deduction.