J Log Lang Inf (2011) 20:397–417 DOI 10.1007/s10849-011-9137-x Syllogistic Logic with Comparative Adjectives Lawrence S. Moss Published online: 26 April 2011 © Springer Science+Business Media B.V. 2011 Abstract This paper adds comparative adjectives to two systems of syllogistic logic. The comparatives are interpreted by transitive and irreflexive relations on the under- lying domain. The main point is to obtain sound and complete axiomatizations of the valid formulas in the logics. Keywords Syllogistic logic · Completeness · Adjectives · Transitive relations 1 Introduction This paper is a contribution to the area of natural logic, the study of inference patterns of sentences in natural language studied as close to the surface as possible, hence without recourse to translation into first-order logic. It is concerned with comparative adjective phrases in syllogistic logics. We shall propose and study logical systems in which one can express arguments such as the following: Every giraffe is taller than every gnu Some gnu is taller than every lion Some lion is taller than some zebra Every giraffe is taller than some zebra (1) We encourage the reader to work out an argument showing that the conclusion of (1) follows from the premises. A picture would do. Any convincing argument must appeal in some way to the transitivity of is taller than; without transitivity, the conclu- sion does not follow. (The argument also depends on the subject-wide scope reading L. S. Moss (B ) Department of Mathematics, Indiana University, Bloomington, IN 47405, USA e-mail: lsm@cs.indiana.edu 123