Journal of Pure and Applied Mathematics: Advances and Applications Volume 18, Number 2, 2017, Pages 75-86 Available at http://scientificadvances.co.in DOI: http://dx.doi.org/10.18642/jpamaa_7100121874 2010 Mathematics Subject Classification: 03B65, 68W01, 03C10. Keywords and phrases: logic of natural languages, algorithms, completeness. Received October 8, 2017  2017 Scientific Advances Publishers ON COMPLETENESS AND INFERENCE ALGORITHMS OF THE LOGIC CARD WITH IA TAHSİN ÖNER and SELÇUK TOPAL Department of Mathematics Ege University Izmir 35100 Turkey e-mail: tahsin.oner@ege.edu.tr Department of Mathematics Bitlis Eren University Bitlis, 13000 Turkey e-mail: s.topal@beu.edu.tr Abstract This paper extends the logic CARD of comparisons between the sizes of sets to the logic CARD with IA which contains intersecting adjectives. We prove completeness of the logics CARD and CARD with IA. We also give algorithmic analysis and some algebraic properties of the logics. 1. Introduction First attempts of proving completeness were made for syllogistic inferences by [1] and [2]. Completeness results of some syllogistic logics containing CARD were given in [3] by Moss after some complexity