Meta-Lambda Calculus and Linguistic Monads ⋆ Daisuke Bekki 1,2,3 and Moe Masuko 1,4 1 Ochanomizu University, Graduate School of Humanities and Sciences ⋆⋆ 2 National Institute of Informatics ⋆⋆⋆ 3 CREST, Japan Science and Technology Agency † 4 Research Fellow of the Japan Society for the Promotion of Science {bekki, masuko.moe}@is.ocha.ac.jp Abstract. Meta-lambda calculus (MLC) is a two-level typed lambda calculus with meta-level types and terms. MLC has been adopted in the analyses of natural language semantics and pragmatics by means of monads and monadic translation in [1][2], however, the soundness of the equational theory in [1] has not been fully proven with respect to the categorical semantics in [1]. In this article, we introduce a revised syntax and an equational theory of MLC with base-level/meta-level substitu- tion and α/β/η-conversions, and prove their soundness with respect to a revised categorical semantics of MLC. 1 Introduction 1.1 Monads in Category Theory and Programming Language The notion of monads originates in homological algebra and category theory: a monad in a category C is a triple 〈T ,η,μ〉 that consists of a functor T : C -→C and two natural transformations: η : Id C ■ -→ T , μ : T 2 ■ -→ T such that the following diagrams commute for any object A in C . T 3 A T μA  μ T A  T 2 A μA  T 2 A μA  T A T A η T A  Id C          T 2 A μA  T A T ηA  Id C         T A ⋆ Our sincere thanks to the participants at LENLS5 and LENLS6, especially Eric McCready, Robert van Rooij, Kei Yoshimoto, Masahiko Sato, and the late Norry Ogata, and also to the participants at FLOPS2010, especially Olivier Danvy, Ian Zerney, Chung-chieh Shan, Oleg Kiselyov, Yukiyoshi Kameyama for their valuable comments. Our special thanks go to Kenichi Asai for many discussions in our re- search group. We also thank the anonymous reviewers for their insightful comments. Daisuke Bekki is partially supported by a Grant-in-Aid for Young Scientists (A) (No. 22680013) from the Ministry of Education, Science, Sports and Culture. ⋆⋆ 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan. ⋆⋆⋆ 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan. † 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan.