Chapter 2 Monotonic Reasoning from a Proof-Theoretic Perspective RAFFAELLA BERNARDI ABSTRACT. The article presents the first results we have obtained studying natural reasoning from a proof-theoretic perspective. In particular, we focus our attention on monotonicity reason- ing: Inferences are made using structurally parsed sentences on which monotonic positions are displayed. The monotonicity markers are propagated through the proofs via the combined struc- tural and logical rules for the unary operators of Multimodal Categorial Grammar (MMCG). We have chosen to work with such an expressive ‘grammar logic’ in order to avoid both the use of extra-logical marking devices as made in [ SV91] and a too complex lexicon [ Dow94]. With MMCG as the parser, the system is able to make the derivations fully within the logic. 2.1 Introduction: ‘Natural Logic’ The task of accounting for the role of language in drawing inferences has been commonly considered to belong to the domain of formal semantics. Most of the literature in natural reasoning assumes a model-theoretic perspective and uses a formal language as an intermediate step into which natural language expressions are translated. However, within the generalized type-logical framework [ Moo97] the chal- lenge is to assume a direct proof-theoretic perspective; this entails that instead of employing logical forms as vehicles of inference, natural language expressions are used directly, and instead of taking models into account, the validity of an infer- ence is read off the derivation. A system satisfying these criteria will be called a Natural Logic. We restrict our attention to monotonicity inference in natural language, i.e., inferences that involve replacing an expression with an expression the denotation of which is a subset or a superset of the denotation of the original expression. Our goal is to have a system which automatically accounts for this kind of inference using (1) A formal grammar which calculates and marks the monotonicity positions in a sentence while it is (syntactically) parsing it; and (2) a logic which derives the inferences using the marked output. A similar approach has been assumed by Sanchez in [ SV91], where linguistic expressions are analyzed by a non-directional Categorial Grammar – i.e., Lambek 13 Proceedings of Formal Grammar 1999 Geert-Jan M. Kruijff & Richard T. Oehrle (editors). Chapter 2, Copyright c 1999, Raffaella Bernardi.