Reasoning with Polarity in Categorial Type Logic

The research presented in this thesis follows the parsing as deduction approach to lin- guistics. We use the tools of Categorial Type Logic (CTL) to study the interface of natural language syntax and semantics. Our aim is to investigate the mathematical structure of CTL and explore the possibilities it offers for analyzing natural language structures and their interpretation. The thesis is divided into three parts. Each of them has an introductory chapter. In Chapter 1, we introduce the background assumptions of the categorial approach in linguistics, and we sketch the developments that have led to the introduction of CTL. We discuss the motivation for using logical methods in linguistic analysis. In Chapter 3, we propose our view on the use of unary modalities as `logical features'. In Chapter 5, we set up a general notion of grammatical composition taking into account the form and the meaning dimensions of linguistic expressions. We develop a logical theory of licensing and antilicensing relations that cross-cuts the form and meaning dimensions. Throughout the thesis we focus attention on polarity. This term refers both to the polarity of the logical operators of CTL and to the polarity items one finds in natural language, which, furthermore, are closely connected to natural reasoning. Therefore, the title of this thesis Reasoning with Polarity in Categorial Type Logic is intended to express three meanings. Firstly, we reason with the polarity of the logical operators of CTL and study their derivability patterns. In Chapter 2, we explore the algebraic principles that govern the behavior of the type-forming operations of the Lambek calculus. We extend the categorial vocabulary with downward entailing unary operations obtaining the full tool- kit that we use in the rest of the thesis. We employ unary operators to encode and compute monotonicity information (Chapter 4), to account for the different ways of scope taking of generalized quantifiers (Chapter 6), and to model licensing and antilicensing relations (Chapter 7). Secondly, in Chapter 4, we model natural reasoning inferences drawn from structures suitable for negative polarity item occurrences. In particular, we describe a system of inference based on CTL. By decorating functional types with unary operators we encode the semantic distinction between upward and downward monotone functions. Moreover, we study the advantages of this encoding by exploring the contribution of v monotone functions to the study of natural reasoning and to the analysis of the syntactic distribution of negative polarity items. Thirdly, in Chapter 7, we study the distribution of polarity-sensitive expressions. We show how our theory of licensing and antilicensing relations successfully differentiates between negative polarity items, which are `attracted' by their triggers, and positive polarity items, which are `repelled' by them. We investigate these compatibility and incompatibility relations from a cross-linguistic perspective, and show how we reduce distributional differences between polarity-sensitive items in Dutch, Greek and Italian to differences in the lexical type assignments of these languages

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

Meaning versus Grammar

This volume investigates the complicated relationship between grammar, computation, and meaning in natural languages. It details conditions under which meaning-driven processing of natural language is feasible, discusses an operational and accessible implementation of the grammatical cycle for Dutch, and offers analyses of a number of further conjectures about constituency and entailment in natural language

Extracting proofs from documents

Often, theorem checkers like PVS are used to check an existing proof, which is part of some document. Since there is a large difference between the notations used in the documents and the notations used in the theorem checkers, it is usually a laborious task to convert an existing proof into a format which can be checked by a machine. In the system that we propose, the author is assisted in the process of converting an existing proof into the PVS language and having it checked by PVS. 1 Introduction The now-classic ALGOL 60 report [5] recognized three different levels of language: a reference language, a publication language and several hardware representations, whereby the publication language was intended to admit variations on the reference language and was to be used for stating and communicating processes. The importance of publication language ---often referred to nowadays as "pseudo-code"--- is difficult to exaggerate since a publication language is the most effective way..