Recent Progress on Monotonicity

Recent Progress on Monotonicity Thomas F. Icard, III1 and Lawrence S. Moss2 This paper serves two purposes. It is a summary of much work con- cerning formal treatments of monotonicity and polarity in natural lan- guage, and it also discusses connections to related work on exclusion relations, and connections to psycholinguistics and computational lin- guistics. The second part of the paper presents a summary of some new work on a formal Monotonicity Calculus. 1 Introduction It has been known since Aristotle that many entailment patterns be- tween linguistic expressions can be derived by simple predicate replace- ment. Monotonicity reasoning, involving a particular sort of predicate replacement, has played an especially important role in the develop- ment of logic and semantics, and continues to do so today. Monotonicity is a pervasive feature of natural language, and it has been linked to many fundamental aspects of linguistic processing, rea- soning, and even grammar. In the late 1980s van Benthem (1986, 1991) and S´anchez-Valencia (1991) defined proof systems for reasoning about entailment using monotonicity in higher-order languages. Working in a simply-typed language and building on the long line of work in cate- gorial grammars, the idea behind the so-called Monotonicity Calculus was to mark expressions of functional type with monotonicity informa- tion, and use this in a proof system. Work on this topic has recently been revived by computational linguists, and it has found its way into 1Department of Philosophy, Stanford University. 2Department of Mathematics, Indiana University, Bloomington. This work was partially supported by a grant from the Simons Foundation (#245591 to Lawrence Moss). 167 LiLT Volume 9 Perspectives on Semantic Representations for Textual Inference. Copyright c ⃝ 2014, CSLI Publications. 168 / Thomas F. Icard, III and Lawrence S. Moss important applications (see Section 4). This paper o↵ers an overview of recent developments in the study of monotonicity in natural language. Strictly speaking, monotonicity (and its opposite antonicity) is a property of functions, and is therefore a semantic notion. However, we will also be concerned with what is usually called polarity, a syntactic notion. In fact, the Monotonicity Calculus and related logical systems can be thought of as exploiting the close connection between polarity and monotonicity. We use syntactic markings to define proof systems whose soundness is guaranteed by the correspondences between positive polarity and monotonicity on the one hand, and negative polarity and antitonicity on the other. One of the main contributions of this paper is to make these notions fully precise, and to present a completeness result for the Monotonicity Calculus. The outline of the paper is as follows. Section 2 is a less technical introduction to the fundamental concepts involved. In Sections 2.1-2.3, we introduce the notions of monotonicity, antitonicity, and polarity. Using examples from natural language semantics and basic algebra, we illustrate the use of marked types, and explain how expressions of these types are interpreted model-theoretically. We also outline the fun- damental algorithms used for marking parsed expressions with polarity information. In Section 2.5, we discuss a …

Content truncated to 500 words