Second order inference in natural language semantics

Abstract

In this paper I look at a number of apparently trivial validinferences (as well as some invalid and missing inferences) associatedwith the possessive construction and with different types ofadjectival modification of nouns. In the case of possessives, allanalyses I know of, whether implemented or not, systematicallysanction invalid inferences. In the case of adjectives, there are somemodel-theoretic linguistic analyses that are adequate at a theoretical level, but no satisfactory practical computational implementations that I am aware of which capture the correct inference patterns.A common thread between the possessive and the adjectivalconstructions is that to derive the correct inferences we need secondorder quantification. This is an uncontroversial move withinmodel-theoretic formal semantics but a problem for computationalsemantics, since we have no fully automated theorem provers foranything other than first order logic (and only for subsets of firstorder logic do we have provers that are both fully decidable andefficient). I explore what is needed to provide a proof-theoreticaccount of the relevant inference patterns, and suggest some analysesrequiring second order axioms. In order to make this a practicalcomputational possibility I go on to propose two techniques forapproximating such inferences in a first order setting. The suggestedanalyses have been fully implemented, and in an appendix I provide asmall FraCaS-like corpus of relevant examples, all of which arehandled correctly by the implementation.