Comparing exhaustivity operators

Benjamin Spector

doi:10.3765/sp.9.11

Abstract

In this paper, I investigate the formal relationships between two types of exhaustivity operators that have been discussed in the literature, one based on minimal worlds/models, noted exh-mw (van Rooij & Schulz 2004, Schulz & van Rooij 2006, Spector 2003, 2006, with roots in Szabolcsi 1983, Groenendijk & Stokhof 1984), and one based on the notion of innocent exclusion, noted exh-ie (Fox 2007). Among others, I prove that whenever the set of alternatives relative to which exhaustification takes place is semantically closed under conjunction, the two operators are necessarily equivalent. Together with other results, this provides a method to simplify, in some cases, the computation associated with exh-ie, and, in particular, to drastically reduce the number of alternatives to be considered. Besides their practical relevance, these results clarify the formal relationships between both types of operators. BibTeX info

Highlights

In the literature on scalar implicatures and exhaustivity effects, it has proved useful to define so-called exhaustivity operators
A world assigns a denotation to every non-logical atomic expression of the language, and, through compositional semantic rules, a truth-value to every sentence in the language
Comparing exhaustivity operators of pairs of members of E, and we do the same thing with the resulting set, which yields a new set, ad infinitum

Summary

Introduction

In the literature on scalar implicatures and exhaustivity effects, it has proved useful to define so-called exhaustivity operators Such operators take two arguments, a proposition φ and a set of propositions ALT (the alternatives of φ), and return a proposition that entails φ, which corresponds to the pragmatically strengthened meaning of φ (the conjunction of φ and its quantity implicatures). In words: exhkrifka(ALT, φ) states that φ is true and that every member of ALT that is true is entailed by φ, i.e., every non-entailed member of ALT is false This operator has been known for quite some time to be inadequate, for disjunctive sentences. I adopt the set-theoretic notation φ ⊆ ψ to mean that φ entails ψ

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Conclusion