Conventional implicature

This chapter presents a monadic analysis of conventional implicatures. These expressions are compositionally challenging and also seem to challenge the traditional semantics/pragmatics divide by straddling it. This chapter first introduces two main sorts of conventional implicature, appositives and expressives. It reviews one standard approach to capturing the dual nature of conventional implicatures, multidimensional semantic representations. It then reviews some challenges and argues that they do not entail abandoning multidimensionality. The chapter introduces a new multidimensional analysis using monads. Two examples are analysed in detail. The first is an example of a conventional implicature arising from an appositive. The second is an example of a conventional implicature arising from an expressive predicate, which is a more controversial case. The chapter shows that the enriched meaning analysis naturally extends to this case without imperilling the intuition behind multidimensionality. The chapter ends with some exercises to aid understanding.

Category theory

This chapter aims to introduce sufficient category theory to enable a formal understanding of the rest of the book. It first introduces the fundamental notion of a category. It then introduces functors, which are maps between categories. Next it introduces natural transformations, which are natural ways of mapping between functors. The stage is then set to at last introduces monads, which are defined in terms of functors and natural transformations. The last part of the chapter provides a compositional calculus with monads for natural language semantics (in other words, a logic for working with monads) and then relates the compositional calculus to Glue Semantics and to a very simple categorial grammar for parsing. The chapter ends with some exercises to aid understanding.

Monad combinatorics

This chapter considers how the monadic formalization deals with interactions between the phenomena from the second part of the book by examining the pairwise interaction of all three phenomena. Distributive laws are introduced to combine monads. The chapter shows that only certain combinations of the monads from the second part have definable distributive laws. These results comport with linguistic intuitions. The option of using the related category-theoretic concepts of functors and applicative functors instead of monads is also considered. The chapter shows that functors are not powerful enough and that applicative functors introduce a tracking/layering problem that is inelegant. Also, the closure property of applicative functor composition, whereby the composition of any two applicative functors is also an applicative functor, overgenerates with respect to the data. Monads are therefore argued to be empirically superior to applicative functors in this domain precisely because they lack the closure property. Some exercises are provided to aid understanding.

Enriched meanings in semantics and pragmatics

This chapter briefly introduces the notion of enriched meanings from a general, pretheoretical perspective, without reference to the apparatus used in subsequent chapters to capture the notion formally. The chapter begins by discussing the traditional distinction between semantics and pragmatics, in light of the distinction between truth-conditional and conventional aspects of meaning. It then introduces and defines enriched meanings. It finishes with a brief overview of the various phenomena whose analyses constitute the case studies in the second part of the book (conventional implicatures, substitution puzzles, and conjunction fallacies).

Introduction

This chapter introduces and motivates the book. It introduces monads as a way to model enriched meanings and motivates enriched meanings as a way to avoid generalizing to the worst case in natural language interpretation. It reviews the three goals of the book: 1. to provide background on the theory of enriched meanings and how to model meaning enrichment formally using category theory, in particular monads; 2. to show the usefulness of the theory by providing new compositional analyses of the three phenomena; and 3. to explore the compositional possibilities for combining the three monads used in these analyses. The chapter also discusses the place of this kind of research in cognitive science. It lists some related literature on monads for natural language interpretation. It also introduces the computational tools and exercises.

Perspectival reference

This chapter is about substitutability of co-referential terms and argues that what is required is a general semantics of perspectives. It begins by reviewing the standard sort of puzzle, which involves embedding distinct terms under a propositional attitude verb. It then reviews evidence that embedding is in fact not necessary for a substitution puzzle to arise. It also reviews a related puzzle that shows that similar problems can arise without distinct terms. It builds on this to illustrate a limiting case of lack of substitutability, which involves an identity statement. Two formalizations are compared. A Logical Form analysis is shown to have various issues, including generalizing meanings to the worst case. A contrasting monadic analysis is presented which avoids this problem and is demonstrated with four key examples. The chapter ends with a generalization to non-referential expressions. Some exercises are provided to aid understanding.

Uncertainty and conjunction fallacies

This chapter examines conjunction fallacies. This phenomenon is a topic in the psychology of reasoning and is not strictly linguistic, but it is related to pragmatics. Monads are shown to capture conjunction fallacies compositionally, which has eluded prominent prior theories. The chapter contrasts a monad built around the probability semiring with one built around a simpler semiring, the one semiring. The choice between the probability and one semirings partially predicts experimental participants’ behaviour. This points to an explanation in terms of the satisficing heuristic, rather than the representativeness heuristic. The chapter explores two options for fully predicting the results. The first is to use Gricean pragmatics in addition to the one semiring. The second is to use alternative underlying semirings for the monad: tropical semirings. This alternative compositional solution achieves a highly satisfying fit with aggregate psychological data and preserves an interesting duality between the logical operations of conjunction and disjunction. Some exercises are provided to aid understanding.

Conclusion

This chapter is a short conclusion. It briefly reviews the notion of enriched meanings. It then summarizes the main results of the book, chapter by chapter.