Reinhard Muskens

Combining Montague semantics and discourse representation

This paper embeds the core part of Discourse Representation Theory in the classical theory of types plus a few simple axioms that allow the theory to express key facts about variables and assignments on the object level of the logic. It is shown how the embedding can be used to combine core analyses of natural language phenomena in Discourse Representation Theory with analyses that can be obtained in Montague Semantics.

Language, lambdas, and logic

In Van Benthem, 1986 it was observed that the Curry-Howard correspondence between proofs and λ-terms can be exploited to obtain a very elegant and principled match between Lambek Categorial Grammar and Montague Semantics. The correspondence associates each proof of the calculus with a λ-term and Van Benthem shows how such terms can be used as a recipe for obtaining the meaning of a complex expression in terms of the meanings of its parts. The method is easily extended to various other forms of Lambek calculi, including multimodal calculi (see Moortgat, 1997 and references therein).

Talking about Trees and Truth-Conditions

We present Logical Description Grammar (LDG), a model ofgrammar and the syntax-semantics interface based on descriptions inelementary logic. A description may simultaneously describe the syntacticstructure and the semantics of a natural language expression, i.e., thedescribing logic talks about the trees and about the truth-conditionsof the language described. Logical Description Grammars offer a naturalway of dealing with underspecification in natural language syntax andsemantics. If a logical description (up to isomorphism) has exactly onetree plus truth-conditions as a model, it completely specifies thatgrammatical object. More common is the situation, corresponding tounderspecification, in which there is more than one model. A situation inwhich there are no models corresponds to an ungrammatical input.

Categorial Grammar and Discourse Representation Theory

In this paper it is shown how simple texts that can be parsed in a Lambek Categorial Grammar can also automatically be provided with a semantics in the form of a Discourse Representation Structure in the sense of Kamp [1981]. The assignment of meanings to texts uses the Curry-Howard-Van Benthem correspondence.

Anaphora and the Logic of Change

This paper shows how the dynamic interpretation of natural language introduced in work by Hans Kamp and Irene Heim can be modeled in classical type logic. This provides a synthesis between Richard Montagues theory of natural language semantics and the work by Kamp and Heim.

Chapter 10 – Dynamics

Publisher Summary This chapter provides an overview of some important dynamic theories in linguistics and artificial intelligence and shows the way in which these fit into the general perspective on communication. It contains some more general logical considerations on dynamic phenomena, discussing various ways to model their logic and the way the logic that emerges is related to its classical static predecessors. Natural language involves different dynamic mechanisms. Discourse representation theory and dynamic predicate logic highlight changing anaphoric bindings, whereas Veltmans Update semantics focuses on information flow and epistemic statements about its stages. The basic ideas of Heims file change semantics (FCS) and Kamps discourse representation theory (DRT) are very much the same. While in Heims theory the readers or hearers of a text represent the information that they have obtained by means of a file, DRT lets them keep track of that information with the help of a discourse representation structure.

New Directions in Type-Theoretic Grammars

This paper argues for the idea that in describing language we should follow Haskell Curry in distinguishing between the structure of an expression and its appearance or manifestation. It is explained how making this distinction obviates the need for directed types in type-theoretic grammars and a simple grammatical formalism is sketched in which representations at all levels are lambda terms. The lambda term representing the abstract structure of an expression is homomorphically translated to a lambda term representing its manifestation, but also to a lambda term representing its semantics.

A squib on anaphora and coindexing

There are two kinds of semantic theories of anaphora. Some, such as Heim’s File Change Semantics, Groenendijk and Stokhof’s Dynamic Predicate Logic, or Muskens’ Compositional DRT (CDRT), seem to require full coindexing of anaphora and their antecedents prior to interpretation. Others, such as Kamp’s Discourse Representation Theory (DRT), do not require this coindexing and seem to have an important advantage here. In this squib I will sketch a procedure that the first group of theories may help themselves to so that they can interleave interpretation and coindexing in DRT’s way.

A Gentzen Calculus for Nothing but the Truth

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.

From Bi-facial Truth to Bi-facial Proofs

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two “semi-classical negations” on them. The resulting semantics is used to define three novel logics which are closely related to Belnap’s well-known four valued logic. A syntactic characterization of these logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap’s logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic.