•  151
    Higher Order Modal Logic
    In Patrick Blackburn, Johan Van Benthem & Frank Wolter (eds.), Handbook of Modal Logic, Elsevier. pp. 621-653. 2006.
    A logic is called higher order if it allows for quantification over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14].1 While classical type theory has since long been overshadowed by set theory as a foundation of mathemat…Read more
  •  148
    Language, Lambdas, and Logic
    In R. Oehrle & J. Kruijff (eds.), Resource Sensitivity, Binding, and Anaphora (Studies in Linguistics and Philosophy 80), Kluwer Academic Publishers. pp. 23--54. 2003.
    The paper develops Lambda Grammars, a form of categorial grammar that, unlike other categorial formalisms, is non-directional. Linguistic signs are represented as sequences of lambda terms and are combined with the help of linear combinators.
  •  379
    Combining Montague semantics and discourse representation
    Linguistics and Philosophy 19 (2). 1996.
    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.
  •  52
    Hyperfine-grained meanings in classical logic
    Logique Et Analyse 133 159-176. 1991.
    This paper develops a semantics for a fragment of English that is based on the idea of `impossible possible worlds'. This idea has earlier been formulated by authors such as Montague, Cresswell, Hintikka, and Rantala, but the present set-up shows how it can be formalized in a completely unproblematic logic---the ordinary classical theory of types. The theory is put to use in an account of propositional attitudes that is `hyperfine-grained', i.e. that does not suffer from the well-known problems …Read more
  •  111
    Intensional models for the theory of types
    Journal of Symbolic Logic 72 (1): 98-118. 2007.
    In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly,…Read more
  •  142
    Existence predicate
    In R. E. Asher & J. M. Y. Simpson (eds.), The Encyclopedia of Language and Linguistics, Pergamon. pp. 1191. 1993.
    Kant said that existence is not a predicate and Russell agreed, arguing that a sentence such as ‘The king of France exists’, which seems to attribute existence to the king of France, really has a logical form that is not reflected in the surface structure of the sentence at all. While the surface form of the sentence consists of a subject and a predicate, the underlying logical form, according to Russell, is the formula given in. This formula obviously has no subjectpredicate form and in fact ha…Read more
  •  118
    Perception Verbs
    In R. E. Asher & J. M. Y. Simpson (eds.), The Encyclopedia of Language and Linguistics, Pergamon Press. pp. 6--2999. 1993.
    The semantics of a sentence containing a perception verb such as see or hear depends to a high degree on the exact syntactic form of the perception verb’s complement. Let us compare sentence (1), where the complement is tenseless, with (2), where the complement is a tensed clause
  •  188
    Categorial Grammar and Lexical-Functional Grammar
    In Miriam Butt & Tracey Holloway King (eds.), Proceedings of the LFG01 Conference, University of Hong Kong, Csli Publications. pp. 259-279. 2001.
    This paper introduces λ-grammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, λ-grammars are multi-dimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of linear combinators, closed pure λ-terms in which each abstractor binds exactly one variable. Mathematically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the metho…Read more
  •  118
    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.
  •  90
    A relational formulation of the theory of types
    Linguistics and Philosophy 12 (3). 1989.
    This paper developes a relational---as opposed to a functional---theory of types. The theory is based on Hilbert and Bernays' eta operator plus the identity symbol, from which Church's lambda and the other usual operators are then defined. The logic is intended for use in the semantics of natural language.
  •  52
    Partial Information
    In R. E. Asher & J. M. Y. Simpson (eds.), The Encyclopedia of Language and Linguistics, Pergamon Press. pp. 6--2952. 1993.
    the world of phenomena is immensely large this means we can perceive only part of the world. We see, feel and hear parts of reality, not the whole of it, and it seems that a sentence containing a verb of perception like 'John sees a house burn' is most naturally treated as saying that the subject sees an incomplete world in which the embedded sentence is true (see Barwise (1981) for this analysis). But if we want to analyse perception verbs thus, we must introduce some form of incompleteness int…Read more
  •  130
    An Analytic Tableau System for Natural Logic
    In Maria Aloni, H. Bastiaanse, T. De Jager & Katrin Schulz (eds.), Logic, Language and Meaning, Springer. pp. 104-113. 2010.
    Logic has its roots in the study of valid argument, but while traditional logicians worked with natural language directly, modern approaches first translate natural arguments into an artificial language. The reason for this step is that some artificial languages now have very well developed inferential systems. There is no doubt that this is a great advantage in general, but for the study of natural reasoning it is a drawback that the original linguistic forms get lost in translation. An alterna…Read more
  • Games, Logic, and Constructive Sets (edited book)
    with Grigori Mints
    Center for the Study of Language and Inf. 2003.
    Mathematical game theory has been embraced by a variety of scholars: social scientists, biologists, linguists, and now, increasingly, logicians. This volume illustrates the recent advances of game theory in the field. Logicians benefit from things like game theory's ability to explain informational independence between connectives; meanwhile, game theorists have even begun to benefit from logical epistemic analyses of game states. In concert with such pioneering work, this volume also present su…Read more
  •  142
    A squib on anaphora and coindexing
    Linguistics and Philosophy 34 (1): 85-89. 2011.
    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 t…Read more
  •  59
    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 Montague's theory of natural language semantics and the work by Kamp and Heim.
  •  107
    Construction by Description in Discourse Representation
    with Noor van Leusen
    In Jaroslav Peregrin (ed.), Meaning: The Dynamic Turn, Elsevier. pp. 33-65. 2003.
    This paper uses classical logic for a simultaneous description of the syntax and semantics of a fragment of English and it is argued that such an approach to natural language allows procedural aspects of linguistic theory to get a purely declarative formulation. In particular, it will be shown how certain construction rules in Discourse Representation Theory, such as the rule that indefinites create new discourse referents and definites pick up an existing referent, can be formulated declarative…Read more
  • Interfaces: Explorations in Logic, Language and Computation (edited book)
    with T. Icard
    Springer Berlin. 2010.
    The European Summer School in Logic, Language and Information (ESSLLI) takes place every year, each time at a different location in Europe. With its focus on the large interdisciplinary area where linguistics, logic and computation converge, it has become very popular since it started in 1989, attracting large crowds of students. ESSLLI is where everyone in the field meets, teaches, takes courses, gives talks, dances all night, and generally has a good time. One of the enjoyable features of the …Read more