
1198Combining Montague semantics and discourse representationLinguistics 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.

1066TypeLogical SemanticsRoutledge Encyclopedia of Philosophy Online. 2011.Typelogical semantics studies linguistic meaning with the help of the theory of types. The latter originated with Russell as an answer to the paradoxes, but has the additional virtue that it is very close to ordinary language. In fact, type theory is so much more similar to language than predicate logic is, that adopting it as a vehicle of representation can overcome the mismatches between grammatical form and predicate logical form that were observed by Frege and Russell. The grammatical forms…Read more

658Lambda Grammars and the SyntaxSemantics InterfaceIn Robert Van Rooij & Martin Stokhof (eds.), Proceedings of the Thirteenth Amsterdam Colloquium, Illc. pp. 150155. 2001.In this paper we discuss a new perspective on the syntaxsemantics interface. Semantics, in this new setup, is not ‘read off’ from Logical Forms as in mainstream approaches to generative grammar. Nor is it assigned to syntactic proofs using a CurryHoward correspondence as in versions of the Lambek Calculus, or read off from fstructures using Linear Logic as in LexicalFunctional Grammar (LFG, Kaplan & Bresnan [9]). All such approaches are based on the idea that syntactic objects (trees, proof…Read more

644Sense and the computation of referenceLinguistics and Philosophy 28 (4). 2004.The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends propositions to truth values or sets of possible worlds in Thomason (1980) must be replaced by a relation and the meaning postulates governing the behaviour of this relation must be given in the form…Read more

609Intensional models for the theory of typesJournal of Symbolic Logic 72 (1): 98118. 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 cutfree 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

604Context Update for Lambdas and VectorsIn Maxime Amblard, Philippe de Groote, Sylvain Pogodalla & Christian Rétoré (eds.), Logical Aspects of Computational Linguistics. Celebrating 20 Years of LACL (1996–2016), Springer. pp. 247254. 2016.Vector models of language are based on the contextual aspects of words and how they cooccur in text. Truth conditional models focus on the logical aspects of language, the denotations of phrases, and their compositional properties. In the latter approach the denotation of a sentence determines its truth conditions and can be taken to be a truth value, a set of possible worlds, a context change potential, or similar. In this short paper, we develop a vector semantics for language based on the si…Read more

591Static and dynamic vector semantics for lambda calculus models of natural languageJournal of Language Modelling 6 (2): 319351. 2018.Vector models of language are based on the contextual aspects of language, the distributions of words and how they cooccur in text. Truth conditional models focus on the logical aspects of language, compositional properties of words and how they compose to form sentences. In the truth conditional approach, the denotation of a sentence determines its truth conditions, which can be taken to be a truth value, a set of possible worlds, a context change potential, or similar. In the vector models, t…Read more

554A Theory of Names and True IntensionalityIn Maria Aloni, V. Kimmelman, Floris Roelofsen, G. Weidman Sassoon, Katrin Schulz & M. Westera (eds.), Logic, Language and Meaning: 18th Amsterdam Colloquium, Springer. pp. 441449. 2012.Standard approaches to proper names, based on Kripke's views, hold that the semantic values of expressions are (settheoretic) functions from possible worlds to extensions and that names are rigid designators, i.e.\ that their values are \emph{constant} functions from worlds to entities. The difficulties with these approaches are wellknown and in this paper we develop an alternative. Based on earlier work on a higher order logic that is \emph{truly intensional} in the sense that it does not val…Read more

511Analytic Tableaux for all of SIXTEEN 3Journal of Philosophical Logic 44 (5): 473487. 2015.In this paper we give an analytic tableau calculus P L 1 6 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧ t, ⊧ f, ⊧ i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment relations will in general require devel…Read more

509A squib on anaphora and coindexingLinguistics and Philosophy 34 (1): 8589. 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

487Categorial Grammar and LexicalFunctional GrammarIn Miriam Butt & Tracey Holloway King (eds.), Proceedings of the LFG01 Conference, University of Hong Kong, Csli Publications. pp. 259279. 2001.This paper introduces λgrammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, λgrammars are multidimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of linear combinators, closed pure λterms 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

473Higher Order Modal LogicIn Patrick Blackburn, Johan van Benthem & Frank Wolter (eds.), Handbook of Modal Logic, Elsevier. pp. 621653. 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

448Language, Lambdas, and LogicIn R. Oehrle & J. Kruijff (eds.), resource sensitivity, binding, and anaphora, Kluwer. pp. 2354. 2003.The paper develops Lambda Grammars, a form of categorial grammar that, unlike other categorial formalisms, is nondirectional. Linguistic signs are represented as sequences of lambda terms and are combined with the help of linear combinators.

443New directions in typetheoretic grammarsJournal of Logic, Language and Information 19 (2): 129136. 2010.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 typetheoretic 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…Read more

437Existence predicateIn 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

421Interpolation in 16Valued Trilattice LogicsStudia Logica 106 (2): 345370. 2018.In a recent paper we have defined an analytic tableau calculus PL_16 for a functionally complete extension of Shramko and Wansing's logic based on the trilattice SIXTEEN_3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logicsuch as the relations =_t, =_f, and =_i that each correspond to a lattice order in SIXTEEN_3; and =, the intersection of =_t and =_f,. It turns out that our method of characterising these sema…Read more

420A Gentzen Calculus for Nothing but the TruthJournal of Philosophical Logic 45 (4): 451465. 2016.In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic, an interesting variation upon the fourvalued logic for firstdegree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbertstyle 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 BelnapDunn l…Read more

420On Partial and Paraconsistent LogicsNotre Dame Journal of Formal Logic 40 (3): 352374. 1999.In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of NonContradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the CraigLyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent …Read more

416Tense and the logic of changeIn Talking about Trees and TruthConditions, Springer Verlag. pp. 147183. 1991.In this paper it is shown how the DRT (Discourse Representation Theory) treatment of temporal anaphora can be formalized within a version of Montague Semantics that is based on classical type logic.

414An Analytic Tableau System for Natural LogicIn Maria Aloni, H. Bastiaanse, T. De Jager & Katrin Schulz (eds.), Logic, Language, and Meaning: Selected Papers from the 17th Amsterdam Colloquium, Springer. pp. 104113. 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

403From Bifacial Truth to Bifacial ProofsStudia Logica 103 (3): 545558. 2015.In their recent paper Bifacial 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 “semiclassical negations” on them. The resulting semantics is used to define three novel logics which are closely related to Belnap’s wellknown four valued logic…Read more

385Separating syntax and combinatorics in categorial grammarResearch on Language and Computation 5 (3): 267285. 2007.The ‘syntax’ and ‘combinatorics’ of my title are what Curry (1961) referred to as phenogrammatics and tectogrammatics respectively. Tectogrammatics is concerned with the abstract combinatorial structure of the grammar and directly informs semantics, while phenogrammatics deals with concrete operations on syntactic data structures such as trees or strings. In a series of previous papers (Muskens, 2001a; Muskens, 2001b; Muskens, 2003) I have argued for an architecture of the grammar in which ﬁnite…Read more

383Underspecified semanticsIn Klaus von Heusinger & Urs Egli (eds.), Reference and Anaphoric Relations, Kluwer Academic Publishers. pp. 311338. 2000.Ambiguities in natural language can multiply so fast that no person or machine can be expected to process a text of even moderate length by enumerating all possible disambiguations. A sentence containing $n$ scope bearing elements which are freely permutable will have $n!$ readings, if there are no other, say lexical or syntactic, sources of ambiguity. A series of $m$ such sentences would lead to $(n!)^m$ possibilities. All in all the growth of possibilities will be so fast that generating read…Read more

376Talking about trees and truthconditionsJournal of Logic, Language and Information 10 (4): 417455. 2001.We present Logical Description Grammar (LDG), a model ofgrammar and the syntaxsemantics 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 truthconditionsof the language described. Logical Description Grammars offer a naturalway of dealing with underspecification in natural language syntax andsemantics. If a logical …Read more

372Categorial grammar and discourse representation theoryIn Yorick Wilks (ed.), Proceedings of COLING 94, . pp. 508514. 1994.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 CurryHowardVan Benthem correspondence.

328Construction by Description in Discourse RepresentationIn Jaroslav Peregrin (ed.), Meaning: the dynamic turn, Elsevier Science. pp. 3365. 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

304Description theory, LTAGs and Underspecified SemanticsIn Anne Abeillé, Tilman Becker, Giorgio Satta & K. VijayShanker (eds.), Fourth International Workshop on Tree Adjoining Grammars and Related Frameworks, Institute For Research in Cognitive Science. pp. 112115. 1998.An attractive way to model the relation between an underspecified syntactic representation and its completions is to let the underspecified representation correspond to a logical description and the completions to the models of that description. This approach, which underlies the Description Theory of Marcus et al. 1983 has been integrated in VijayShanker 1992 with a pure unification approach to Lexicalized TreeAdjoining Grammars (Joshi et al. 1975, Schabes 1990). We generalize Description Th…Read more

285Orderindependence and UnderspecificationIn Hans Kamp & Barbara Hall Partee (eds.), Contextdependence in the analysis of linguistic meaning, Elsevier. pp. 11239. 2004.

279Synonymy, common knowledge, and the social construction of meaningIn Paul Dekker & Michael Franke (eds.), Proceedings of the Fifteenth Amsterdam Colloquium, Illc. pp. 161166. 2005.In this paper it is shown how a formal theory of interpretation in Montague’s style can be reconciled with a view on meaning as a social construct. We sketch a formal theory in which agents can have their own theory of interpretation and in which groups can have common theories of interpretation. Frege solved the problem how different persons can have access to the same proposition by placing the proposition in a Platonic realm, independent from all language users but accessible to all of them. …Read more

253A calculus for Belnap's logic in which each proof consists of two treesLogique Et Analyse 220 643656. 2012.In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation…Read more
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Logic and Philosophy of Logic 
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Semantic Theories 