
16TaP: A Toy Tableau Theorem Prover for 16Valued Trilattice LogicsA Programming Road to Logic, Maths, Language, and Philosophy : A Tribute to Jan van Eijck on the Occasion of His Retirement. 2017.A short description of a toy theorem prover for 16valued trilattice logics. Written for the occasion of my friend's Jan van Eijck's retirement. With a link to a swish interface to the prolog prover.

Context Update for Lambdas and VectorsIn Maxime Amblard, Philippe de Groote, Sylvain Pogodalla & Christian Retoré (eds.), Logical Aspects of Computational Linguistics. Celebrating 20 Years of LACL (19962016), 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

A 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

DynamicsIn Johan Van Benthem & Alice Ter Meulen (eds.), Handbook of Logic and Language, Elsevier. pp. 587648. 1997.

Meaning and PartialityCSLI Publications. 1995.This book radically simplifies Montague Semantics and generalizes the theory by basing it on a partial higher order logic. The resulting theory is a synthesis of Montague Semantics and Situation Semantics. In the late sixties Richard Montague developed the revolutionary idea that we can understand the concept of meaning in ordinary languages much in the same way as we understand the semantics of logical languages. Unfortunately, however, he formalized his idea in an unnecessarily complex way  t…Read more

Review of 'Exploring Logical Dynamics' by Johan van Benthem, 1998 (review)Algemeen Nederlands Tijdschrift voor Wijsbegeerte 90 (1): 8486. 1998.Veel Nederlandse woorden (dans, zet, oordeel, assertie, ...) duiden zowel een handeling aan als het resultaat van die handeling. Het fenomeen doet zich in vrijwel alle talen voor en het lijkt erop dat het menselijke cognitieve apparaat er niet zoveel moeite mee heeft te wisselen tussen een statisch perspectief dat resultaten ziet en een dynamisch perspectief dat vooral gericht is op de processen die tot die resultaten geleid hebben. De filosofie heeft meer moeite met het wisselen tussen een stat…Read more

From 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

CoreferenceIn R. E. Asher & J. M. Y. Simpson (eds.), The Encyclopedia of Language and Linguistics, Pergamon. pp. 769. 1993.In mathematical languages and in predicate logic coreferential terms can be interchanged in any sentence without altering the truth value of that sentence. Replacing 3 + 5 by 12 − 4 in any formula of arithmetic will never lead from truth to falsity or from falsity to truth. But natural languages are different in this respect. While in some contexts it is always allowed to interchange coreferential terms, other contexts do not admit this. An example of the first sort of context is likes bananas: …Read more

TypeLogical 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

A 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

Tense and the logic of changeIn Urs Egli, Peter Pause, Christoph Schwarze, Arnim von Stechow & Götz Wienold (eds.), Lexical Knowledge in the Organization of Language, John Benjamins. pp. 147183. 1995.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.

Interpolation Methods for Dunn Logics and Their ExtensionsStudia Logica 105 (6): 13191347. 2017.The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of firstdegree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the \ calculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In t…Read more

Underspecified 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

A 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

Program semantics and classical logicIn CLAUS Report Nr 86, University of the Saarland. pp. 127. 1997).In the tradition of Denotational Semantics one usually lets program constructs take their denotations in reflexive domains, i.e. in domains where selfapplication is possible. For the bulk of programming constructs, however, working with reflexive domains is an unnecessary complication. In this paper we shall use the domains of ordinary classical type logic to provide the semantics of a simple programming language containing choice and recursion. We prove that the rule of {\em Scott Induction\/}…Read more

Separating 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

New 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

Propositional AttitudesIn R. E. Asher & J. M. Y. Simpson (eds.), The Encyclopedia of Language and Linguistics, Pergamon Press. 1993.Verbs such as know, believe, hope, fear, regret and desire are commonly taken to express an attitude that one may bear towards a proposition and are therefore called verbs of propositional attitude. Thus in (1) below the agent Cathy is reported to have a certain attitude

Talking 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

Orderindependence and UnderspecificationIn Hans Kamp & Barbara Hall Partee (eds.), ContextDependence in the Analysis of Linguistic Meaning, Elsevier. pp. 11239. 2004.

Synonymy, 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

The dynamics of discourse situations (extended abstract)In Paul Dekker, Martin Stokhof & Yde Venema (eds.), Proceedings of the Eleventh Amsterdam Colloquium, University of Amsterdam. pp. 247252. 1997.The effects of utterances such as cue phrases, keepturn markers, and grounding signals cannot be characterized as changes to a shared record of the propositions under discussed: the simplest (and arguably most natural) way of characterizing the meaning of these utterances is in terms of a theory in which the conversational score is seen as a record of the discourse situation, or at least of the speech acts that have been performed. The problem then becomes to explain how discourse entities are …Read more

On 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

Sense 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

Interpolation 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

Description 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

Higher 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

Language, Lambdas, and LogicIn R. Oehrle & J. Kruijff (eds.), Resource Sensitivity, Binding, and Anaphora (Studies in Linguistics and Philosophy 80), Kluwer Academic Publishers. 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.

Combining 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.