An electronic poster from "Polarity from Different Perspectives," New York University, 2005. The authors present an experiment that investigated to what extent six negative polarity items (slept a wink, in ages, ever, much, at all, and yet) are licensed by 9 potential licensers.

Standard theories of scope are semantically blind. They employ a single logico-syntactic rule of scope assignment quantifying in Quantifier Raising, storage, or type change etc which roughly speaking prefixes an expression \aplha.

The main claim of this paper is that a general theory of negative concord (NC) should allow for the possibility of NC involving scoping of a universal quantifier above negation. I propose that Greek NC instantiates this option. Greek n-words will be analyzed as polarity sensitive universal quantifiers which need negation in order to be licensed, but must raise above negation in order to yield the scoping ∀¬. This gives the correct interpretation of NC structures as general negative statements. The (...) effect is achieved by application of QR, and the account is fully compositional, as only sentence negation is the vehicle of logical negation ¬. Greek n-words are also compared to nwords in Romance, Slavic, and Hungarian. This analysis, if correct, has two important.. (shrink)

In these essays the minimalist approach to linguistic theory is formulated and progressively developed.

The focus of this paper is the syntax of inverse scope in Hungarian, a language that largely disambiguates quantifier scope at spell-out. Inverse scope is attributed to alternate orderings of potentially large chunks of structure, but with appeal to base-generation, as opposed to nonfeature-driven movement as in Kayne 1998. The proposal is developed within mirror theory and conforms to the assumption that structures are antisymmetrical. The paper also develops a matching notion of scope in terms of featural domination, as opposed (...) to c-command, and applies it to otherwise problematic cases of pied piping. Finally, the interaction of different quantifier types is examined and the patterns are explained invoking morphological considerations on one hand and A-bar reconstruction on the other. (shrink)

This book sets out the foundations, methodology, and practice of a formal framework for the description of language. The approach embraces the trends of lexicalism and compositional semantics in computational linguistics, and theoretical linguistics more broadly, by developing categorial grammar into a powerful and extendable logic of signs. Taking Montague Grammar as its point of departure, the book explains how integration of methods from philosophy (logical semantics), computer science (type theory), linguistics (categorial grammar) and meta-mathematics (mathematical logic ) provides a (...) categorial foundation with coverage including intensionality, quantification, featural polymorphism, domains and constraints. For the first time, the book systematises categorial thinking into a unified program which is at once both logically secured, and a practical tool for pure lexical grammar development with type-theoretic semantics. It should be of interest to all those active in computational linguistics and formal grammar and is suitable for use at advanced undergraduate, postgraduate, and research levels. (shrink)

Using the programming-language concept of continuations, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of in-situ quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, type-logical way to model evaluation order and side effects in natural language. We illustrate with an (...) improved account of quantificational binding, weak crossover, wh-questions, superiority, and polarity licensing. (shrink)

This paper gives a simple method for providing categorial brands of feature-based unification grammars with a model-theoretic semantics. The key idea is to apply the paradigm of fibred semantics (or layered logics, see Gabbay (1990)) in order to combine the two components of a feature-based grammar logic. We demonstrate the method for the augmentation of Lambek categorial grammar with Kasper/Rounds-style feature logic. These are combined by replacing (or annotating) atomic formulas of the first logic, i.e. the basic syntactic types, by (...) formulas of the second. Modelling such a combined logic is less trivial than one might expect. The direct application of the fibred semantics method where a combined atomic formula like np (num: sg & pers: 3rd) denotes those strings which have the indicated property and the categorial operators denote the usual left- and right-residuals of these string sets, does not match the intuitive, unification-based proof theory. Unification implements a global bookkeeping with respect to a proof whereas the direct fibring method restricts its view to the atoms of the grammar logic. The solution is to interpret the (embedded) feature terms as global feature constraints while maintaining the same kind of fibred structures. For this adjusted semantics, the anticipated proof system is sound and complete. (shrink)

Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value Structures unify amounts to testing for modal satisfiability. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalisations, and the ability to express recursive constraints. Two main (...) techniques are used: either Kripke models with desirable properties are constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic. Further possibilities for the application of modal logic in computational linguistics are noted. (shrink)

Even though residuation is at the core of Categorial Grammar (Lambek, 1958), it is not always immediate to realize how standard logical systems like Multi-modal Categorial Type Logics (MCTL) (Moortgat, 1997) actually embody this property. In this paper, we focus on the basic system NL (Lambek, 1961) and its extension with unary modalities NL() (Moortgat, 1996), and we spell things out by means of Display Calculi (DC) (Belnap, 1982; Goré, 1998). The use of structural operators in DC permits a sharp (...) distinction between the core properties we want to impose on the logical system and the way these properties are projected into the logical operators. We will show how we can obtain Lambek residuated triple \, / and of binary operators, and how the operators and introduced by Moortgat (1996) are indeed their unary counterpart.In the second part of the paper we turn to other important algebraic properties which are usually investigated in conjunction with residuation (Birkhoff, 1967): Galois and dual Galois connections. Again, DC let us readily define logical calculi capturing them. We also provide preliminary ideas on how to use these new operators when modeling linguistic phenomena. (shrink)

Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also (...) have a 'cyclic' counterpart. These logics have been studied extensively and are quite well understood.Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and some even have a 'cyclic' counterpart. These extensions of Bi-Lambek logic are not so well understood. Cut-elimination for Classical Bi-Lambek logic, for example, is not completely clear since some cut rules have side conditions requiring that certain constituents be empty or non-empty.The display logic of Nuel Belnap is a general Gentzen-style proof theoretical framework designed to capture many different logics in one uniform setting. The beauty of display logic is a general cut-elimination theorem, due to Belnap, which applies whenever the rules of the display calculus obey certain, easily checked, conditions. The original display logic, and its various incarnations, are not suitable for capturing bi-intuitionistic and bi-classical logics in a uniform way. We remedy this situation by giving a single Display calculus for the Bi-Lambek Calculus, from which all the well-known extensions are obtained by the incremental addition of structural rules to a constant core of logical introduction rules.We highlight the inherent duality and symmetry within this framework obtaining 'four proofs for the price of one'. We give algebraic semantics for the Bi-Lambek logics and prove that our calculi are sound and complete with respect to these semantics. We show how to define an alternative display calculus for bi-classical substructural logics using negations, instead of implications, as primitives.Borrowing from other display calculi, we show how to extend our display calculus to handle bi-intuitionistic or bi-classical substructural logics containing the forward and backward modalities familiar from tense logic, the exponentials of linear logic, the converse operator familiar from relation algebra, four negations, and two unusual modalities corresponding to the non-classical analogues of Sheffer's 'dagger' and 'stroke', all in a modular way.Using the Gaggle Theory of Dunn we outline relational semantics for the binary and unary intensional connectives, but make no attempt to do so for the extensional connectives, or the exponentials.Finally, we flesh out a suggestion of Lambek to embed intuitionistic logic using two unusual 'exponentials', and show that these 'exponentials' are essentially tense logical modalities, quite at odds with the usual exponentials.Using a refinement of the display property, you can pick and choose from these possibilities to construct a display calculus for your needs. (shrink)