Polarity Items in Resource Logics

Catherine Piliere, 2000

Even though residuation is at the core of Categorial Grammar , 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(3) , and we spell things out by means of Display Calculi (DC) . 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 3 and 2 ↓ introduced by Moortgat (1996) are indeed their unary counterpart.

Journal of Logic, Language and Information, 2008

This paper uses a partially ordered set of syntactic categories to accommodate optionality and licensing in natural language syntax. A complex but well-studied data set pertaining to the syntax of quantifier scope and negative polarity licensing in Hungarian is used to illustrate the proposal. The presentation is geared towards both linguists and logicians. The paper highlights that the main ideas can be implemented in different grammar formalisms, and discusses in detail an implementation where the partial ordering on categories is given by the derivability relation of a calculus with residuated and Galois-connected unary operators.

Categorial grammars in the tradition of Lambek are asymmetric: sequent statements are of the form Γ ⇒ A, where the succedent is a single formula A, the antecedent a structured configuration of formulas A 1 , . . . , A n . The absence of structural context in the succedent makes the analysis of a number of phenomena in natural language semantics problematic. A case in point is scope construal: the different possibilities to build an interpretation for sentences containing generalized quantifiers and related expressions. In this paper, we explore a symmetric version of categorial grammar, based on work by Grishin . In addition to the Lambek product, left and right division, we consider a dual family of type-forming operations: coproduct, left and right difference. Communication between the two families is established by means of structurepreserving distributivity principles. We call the resulting system LG. We present a Curry-Howard interpretation for LG(/, \, , ) derivations, based on Curien and Herbelin's λµ µ calculus [10]. We discuss continuation-passing-style (CPS) translations mapping LG derivations to proofs/terms of Intuitionistic Multiplicative Linear Logic -the categorial system LP which serves as the logic for natural language meaning assembly. We show how LG, thus interpreted, associates sentences with quantifier phrases with the appropriate range of meanings, thus overcoming the expressive limitations of asymmetric categorial grammars in this area.

The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are order-preserving; in this paper instead, we consider the addition of order-reversing, Galois connected operators. In §2 we do the basic model-theoretic and proof-theoretic groundwork. In §3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quantifier expressions, and to describe a typology of polarity items.

In this paper we present a logical system able to compute the semantics of both declarative and interrogative sentences. Our proposed analysis takes place at both the sentential and at the discourse level. We use syntactic inference on the sentential level for declarative sentences, while the discourse level comes into play for our treatment of questions. Our formalisation uses a type logic sensitive to both the syntactic and semantic properties of natural language. We will show how an account of the linguistic data follows naturally from the logical relations inherent in the type logic.

In this chapter, we discuss the distribution and lexical properties of common varieties of negative polarity items (NPIs) and positive polarity items (PPIs). We establish first that NPIs can be licensed in negative, downward entailing, and nonveridical environments. Then we examine if the scalarity approach (originating in Kadmon and Landman 1993) can handle the attested NPI distribution and empirical variation. By positing a unitary lexical source for NPIs—widening, plus EVEN— scalarity fails to capture the fact that a significant number of NPIs are not scalar, and does not predict correctly NPI distribution in nonveridical contexts. It also misses the variation within the scalar class between broader (any) and narrow NPIs (either). Finally, scalarity predicts weaker effects (contradictions, presupposition failures) with ill-formed NPIs than is actually the case. The variation approach (Giannakidou 1998, 2001, 2007), on the other hand, posits that besides scalarity, NPIs can be cre...

This thesis investigates the multifaceted nature of negation on a theoretical account and discusses its representation in non-standard and standard varieties of English and German. In natural languages, negation is understood as being marked by the presence of certain morpho-syntactic elements, qua negation markers. In simple terms, logical negation can be treated as merely symbolizing a two-valued operator indicating the truth category of an utterance, which strictly surfaces as being either true or false. From a theoretical viewpoint, negation is regarded as more complex when its interaction with various principles of logical form, syntax, compositional semantics and processes of language acquisition are considered. This thesis attempts an integrated analysis of negative expressions across a fragment of English and German varieties. The empirical part of this thesis is mainly engaged with the phenomenon of multiple negation and distinct approaches that deal with clusters of negative markers. The nuances of negativity in this respect can range from preserving the complete falsity of an utterance to elevating it to the exact opposite, namely positive semantics. While affirmatives can be distinguished from negative utterances by the occurrence of n-words, the semantical and logical analysis of multiple negation encompasses a wider spectrum of multifunctional meanings, and thus has triggered an ongoing debate in the field. To contribute to this discussion, a study of native speakers of an Austrian variety of German has been conducted in order to explore form, function and acquisition of multiply-marked negation in this lesser examined non-standard variety of German.