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 created because of the presence of a variable in the NPI that cannot be interpreted deictically. This approach, by allowing more lexical sources for NPIs, is consistent with the diversity of NPIs, and extends easily to PPIs, which, as a class, appear to be non-scalar (Szabolcsi 2004 for some). Here one can again advance the argument that PPI-status is due to various sources including referentiality (some), and speaker commitment (with PPI adverbs; Ernst 2008). (shrink)

This paper provides a foundation for the polarity marking technique introduced by David Dowty [3] in connection with monotonicity reasoning in natural language and in linguistic analyses of negative polarity items based on categorial grammar. Dowty's work is an alternative to the better-known algorithmic approach first proposed by Johan van Benthem [11], and elaborated by Víctor Sánchez Valencia [10]. Dowty's system internalized the monotonicity/polarity markings by generating strings using a categorial grammar whose categories already contain the markings that the earlier (...) system would obtain by separate steps working on an already-derived string. Despite the linguistic advantages of the internalized system, no soundness proof has yet been given for it. This paper offers an account. The leading mathematical idea is to interpret categorial types as preorders (in order to talk about monotonicity in the first place), and then to add a primitive operation to the type hierarchy of taking the opposite of a preorder (in order to capture monotone decreasing functions). At the same time, the use of internalized categories also raises issues. Although these will not be addressed in full, the paper points out possible approaches to them. (shrink)

This paper develops an inference system for natural language within the ‘Natural Logic’ paradigm as advocated by van Benthem, Sánchez and others. The system that we propose is based on the Lambek calculus and works directly on the Curry-Howard counterparts for syntactic representations of natural language, with no intermediate translation to logical formulae. The Lambek -based system we propose extends the system by Fyodorov et~al., which is based on the Ajdukiewicz/Bar-Hillel calculus Bar Hillel,. This enables the system to deal with (...) new kinds of inferences, involving relative clauses, non-constituent coordination, and meaning postulates that involve complex expressions. Basing the system on the Lambek calculus leads to problems with non-normalized proof terms, which are treated by using normalization axioms. (shrink)

We propose an approach to natural language inference based on a model of natural logic, which identifies valid inferences by their lexical and syntactic features, without full semantic interpretation. We greatly extend past work in natural logic, which has focused solely on semantic containment and monotonicity, to incorporate both semantic exclusion and implicativity. Our system decomposes an inference problem into a sequence of atomic edits linking premise to hypothesis; predicts a lexical entailment relation for each edit using a statistical classifier; (...) propagates these relations upward through a syntax tree according to semantic properties of intermediate nodes; and composes the resulting entailment relations across the edit sequence. We evaluate our system on the FraCaS test suite, and achieve a 27% reduction in error from previous work. We also show that hybridizing an existing RTE system with our natural logic system yields significant gains on the RTE3 test suite. (shrink)

The perfect fit of syntactic derivability and logical consequence in first-order logic is one of the most celebrated facts of modern logic. In the present flurry of attention given to the semantics of natural language, surprisingly little effort has been focused on the problem of logical inference in natural language and the possibility of its completeness. Even the traditional theory of the syllogism does not give a thorough analysis of the restricted syntax it uses.My objective is to show how a (...) theory of inference may be formulated for a fragment of English that includes a good deal more than the classical syllogism. The syntax and semantics are made as formal and as explicit as is customary for artificial formal languages. The fragment chosen is not maximal but is restricted severely in order to provide a clear overview of the method without the cluttering details that seem to be an inevitable part of any grammar covering a substantial fragment of a natural language. (Some readers may feel the details given here are too onerous.). (shrink)