The Σ 2 theory of D h ( ⩽ h O ) as an uppersemilattice with least and greatest element is decidable

The decidability of the two quantifier theory of the hyperarithmetic degrees below Kleene’s O in the language of uppersemilattices with least and greatest element is established. This requires a new kind of initial segment result and a new extension of embeddings result both in the hyperarithmetic setting.

Partial fuzzy quantifiers and their computation

Background: In computer science, one often meets the requirement to deal with partial functions. They naturally raise, for example, when a mistake such as the square root of a negative number or division by zero occurs, or when we want to express the semantics of the expression “Czech president in 18th century” because there was no such president before 1918. Method: In this paper, we will extend the theory of intermediate quantifiers (i.e., expressions such as “most, almost all, many, a few”, etc.) to deal with partially defined fuzzy sets. First, we extend algebraic operations that are used in fuzzy logic by additional value “undefined”. Then we will introduce intermediate quantifiers using the former. The theory of intermediate quantifiers has been usually developed as a special theory of higher-order fuzzy logic. Results: In this paper, we introduce the quantifiers semantically and show how they can be computed. The latter is also demonstrated in three illustrative examples. Conclusion: The paper contributes to the development of fuzzy quantifier theory by its extension by undefined values and suggests methods for computation of their truth values.

A generalised quantifier theory of natural language in categorical compositional distributional semantics with bialgebras

Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, and then instantiate the abstract setting to sets and relations and to finite-dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics.

Determiners

Determiners are a nominal syntactic category distinct from both adjectives and nouns; they constitute a functional (aka closed or ‘minor’) category and they are typically located high inside the nominal phrasal structure. From a syntactic point of view, the category of determiners is commonly understood to comprise the word classes of article, demonstrative, and quantifier, as well as non-adjectival possessives and some nominal agreement markers. From a semantic point of view, determiners are assumed to function as quantifiers, especially within research informed by Generalized Quantifier Theory. However, this is a one-way entailment: although determiners in natural language are quantificational, their class contains only a subset of the logically possible quantifiers; this class is restricted by conservativity and other factors. The tension between the ‘syntactic’ and the ‘semantic’ perspective on determiners results to a degree of terminological confusion: it is not always clear which lexical items the Determiner category includes or what the function of determiners is; moreover, there exists a tendency among syntacticians to view ‘Determiner’ as naming not a class, but a fixed position within a nominal phrasal template. The study of determiners rose to prominence within grammatical theory during the ’80s both due to advances in semantic theorizing, primarily Generalized Quantifier Theory, and due to the generalization of the X' phrasal schema to functional (minor) categories. Some issues in the nature and function of determiners that have been addressed in theoretical and typological work with considerable success include the categorial status of determiners, their (non-)universality, their structural position and feature makeup, their role in argumenthood and their interaction with nominal predicates, and their relation to pronouns. Expectedly, issues in (in)definiteness, quantification, and specificity also figure prominently in research work on determiners.

Determiners

Determiners are a nominal syntactic category distinct both from adjectives and nouns, despite the close affinity among them. They are commonly understood to comprise the word classes of article, demonstrative, and quantifier, as well as some possessives and some nominal agreement markers. Determiners became a prominent topic of study in grammatical theory during the 1980s, due both to advances in semantic theorizing, such as Generalized Quantifier Theory, and to the generalization of the X’ phrasal schema to minor (functional) categories, to which determiners are posited to belong. The main questions that have been the focus of theoretical and typological inquiry ever since are the categorial status of determiners (functional or lexical), whether they are universal as a distinct syntactic category, whether they constitute a uniform category or not, their structural position, their feature content, their role in argumenthood and semantic interpretation in general, and their relation to pronouns. Answers to these questions are in part determined by whether one takes determiners to be the nominal equivalent of complementizers (i.e., to constitute the topmost functional layer of the nominal phrase) or to be all quantifiers, defining relations between predicates. At the same time, a wealth of syntactic phenomena involving determiners have been investigated, shedding light not only on the structure of the nominal phrase and on the distribution of nominal features within it, but also on the nature of adjectives, possessives, and nouns.

Lattice initial segments of the hyperdegrees

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, . In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of . Corollaries include the decidability of the two quantifier theory of , and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of ω1ck. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω1. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of .