Pregroups and pregroup grammars were introduced by Lambek in 1999 [14] as an algebraic tool for the syntactic analysis of natural lan-guages. The main focus in that paper was on certain extended pregroup grammars such as pregroups with modalities, product pregroup grammars and tupled pregroup grammars. Their applications to different syntactic structures of natural languages, mainly Polish, are explored/shown here.

Pregroup grammars were developed in 1999 and stayed Lambek’s preferred algebraic model of grammar. The set-theoretic semantics of pregroups, however, faces an ambiguity problem. In his latest book, Lambek suggests that this problem might be overcome using finite dimensional vector spaces rather than sets. What is the right notion of composition in this setting, direct sum or tensor product of spaces?

The paper presents a way to transform pregroup grammars into contextfree grammars using functional composition. The same technique can also be used for the proof-nets of multiplicative cyclic linear logic and for Lambek calculus allowing empty premises.

Every pregroup grammar is shown to be strongly equivalent to one which uses basic types and left and right adjoints of basic types only. Therefore, a semantical interpretation is independent of the order of the associated logic. Lexical entries are read as expressions in a two sorted predicate logic with ∈ and functional symbols. The parsing of a sentence defines a substitution that combines the expressions associated to the individual words. The resulting variable free formula is the translation of (...) the sentence. It can be computed in time proportional to the parsing structure. Non-logical axioms are associated to certain words (relative pronouns, indefinite article, comparative determiners). Sample sentences are used to derive the characterizing formula of the DRS corresponding to the translation. (shrink)

Pregroups are partially ordered monoids in which each element has two “adjoints”. Pregroup grammars provide a computational approach to natural languages by assigning to each word in the mental dictionary a type, namely an element of the pregroup freely generated by a partially ordered set of basic types. In this expository article, the attempt is made to introduce linguists to a pregroup grammar of English by looking at Chomsky’s earliest examples.

The paper presents a generalization of pregroup, by which a freely-generated pregroup is augmented with a finite set of commuting inequations, allowing limited commutativity and cancelability. It is shown that grammars based on the commutation-augmented pregroups generate mildly context-sensitive languages. A version of Lambek’s switching lemma is established for these pregroups. Polynomial parsability and semilinearity are shown for languages generated by these grammars.

This paper investigates the learnability by positive examples in the sense of Gold of Pregroup Grammars. In a first part, Pregroup Grammars are presented and a new parsing strategy is proposed. Then, theoretical learnability and non-learnability results for subclasses of Pregroup Grammars are proved. In the last two parts, we focus on learning Pregroup Grammars from a special kind of input called feature-tagged examples. A learning algorithm based on the parsing strategy presented (...) in the first part is given. Its validity is proved and its properties are examplified. (shrink)

We show that vector space semantics and functional semantics in two-sorted first order logic are equivalent for pregroup grammars. We present an algorithm that translates functional expressions to vector expressions and vice-versa. The semantics is compositional, variable free and invariant under change of order or multiplicity. It includes the semantic vector models of Information Retrieval Systems and has an interior logic admitting a comprehension schema. A sentence is true in the interior logic if and only if the ‘usual’ (...) first order formula translating the sentence holds. The examples include negation, universal quantifiers and relative pronouns. (shrink)

We study mathematical and algorithmic properties of Lambek's pregroups and illustrate them by the French noun phrase. An algorithm of complexity n3 to solve the reduction problem in an arbitrary free pregroup as well as recognition by a pregroup grammar is presented. This algorithm is then specified to run in linear time. A sufficient condition for a language fragment that makes the linear algorithm complete is given.

Pregroup grammars have a cubic recognition algorithm. Here, we define a correct and complete recognition and parsing algorithm and give sufficient conditions for the algorithm to run in linear time. These conditions are satisfied by a large class of pregroup grammars, including grammars that handle coordinate structures and distant constituents.

In this paper, we propose an extension of free pregroups with lower bounds on sets of pregroup elements. Pregroup grammars based on such pregroups provide a kind of an algebraic counterpart to universal quantification over type-variables. In particular, we show how our pregroup extensions can be used for pregroup grammars expressing natural-language coordination and extraction.

We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cut-elimination theorem and a normalization theorem for an extended system of this logic, its P-TIME decidability, its interpretation in L1, and a general construction of (preordered) bilinear algebras and pregroups whose universe is an arbitrary monoid.

One way of coping with agreement of features in French is to perform two parallel computations, one in the free pregroup of syntactic types, the other in that of feature types. Technically speaking, this amounts to working in the direct product of two free pregroups.

Grammar can be formulated as a kind of substructural propositional logic. In support of this claim, we survey bare Gentzen style deductive systems and two kinds of non-commutative linear logic: intuitionistic and compact bilinear logic. We also glance at their categorical refinements.

We explore a computational algebraic approach to grammar via pregroups, that is, partially ordered monoids in which each element has both a left and a right adjoint. Grammatical judgements are formed with the help of calculations on types. These are elements of the free pregroup generated by a partially ordered set of basic types, which are assigned to words, here of English. We concentrate on the object pronoun who(m).

The Distributional Compositional Categorical model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that (...) could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus. (shrink)

We know from the literature in theoretical linguistics that interrogative constructions in Italian have particular syntactic properties, due to the liberal word order and the rich inflectional system. In this paper we show that the calculus of pregroups represents a flexible and efficient computational device for the analysis and derivation of Italian sentences and questions. In this context the distinction between direct vs. indirect statements and questions is explored.

This volume provides a detailed account of the syntax of expressive language, that is, utterances that express, rather than describe, the emotions and attitudes of the speaker... Daniel Gutzmann demonstrates that expressivity has strong syntactic reflexes that interact with the semantic and pragmatic interpretation of these utterances, and argues that expressivity is in fact a syntactic feature on a par with other established features such as tense and gender. Evidence for this claim is drawn from three detailed case studies of (...) expressive adjectives, intensifiers, and vocatives; their puzzling properties are accounted for through a minimalist approach to syntactic features and agreement, which shows that expressivity can partake in agreement operations, trigger movement, and be selected for syntactically. The analysis not only supports the hypothesis of expressive syntax, but also highlights the hidden role that grammar may play in phenomena that are traditionally considered to be solely semantic in nature" -- Publisher. (shrink)

Italian political thinker Paolo Virno argues that the category of "multitude" is a far better tool to analyze contemporary issues than the Hobbesian concept of "people." Globalization is forcing us to rethink some of the categories—such as "the people"—that traditionally have been associated with the now eroding state. Italian political thinker Paolo Virno argues that the category of "multitude," elaborated by Spinoza and for the most part left fallow since the seventeenth century, is a far better tool to analyze contemporary (...) issues than the Hobbesian concept of "people," favored by classical political philosophy. Hobbes, who detested the notion of multitude, defined it as shunning political unity, resisting authority, and never entering into lasting agreements. "When they rebel against the state," Hobbes wrote, "the citizens are the multitude against the people." But the multitude isn't just a negative notion, it is a rich concept that allows us to examine anew plural experiences and forms of nonrepresentative democracy. Drawing from philosophy of language, political economics, and ethics, Virno shows that being foreign, "not-feeling-at-home-anywhere," is a condition that forces the multitude to place its trust in the intellect. In conclusion, Virno suggests that the metamorphosis of the social systems in the West during the last twenty years is leading to a paradoxical "Communism of the Capital.". (shrink)

In recent years, the study of formal semantics and formal pragmatics has grown tremendously showing that core aspects of language meaning can be explained by a few principles. These principles are grounded in the logic that is behind - and tightly intertwined with - the grammar of human language. In this book, some of the most prominent figures in linguistics, including Noam Chomsky and Barbara H. Partee, offer new insights into the nature of linguistic meaning and pave the way for (...) the further development of formal semantics and formal pragmatics. Each chapter investigates various dimensions in which the logical nature of human language manifests itself within a language and/or across languages. Phenomena like bare plurals, free choice items, scalar implicatures, intervention effects, and logical operators are investigated in depth and at times cross-linguistically and/or experimentally. This volume will be of interest to scholars working within the fields of semantics, pragmatics, language acquisition and psycholinguistics. (shrink)

In The Grammar of Society, first published in 2006, Cristina Bicchieri examines social norms, such as fairness, cooperation, and reciprocity, in an effort to understand their nature and dynamics, the expectations that they generate, and how they evolve and change. Drawing on several intellectual traditions and methods, including those of social psychology, experimental economics and evolutionary game theory, Bicchieri provides an integrated account of how social norms emerge, why and when we follow them, and the situations where we are most (...) likely to focus on relevant norms. Examining the existence and survival of inefficient norms, she demonstrates how norms evolve in ways that depend upon the psychological dispositions of the individual and how such dispositions may impair social efficiency. By contrast, she also shows how certain psychological propensities may naturally lead individuals to evolve fairness norms that closely resemble those we follow in most modern societies. (shrink)

pt. 1. The proposition and its sense.--pt. 2. On logic and mathematics.

Universal Grammar (UG) is a theory about the innate linguistic knowledge that child language learners bring to the task of language acquisition. This chapter examines the findings of experimental research on children's knowledge of one principle of UG, called Principle C. It presents the defining properties of Principle C. The chapter reviews empirical evidence showing that children apply Principle C to a range of disparate‐looking phenomena. It also presents empirical findings that document children's assignment of hierarchical structure to strings of (...) words. The empirical evidence supports a conclusion – that child language learners access a level of linguistic representation in addition to the constituent structure that is formed using Merge. This additional component of the language apparatus is called Logical Form. The chapter looks at a string of words where Principle C applies at the level of Logical Form. Cleft sentences are one such case. (shrink)

In this paper we concentrate mainly on the notion of β-pregroups, which are pregroups (first introduced by Lambek [18] in 1999) enriched with modality operators. β-pregroups were first proposed by Fadda [11] in 2001. The motivation to introduce them was to limit (locally) the associativity in the calculus considered. In this paper we present this new calculus in the form of a rewriting system, prove the very important feature of this system - that in a given derivation the non- expanding (...) rules must always proceed non-contracting ones in order the derivation to be minimal (normalization theorem). We also propose a sequent system for this calculus and prove the cut elimination theorem for it. As an illustration we show how to use β-pregroups for linguistical applications. (shrink)

In this paper we concentrate mainly on the notion of β-pregroups, which are pregroups enriched with modality operators. β-pregroups were first proposed by Fadda [11] in 2001. The motivation to introduce them was to limit the associativity in the calculus considered. In this paper we present this new calculus in the form of a rewriting system, prove the very important feature of this system - that in a given derivation the non- expanding rules must always proceed non-contracting ones in order (...) the derivation to be minimal. We also propose a sequent system for this calculus and prove the cut elimination theorem for it. As an illustration we show how to use β-pregroups for linguistical applications. (shrink)

Written by two of the leading figures in the field, this is a lucid and systematic introduction to semantics as applied to transformational grammars of the ...

This seminal, multidisciplinary book shows how mathematics can be used to study the first principles of DNA. Most importantly, it enriches the so-called "Chargaff's grammar of biology" by providing the conceptual theoretical framework necessary to generalize Chargaff's rules. Starting with a simple example of DNA mathematical modeling where human nucleotide frequencies are associated to the Fibonacci sequence and the Golden Ratio through an optimization problem, its breakthrough is showing that the reverse, complement and reverse-complement operators defined over oligonucleotides induce a (...) natural set partition of DNA words of fixed-size. These equivalence classes, when organized into a matrix form, reveal hidden patterns within the DNA sequence of every living organism. Intended for undergraduate and graduate students both in mathematics and in life sciences, it is also a valuable resource for researchers interested in studying invariant genomic properties. (shrink)

Bringing together evidence from natural and social sciences, the work introduces the non-reductionist Instruction Grammar programme. Viewed from within the practicalities of the lifeworld, utterances are described as instructions to simulate perceptions and attributions for action. The approach provides solutions to long-standing philosophical problems of cognitive grammar theories and traditionally puzzling syntactic phenomena.

This book is a collection of essays, weaving together cognitive psychology, psycho-linguistics, developmental psychology, modern philosophy and behavioural sciences. It raises the question, how grammar relates to our remarkable ability to cooperate for future needs and how our thought process is related to grammatical parameters.

“Grammar” is Ludwig Wittgenstein's preferred term for the workings of a language: the system of rules that determine linguistic meaning. A philosophical study of language is a study of “grammar”, in this sense, and insofar as any philosophical investigation is concerned with conceptual details, which manifest themselves in language, it is a grammatical investigation. In the Tractatus Logico‐Philosophicus Wittgenstein offered a mathematical picture of language: presenting language as a calculus. Like a calculus, language was claimed to be governed by syntactic (...) rules: formation rules about the licit combination of names to form elementary propositions; formation rules about the licit combination of elementary propositions to make complex propositions; and finally, truth‐table rules, which enable us to identify logical truths and entailments. Logical grammar, for the young Wittgenstein, is only logical syntax and logical syntax can be determined without paying any attention to the meaning of the signs. (shrink)

Dans cet article nous examinons la syntaxe de l’anglo-normand tardif, en confrontant l’hypothèse d’une « différence fondamentale » entre l’anglo-normand (AN) et le français du continent (Kibbee (1991), à celle de Trotter (2003), selon qui l’AN aurait participé au « continuum dialectal » francophone du Moyen Age. Il est proposé par la même occasion de démontrer la capacité de textes non-littéraires, comme le sont la plupart des textes AN tardifs, à nous renseigner quant à la datation d’évolutions en syntaxe. Une (...) analyse de sources appartenant à la base de textes de l’Anglo-Norman Hub nous a permis de constater que la syntaxe du français insulaire s’est conformée à celle du français du continent sur les points étudiés, observant jusqu’en 1350 environ une distinction entre adverbes invertissants et non-invertissants, qui s’est ensuite perdue, et d’autre part en n’employant l’article partitif de qu’après point, plutôt qu’après mie ou pas. Ces résultats s’expliquent mieux si l’on suit l’hypothèse de l’adhésion de l’AN à l’aire francophone médiévale plutôt que celle d’une différence fondamentale entre le français insulaire et celui du continent. Il est proposé en outre que la base de textes de l’ Anglo-Norman Hub puisse servir de référence quant à la chronologie des changements syntaxiques survenus en français médiéval, en attendant d’être complétée par des sources continentales de genres comparables. (shrink)

In this paper we discuss a new perspective on the syntax-semantics interface. Semantics, in this new set-up, is not ‘read off’ from Logical Forms as in mainstream approaches to generative grammar. Nor is it assigned to syntactic proofs using a Curry-Howard correspondence as in versions of the Lambek Calculus, or read off from f-structures using Linear Logic as in Lexical-Functional Grammar (LFG, Kaplan & Bresnan [9]). All such approaches are based on the idea that syntactic objects (trees, proofs, fstructures) are (...) somehow prior and that semantics must be parasitic on those syntactic objects. We challenge this idea and develop a grammar in which syntax and semantics are treated in a strictly parallel fashion. The grammar will have many ideas in common with the (converging) frameworks of categorial grammar and LFG, but its treatment of the syntax-semantics interface is radically different. Also, although the meaning component of the grammar is a version of Montague semantics and although there are obvious affinities between Montague’s conception of grammar and the work presented here, the grammar is not compositional, in the sense that composition of meaning need not follow surface structure. (shrink)

Much philosophical attention has been devoted to the truth predicates of natural language and their logic. However, lexical truth predicates are neither necessary nor sufficient for a truth-attribution to occur, which warrants closer attention to the grammar of truth attribution. A unified analysis of five constructions is offered here, in two of which the lexical truth predicate occurs (It's true that John left and That John left is true), while in the three remaining, it does not (John left; It seems (...) that John left; and It's that John left). This analysis is philosophically significant for four reasons. First, it explains why speakers of natural language find standard instances of Tarski-inspired equivalences (e.g. That John left is true iff John left) intuitively compelling. Second, it derives the widespread ‘deflationist’ intuition that truth has no substantive content. Third, insofar as the deflationist sees insights on truth as flowing from understanding our practice of truth attribution, it furthers the deflationist agenda through a new analysis of such attributions. Finally, it advances the philosophical project of the ‘naturalization’ of truth by reducing our understanding of truth to our competence in the grammar of truth, as an aspect of our biological endowment. (shrink)

About this book Mr. Burke contributes an introductory and summarizing remark, "What is involved, when we say what people are doing and why they are doing it?

Drawing on everyday telephone and video interactions, this book surveys how English speakers use grammar to formulate responses in ordinary conversation. The authors show that speakers build their responses in a variety of ways: the responses can be longer or shorter, repetitive or not, and can be uttered with different intonational 'melodies'. Focusing on four sequence types: responses to questions, responses to informings, responses to assessments, and responses to requests, they argue that an interactional approach holds the key to explaining (...) why some types of utterances in English conversation seem to have something 'missing' and others seem overly wordy. (shrink)