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.

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)

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?

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 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.

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)

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.

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.

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 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.

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)

Subjects exposed to strings of letters generated by a finite state grammar can later classify grammatical and nongrammatical test strings, even though they cannot adequately say what the rules of the grammar are (e.g., Reber, 1989). The MINERVA 2 (Hintzman, 1986) and Medin and Schaffer (1978) memory‐array models and a number of connectionist outoassociator models are tested against experimental data by deriving mainly parameter‐free predictions from the models of the rank order of classification difficulty of test strings. The (...) importance of different assumptions regarding the coding of features (How should the absence of a feature be coded? Should single letters or digrams be coded?), the learning rule used (Hebb rule vs. delta rule), and the connectivity (Should features be predicted only by previous features in the string, or by all features simultaneously?) is investigated by determining the performance of the models with and without each assumption. Only one class of connectionist model (the simultaneous delta rule) passes all the tests. It is shown that this class of model can be regarded by abstracting a set of representative but incomplete rules of the grammar. (shrink)

In this paper, I outline a grammar of lifting (i.e., resistance training) and compare it to that of language. I approach lifting as a system of generating complex meaning–form correspondences from regularized elements and describe the levels of mental representations and relationships between them that are involved in full command of this system. To be able to do so, I adopt a goal-based conception of meaning, which allows us to talk about mappings from complex goals to complex surface outputs (...) in systems of intentional action, signaling and non-signaling, interactive and non-interactive, in a unified way, and show how it applies in lifting. I then proceed to argue that the grammar of lifting is architecturally very similar to that of language. First, I show that both involve stable (idiomatized/lexicalized) pairings of regularized forms with regularized meanings. Second, I argue that in both lifting and language, meaning–form mapping is mediated by syntax, which, crucially, operates on non-linearized hierarchical structures of abstract objects that include both content morphemes and functional morphemes. I conclude, following and expanding on some insights from prior literature and offering further evidence for them, that neither of these architectural phenomena (idiomatized meaning–form pairings and abstract syntax) is specific to language, with both of them likely emerging in skilled action that does not necessarily involve social interaction, due to considerations of repeatability and reusability of elements in new contexts. (shrink)

The study aims to explore the roles of Massive Open Online Courses based on deep learning in college students’ English grammar teaching. The data are collected using a survey. After the experimental data are analyzed, it is found that students have a low sense of happiness and satisfaction and are unwilling to practice oral English and learn language points in English learning. They think that college English learning only meets the needs of CET-4 and CET-6 and does not take (...) it as the ultimate learning goal. After the necessity and problems in English grammar teaching are discussed, the advantages of flipped classrooms of MOOCs are discussed in English grammar teaching. A teaching platform is constructed to study the foreign language teaching mode under MOOCs, and classroom teaching is combined with the advantages of MOOCs following the principle of “teaching students according to their personalities” to improve the listening, speaking, reading, writing, and translation skills of foreign language majors. The results show that high-quality online teaching resources and the deep learning-based teaching environment can provide a variety of interactive tools, by which students can communicate with their peers and teachers online. Sharing open online communication, classroom discussion, and situational simulation can enhance teachers’ deep learning ability, like the ability to communication and transfer thoughts. Constructivism with interaction as the core can help students grasp new knowledge easily. Extensive communication and interaction are important ways for learning and thinking. The new model provides students with profound learning experience, expands the teaching resources of MOOCs around the world, and maximizes the interaction between online and offline teachers and students, making knowledge widely rooted in the campus and realizing the combination of online resources and campus classroom teaching. Students can learn the knowledge through autonomous learning and discussion before class, which greatly broadens the learning time and space. In the classroom and after class, the internalization and sublimation of knowledge are completed through group cooperation, inquiry learning, scenario simulation, display, and evaluation, promoting students to know about new knowledge and highlighting the dominant position of students. (shrink)

This interdisciplinary book considers the relationship between language and thought from a philosophical perspective, drawing both on the philosophical study of language and the purely formal study of grammar, and arguing that the two should align. The claim is that grammar provides homo sapiens with the ability to think in certain grammatical ways and that this in turn explains the vast cognitive powers of human beings. Evidence is considered from biology, the evolution of language, language disorders, and linguistic (...) phenomena. (shrink)

I seek to provide a systematic and comprehensive framework for the description and analysis of scientific practice—a philosophical grammar of scientific practice, ‘grammar’ as meant by the later Wittgenstein. I begin with the recognition that all scientific work, including pure theorizing, consists of actions, of the physical, mental, and ‘paper-and-pencil’ varieties. When we set out to see what it is that one actually does in scientific work, the following set of questions naturally emerge: who is doing what, why, (...) and how? More specifically, we must arrive at some coherent philosophical accounts of the following elements of scientific practice: the agent—free, embodied, and constantly in second-person interactions with other agents; the purposes and proximate aims of the agent; types of activities that the agent engages in; ontological principles necessarily presumed for the performance of particular activities; instruments and other resources that the agent pulls together for the performance of each activity. After sketching the general framework, I also give some illustrative contrasts between the more traditional descriptions of scientific practice and the kind of descriptions enabled by the proposed framework. (shrink)

In discussing apprehension, assent, and inference in his Grammar of Assent, Newman contrasted “notions” and “things”—terms that distinguish knowledge of the abstract and “unreal” from knowledge of the singular and concrete. This essay proposes that Newman’s contrast between “notions” and “things” is an adverbial distinction, qualifying a person’s mode of engagement with the world, rather than an adjectival distinction, qualifying the metaphysical status of particular terms.

In An Essay in Aid of a Grammar of Assent, John Henry Newman articulates his fundamental philosophical orientation by giving priority to real apprehension over notional apprehension. He distinguishes between the two by saying that notional apprehension hasto do with things internal to the mind and admits of exactness and clarity whereas real apprehension has to do with things external to the mind and does not admit of the same degree of clarity and exactness. I argue that the connection (...) between “inside the mind” and “clarity and exactness” lies in the constructive activity underlying notional thinking. Real apprehension, on the other hand, involves a given apprehension of unity, mainly, the concrete unity of intelligent life, which includes but cannot be reduced to the constructive activity of notional thinking. Thus, I argue, Newman’s realism undercuts any form of modern transcendentalism and evinces a form of classical human realism. (shrink)

Are rules processes or similarity processes the default for acquisition of grammatical knowledge during natural second language acquisition? Whereas Pothos argues similarity processes are the default in the many areas he reviews, including artificial grammar learning and first language development, I suggest, citing evidence, that in second language acquisition of grammatical morphology “rules processes” may be the default.

Introduction 1. Language and meaning Nothing is as easily overlooked, or as easily forgotten, as the most obvious truths. The tenet that language is a tool ...