Categorial Grammar and Lambda Calculus

We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified proof of standardization for beta reduction that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.

The concept of polarity is pervasive in natural language. It relates syntax, semantics and pragmatics narrowly (Giannakidou, in: Maienborn, von Heusinger, Portner (eds.), Semantics: an international handbook of natural language meaning, De Gruyter Mouton, Berlin, 2011; Israel in The grammar of polarity: pragmatics, sensitivity, and the logic of scales, Cambridge studies in linguistics, Cambridge University Press, Cambridge, 2014), it refers to items of many syntactic categories such as nouns, verbs and adverbs. Neutral polarity items appear in affirmative and negative sentences, negative polarity items cannot appear in affirmative sentences, and positive polarity items cannot appear in negative sentences. A way of reasoning in Natural Language is through Natural Logic (van Benthem in Essays in logical semantics, vol. 29 of Studies in linguistics and philosophy, Reidel, Dordrecht, 1986; Language in action: categories, lambdas, and dynamic logic, vol. 130 of Studies in logic, Elsevier, Amsterdam, 1991). This logic is based on the concept of polarity in order to make the meaning of a sentence weaker o stronger without changing its truth value. There exist many proposals to compute polarity in the Natural Logic context, the most widely known are the ones by: van Benthem (1986, 1991), Sanchez-Valencia (Studies on natural logic and categorial grammar, Ph.D. thesis, Universiteit van Amsterdam, 1991), Dowty (Proceedings of the 4th conference on semantics and theoretical linguistics, Cornel University, CLC Publications, Rochester, 1994), and van Eijck (in: ten Cate, Zeevat (eds.), 6th international Tbilisi symposium on logic, language, and computation, Batumi, Georgia, Springer, 2007). If Natural Logic is going to be used, as an inferential mechanism between text fragments, in Natural Language Processing applications such as text summarization, question answering, and information extraction, it is a priority to know what the existing relationship among the aforementioned algorithms is; for example, to implement the most general. We show in this paper the equivalence among the analyzed algorithms, filling a gap in Natural Logic research, particularly in computing polarity, and the soundness of their algorithms.

Three versions of the lambda calculus are discussed: typed lambda calculus, untyped lambda calculus, and polymorphically typed lambda calculus. It is shown that the syntax for these calculi can be described as initial algebras for suitable endofunctors. Models of the first two calculi are described as in [Lambek and Scott 1985] and the Per model of the third is recalled. Algebraic types are described in terms of sketches. A main contribution of the paper is the internalization of the theory of sketches in the category Per. Sketches themselves, models of sketches, categories of models of sketches and the category of all models of all sketches can all be realized in this setting. Next, dinatural transformations are reviewed in the context of the category Per and the relation between certain end formulas and initial algebras for finite signatures is discussed. The point of the theory of algebraic semantics in Per is to provide a setting for thinking about algebraic types and algebraic operations in conjunction with logical types and polymorphic operations. Two specific examples are discussed and a general principle for combining algebraic and polymorphic operations is proposed.KeywordsNatural TransformationCategory TheoryLogical TypeDenotational SemanticGraph ObjectThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the linguistic study of syntax, various formats have been developed for measuring the combinatorial power of natural languages. In particular, there is Chomsky's well-known hierarchy of grammars which can be employed to calibrate syntactic complexity. No similar tradition exists in linguistic semantics, however. Existing powerful formalisms for stating truth conditions, such as Montague Semantics, are usually presented as monoliths like Set Theory or Intensional Type Theory, without any obvious way of expressive fine-tuning. The purpose of this chapter is to show how semantics has its fine-structure too, when viewed from a proper logical angle.

"This paper is about our hobby." That is the first sentence of [MP93], the first report on our formal development of lambda calculus and type theory, written in autumn 1992. We have continued to pursue this hobby on and off ever since, and have developed a substantial body of formal knowledge, including Church-Rosser and standardization<br />theorems for beta reduction, and the basic theory of<br />Pure Type Systems (PTS) leading to the strengthening theorem and type checking algorithms for PTS. Some of this work is reported in [MP93, vBJMP94, Pol94b, Pol95]. In the present paper we survey this work, including some new proofs, and point out what we feel has been learned about the general issues of formalizing mathematics. On the technical side, we describe an abstract, and simplified, proof of standardization for beta reduction, not previously published, that does<br />not mention redex positions or residuals. On the general issues, we emphasize the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts. The LEGO Proof Development System [LP92] was used to check the work in an implementation of the Extended Calculus of Constructions<br />(ECC) with inductive types [Luo94]. LEGO is a refinement style<br />proof checker, publicly available by ftp and WWW, with a User's Manual [LP92] and a large collection of examples. Section 1.3 contains information on accessing the formal development described in this paper. Other interesting examples formalized in LEGO include program specification and data refinement [Luo91], strong normalization of System F [Alt93], synthetic domain theory [Reu95, Reu96], and operational semantics for imperative programs [Sch97].

William E. McMahon: Did Montague Invent «Montague Grammar»? Is Montague Grammar a variant of categorial grammar? After a careful examination of Ajdukiewicz's, Bar Hillel's and Bochenski's categorial theories, the author says in conclusion that Montague Grammar is not reducible to Ajdukiewicz categorial grammar and that it is not simply a categoriaI grammar. There is in Montague Grammar a synthesis of recent developments in logical theory which transcends categorial theory. But we have to make a distinction between assessing its originality and accepting its empirical relevance.

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 Victor Sanchez 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.

We describe experiments in teaching fundamental informatics notions around mathematical structures for formal concepts, and effective algorithms to manipulate them. The major themes of lambda-calculus and type theory served as guides for the effective implementation of functional programming languages and higher-order proof assistants, appropriate for reflecting the theoretical material into effective tools to represent constructively the concepts and formally certify the proofs of their properties. Progressively, a literate programming and proving style replaced informal mathematics in the presentation of the material as executable course notes. The talk will evoke the various stages of (in)completion of the corresponding set of notes along the years, and tell how their elaboration proved to be essential to the discovery of fundamental results.

A compositional theory of verbal argument structure that explores how “noncore” arguments are introduced into argument structures and examines cross-linguistic variation in introducing arguments. This concise work offers a compositional theory of verbal argument structure in natural languages that focuses on how arguments that are not “core” arguments of the verb (arguments that are not introduced by verbal roots themselves) are introduced into argument structures. Liina Pylkkänen shows that the type of argument structure variation that allows additional noncore arguments is a pervasive property of human language and that most languages have verbs that exhibit this behavior. It would be natural to hypothesize that the grammatical elements that allow for this variation are the same in different languages, but Pylkkänen, citing the differences between the inventories of verbs that allow additional arguments in English and Venda, shows the difficulties in this assumption. Either the noncore arguments are introduced by different elements with different distributions, she argues, or the introducing elements are the same and some other factor is responsible for the distributional difference. Distinguishing between these two types of explanations and articulating the properties of argument-introducing elements is the essence of Pylkkänen's theory. Investigating the grammatical elements that allow the addition of noncore arguments, Pylkkänen argues that the introduction of additional arguments is largely carried by seven functional heads. Following Chomsky, she claims that these belong to a universal inventory of functional elements from which a particular language must make its selection. Cross-linguistic variation, she argues, has two sources: selection; and the way a language packages the selected elements into syntactic heads.

Our daily verbal communication – official, formal, informal, about general, professional, or private topics – represents our subjective opinions that mostly include reasoning and arguments which we use in order to prove our point. The linguistic realisation (verbalisation) of our personal position always requires the use of extraordinary syntactic-pragmatic structural elements, such as arguments that add credibility to the statement. The study of argumentative structures allows us to identify the linguistic cues that turn a regular expression into an argumentative statement. Understanding those cues in a particular language helps the speakers improve their communication skills.Researching argumentative structures in the context of natural language processing has become a very active process over the past few years. One of the main focuses is on the computational processing of argumentative texts, which significantly increases the feasibility of automatic processing and the semantic analysis of large volumes of text data. This paper aims to describe and reveal argumentation structures in the spoken Georgian language. The research data comes from a sub-corpus of the Georgian National Corpus, the Georgian Political Text Corpus. The database is thematically diverse and contains politicians’ speeches and TV debates, specifically:● public speeches of the Presidents of Georgia and other politicians (Zviad Gamsaxurdia, Eduard Shevardnadze, Mikheil Saakashvili, Noe Jordania, Giorgi Kvirikashvili, Shalva Natelashvili)● recordings of Parliament and TV debates (Parliament sessions, political TV shows).The corpus of political texts not only contains thematically diverse materials but is also quite representative from a quantitative point of view: This article analyses the constituent parts of an argumentative text such as premises and reasoning and their connections, with the aim to generally characterise the argumentative structures in Georgian on the basis of Janier and Saint-Disier 2019.1) Arguments based on analogy:Premise 1: Case C1 is similar to case C2.Premise 2: A is true (false) in case C1.Conclusion: A is true (false) in case C2. 2) Arguments based on negative consequences:Premise 1: If an action leads to bad consequences, everything else being equal, it should not be exerted.Premise 2: If action A is exerted, bad consequences will occur.Conclusion: A should not be exerted.3) Arguments based on expert opinions:Premise 1: E is a reliable authority in the domain S.Premise 2: A is a proposition contained in S.Premise 3: E asserts that A.Conclusion: A is true.4) Arguments based on examples:Premise 1: Example 1 is an example that supports claim P.Premise 2: Example n is an example that supports claim P.Conclusion: Claim P is true.5) Arguments based on positions to know:Premise 1: Source A is in a position to know something about domain S which contains proposition P.Premise 2: A asserts that P (in domain S) is true (false),Conclusion: P is true (false)6) Arguments based on popular opinions:Premise 1: Everybody does X.Premise 2: X is a good thing to do.Conclusion: X must be the right thing to do.7) Arguments based on causes and effects:Premise 1: Doing X will cause Y to occur, or if X occurs then Y will occur.Premise 2: X is done, or X occurs.Conclusion: Y will occur

This study focuses on the description of argument structure and word order in Saudi Sign language (SSL). The nature of the syntactic level of the grammar of SSL is clarified. Since word order is often considered the most important part of grammar, this study details the various options that are available for the major constituents (Subject, Verb, and Object) in SSL independent of any connection to spoken Arabic syntax. In SSL, like in other languages, the nature of the arguments (Subject, Object) and the kind of verb can impact the word order. To investigate word order in SSL, which is based on Chomsky’s (1995) Minimalist Program (MP), data were collected through derivation of data from narratives (semi-naturalistic-corpus) and an experiment (picture-description task). This research involved a sample of 10 deaf signing participants who are all fluent in SSL. All the participants have lived in Saudi Arabia for at least 10 years. Results indicate that the SSL is as any natural language and from the spoken language in Saudi Arabia. The most common word order in declaratives and the basic constituent order in SSL is SVO. As in virtually all sign languages, Wh-elements in SSL occur at the end of the clause. According to research on the interaction between word order and grammatical use of facial expressions and head locations (nonmanual marking), nonmanual markings have pragmatic purposes and may have syntactic functions.

Powersets, Fixed Points, and Lambda Calculus

The aim of this book is to show how constructive type theory, in the form developed by Per Martin-Löf, helps us to understand the structure of natural language. The study of similar questions within simple (=classical) type theory was inaugurated by Frege and given a standard of technical detail by Montague. The point of shifting to constructive type theory is not in the constructivist criticism against classical reasoning, but in the positive observation that constructive type theory has new things to say. Some of the problems to which grammars based on classical type theory have not found satisfactory solutions will be shown to require precisely these new things. These problems appear in a simple form in what is known, since Geach, as donkey sentences. But they are repeated in other guises, for example, in the functioning of tenses, and in a larger scale, such as in the progressive structure of texts, and in the significance of context in communication.

An abstract framework for structured arguments is presented, which instantiates Dung's (‘On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming, and n-Person Games’, Artificial Intelligence, 77, 321–357) abstract argumentation frameworks. Arguments are defined as inference trees formed by applying two kinds of inference rules: strict and defeasible rules. This naturally leads to three ways of attacking an argument: attacking a premise, attacking a conclusion and attacking an inference. To resolve such attacks, preferences may be used, which leads to three corresponding kinds of defeat: undermining, rebutting and undercutting defeats. The nature of the inference rules, the structure of the logical language on which they operate and the origin of the preferences are, apart from some basic assumptions, left unspecified. The resulting framework integrates work of Pollock, Vreeswijk and others on the structure of arguments and the nature of defeat and extends it in several respects. Various rationality postulates are proved to be satisfied by the framework, and several existing approaches are proved to be a special case of the framework, including assumption-based argumentation and DefLog.

There has recently been a resurgence of scientific interest concerning the importance of various types of phrase structure in human language, and their potential presence in other species (Hauser, Chomsky, and Fitch 2002; Fitch and Hauser 2004; Everett 2005; Fitch, Hauser, and Chomsky 2005; Pinker and Jackendoff 2005; Gentner et al. 2006). Following Hauser, Chomsky, and Fitch (2002, HCF hereafter), many of these recent discussions have used the term "recursion," but the term has rarely been defined explicitly. There are several possible interpretations of this word, which is used somewhat differently in different disciplines, without there being one universally accepted scientific definition. With the recent advent of intense inter-disciplinary discussion of these issues, it has become clear that several different interpretations of the term are being used interchangeably. We seem to have reached a point where serious misunderstandings are in danger of propagating through the literature. The main purpose of this paper is to clarify these different meanings, and in the process to examine the implications of recent and ongoing experiments for different types of grammars in animals and humans.