The classification of grammars that became known as the Chomsky hierarchy was an exploration of what kinds of regularities could arise from grammars that had various conditions imposed on their structure. Intersubstitutability is closely related to the way different levels on the Chomsky hierarchy correspond to different kinds of memory. This chapter deals with the general concept of a string‐rewriting grammar, which provides the setting in which the Chomsky hierarchy can be formulated. An unrestricted rewriting grammar works with a specified (...) set of nonterminal symbols, and specified set of terminal symbols. From the very outset there were doubts about about whether context‐free grammars (CFGs) could form the basis of a theory of natural language syntax. The grammar's rewrite rules correspond to the automaton's transitions. Chomsky argued that even if the generative capacity of CFGs turned out to be sufficient for English, the resulting grammars would be unreasonably complex. (shrink)

Pāṇini's fourth century BC Sanskrit grammar uses rewrite rules utilizing an explicit formal language defined through a semi-formal metalanguage. The grammar is generative, meaning that it is capable of expressing a potential infinity of well-formed Sanskrit sentences starting from a finite symbolic inventory. The grammar's operational rules involve extensive use of auxiliary markers, in the form of Sanskrit phonemes, to control grammatical derivations. Pāṇini's rules often utilize a generic context-sensitive format to identify terms used in (...) replacement, modification or deletion operations. The context-sensitive rule format is itself defined using Pāṇini's more general method of auxiliary markers, the latter used to define many dozens of linguistic categories and rules controlling derivations of Sanskrit sentences through the manipulation of ‘non-terminal’ and ‘terminal’ symbols. This technique for controlling formal derivations was rediscovered by Emil Post in the 1920s and later shown by him to be ca... (shrink)

This chapter contains sections titled: The Tractatus and rules of logical syntax From logical syntax to philosophical grammar Rules and rule‐formulations Philosophy and grammar The scope of grammar Some morals.

The influential Italian thinker offers three essays in the political philosophy of language. Multitude between Innovation and Negation by Paolo Virno translated by James Cascaito. The publication of Paolo Virno's first book in English, Grammar of the Multitude, by Semiotext in 2004 was an event within the field of radical political thought and introduced post-'68 currents in Italy to American readers. Multitude between Innovation and Negation, written several years later, offers three essays that take the reader on a journey through (...) the political philosophy of language. “Wit and Innovative Action” explores the ambivalence inevitably arising when the semiotic and the semantic, grammar and experience, rule and regularity, and right and fact intersect. Virno unravels the infinite potential and wonders of everyday linguistic praxis and ambiguity. Wit, he argues, is a public performance, and its modus operandi characterizes human action in a state of emergency; it is a reaction, an articulate response, and a possible solution to a state of crisis. “Mirror Neurons, Linguistic Negation, and Mutual Recognition” examines the relationship of language and intersubjective empathy: without language, would human beings be able to recognize other members of their species? And finally, in “Multitude and Evil,” Virno challenges the distinction between the state of nature and civil society and argues for a political institution that resembles language in its ability to be at once nature and history. Few thinkers take the risks required by innovation. Like a philosophical entrepreneur, Virno is engaged in no less than rewriting the dictionary of political theory, an urgent and ambitious project when language, caught in a permanent state of emergency impossible to sustain, desperately needs to articulate and enact new practices of freedom for the multitude. Paolo Virno is the author of several books, including A Grammar of the Multitude, 2004). (shrink)

Multitude between Innovation and NegationPaolo Virnotranslated by James CascaitoThe publication of Paolo Virno's first book in English, Grammar of the Multitude, by Semiotext in 2004 was an event within the field of radical political thought and introduced post-'68 currents in Italy to American readers. Multitude between Innovation and Negation, written several years later, offers three essays that take the reader on a journey through the political philosophy of language."Wit and Innovative Action" explores the ambivalence inevitably arising when the semiotic and (...) the semantic, grammar and experience, rule and regularity, and right and fact intersect. Virno unravels the infinite potential and wonders of everyday linguistic praxis and ambiguity. Wit, he argues, is a public performance, and its modus operandi characterizes human action in a state of emergency; it is a reaction, an articulate response, and a possible solution to a state of crisis. "Mirror Neurons, Linguistic Negation, and Mutual Recognition" examines the relationship of language and intersubjective empathy: without language, would human beings be able to recognize other members of their species? And finally, in "Multitude and Evil," Virno challenges the distinction between the state of nature and civil society and argues for a political institution that resembles language in its ability to be at once nature and history.Few thinkers take the risks required by innovation. Like a philosophical entrepreneur, Virno is engaged in no less than rewriting the dictionary of political theory, an urgent and ambitious project when language, caught in a permanent state of emergency impossible to sustain, desperately needs to articulate and enact new practices of freedom for the multitude.Paolo Virno is the author of several books, including A Grammar of the Multitude, 2004). (shrink)

Logic is traditionally considered to be a purely syntactic discipline, at least in principle. However, prof. David Isles has shown that this ideal is not yet met in traditional logic. Semantic residue is present in the assumption that the domain of a variable should be fixed in advance of a derivation, and also in the notion that a numerical notation must refer to a number rather than be considered a mathematical object in and of itself. Based on his work, the (...) central question of this thesis is what kind of logic, if any, results from removing this semantic residue from traditional logic.We differ from traditional logic in two significant ways. The first is that the assumption that a numerical notation must refer to a number is denied. Numerical notations are considered as mathematical objects in their own right, related to each other by means of rewrite rules. The traditional notion of reference is then replaced by the notion of reduction (by means of the rewrite rules) to a normal form. Two numerical notations that reduce to the same normal form would traditionally be considered identical, as they would refer to the same number, and hence they would be interchangeable salva veritate. In the new system, called Buridan-Volpin (BV), the numerical notations themselves are the elements of the domains of variables, and two numerical notations that reduce to the same normal form need not be interchangeable salva veritate, except when they are syntactically identical (i.e., have the same Gödel number).The second is that we do away with the assumption that the domains of variables need to be fixed in advance of a derivation. Instead we focus on what is needed to guarantee preservation of truth in every step of a derivation. These conditions on the domains of the variables, accumulated in the course of a derivation, are combined in a reference grammar. Whereas traditionally a derivation is considered valid when the conclusion follows from the premisses by way of the derivation rules (and possibly axioms), in the BV system a derivation must meet the extra condition that no inconsistency occurs within the reference grammar. For if the reference grammar were to give rise to inconsistency (i.e., it would be impossible to assign domains to all the variables without breaking at least one of the conditions placed on them in the reference grammar), there is no longer a guarantee that truth has been preserved in every step of the derivation, and hence the truth of the conclusion is not guaranteed by the derivation.In Chapter 2 the BV system is introduced in some formal detail. Chapter 3 gives some examples of derivations, notably totality of addition, multiplication and exponentiation, as well as a lemma needed for the proof of Euclid’s Theorem. These examples, taken from prof. Isles’ First-Order Reasoning and Primitive Recursive Natural Number Notations, show that there is a real proof-theoretical difference between traditional logic and the BV system. Here we also find the first major point of departure between myself and prof. Isles, centered on the notion of inheritance of conditions in the reference grammar by way of lemmata. These different points of view are best illustrated in the sections on the totality of exponentiation and on Euclid’s Lemma: prof. Isles maintains that the proof of totality of exponentiation is not BV valid, while I maintain that it is. But I do agree with him that the traditional proof of Euclid’s Lemma is not BV valid. Chapter 6 also expands the arguments for my choice in this matter.Now that it has been shown that there is a difference between traditional logic and BV, the properties of BV need to be examined. In Chapter 4 we give a proof of Cut-elimination for BV minus induction and the subformula property for BV, which allows us to prove the consistency of BV minus induction. We also expand on the reasons for excluding induction. In Chapter 5 we consider in detail the proof of a finite analog to the Löwenheim–Skolem theorem given by prof. Isles in his article with the same title. He proves that under certain conditions it is always possible, given the existence of a (possibly uncountable) model for a derivation, to give a finite model for this derivation. The system he considers deviates from BV as considered in this thesis in two significant ways: it does not contain the induction rule and the domains contain numbers instead of numerical notations. We then go on to show that it is possible to extend the result to include induction, in the sense that the existence of a possibly uncountable model for a derivation guarantees the existence of a model that is at most countable. We also consider the complications that arise from taking numerical expressions instead of numbers as the elements of domains.Finally, in Chapter 6 we consider the philosophical consequences of the BV system, informed by the formal results from the previous chapters. In particular we discuss the relation between reduction and reference, the status of reference grammars, the notion of induction and its function within BV, and some brief considerations on the consequences of the BV system for the discussion regarding nominalism and realism with regard to mathematical objects. The object of the chapter is twofold. On the one hand applying the formal results to philosophical questions, on the other hand arguing that BV is not just a theoretically acceptable alternative to traditional logic, but is in fact deserving of further development and research into its properties. The latter will probably appeal most to those of a nominalist and/or finitist bent.Abstract prepared by Harrie de Swart and Sven Storms.E-mail:[email protected]:https://research.tilburguniversity.edu/files/6170 1294/Storms_The_Buridan_Volpin_04_05_2022.pdf. (shrink)

This is the second volume of analytical commentary on Wittgenstein's masterpiece, the Philosophical Investigations. Like the first, it consists of philosophical essays and critical exegesis. The six essays deal comprehensively with various themes in Wittgenstein''s philosophy: the relationship between his mathematics and his philosophy of mind; his conception of grammar and rules of grammar; the relation between a rule and what accords with a rule; the characterization of rule-following as mastery of a technique manifest in practice; his notion of (...) a form of life, and of agreement in definitions and judgements; a comprehensive investigation into his account of logical and mathematical necessity and other topics. (shrink)

The Second Edition of _Wittgenstein: Rules, Grammar and Necessity_ now includes extensively revised and supplemented coverage of the Wittgenstein's complex and controversial remarks on following rules. Includes thoroughly rewritten essays and the addition of one new essay on communitarian and individualist conceptions of rule-following Includes a greatly expanded essay on Wittgenstein’s conception of logical, mathematical and metaphysical necessity Features updates to the textual exegesis as the result of taking advantage of the search engine for the Bergen edition of (...) the _Nachlass_ Reflects the results of scholarly debates on rule-following that have raged over the past 20 years. (shrink)

The Second Edition of _Wittgenstein: Rules, Grammar and Necessity_ now includes extensively revised and supplemented coverage of the Wittgenstein's complex and controversial remarks on following rules. Includes thoroughly rewritten essays and the addition of one new essay on communitarian and individualist conceptions of rule-following Includes a greatly expanded essay on Wittgenstein’s conception of logical, mathematical and metaphysical necessity Features updates to the textual exegesis as the result of taking advantage of the search engine for the Bergen edition of (...) the _Nachlass_ Reflects the results of scholarly debates on rule-following that have raged over the past 20 years. (shrink)

This volume gathers papers, which were read at the congress held at the University of Castilla-La Mancha in Toledo (Spain), in September 2007, under the general subject of phenomenology. The book is devoted to Wittgenstein s thoughts on phenomenology. One of its aims is to consider and examine the lasting importance of phenomenology for philosophic discussion. For E. Husserl phenomenology was a discipline that endeavoured to describe how the world is constituted and experienced through a series of conscious acts. His (...) fundamental concept was that of intentional consciousness. What did drag Wittgenstein into working on phenomenology? In his 'middle period' work, Wittgenstein used the headline 'Phenomenology is Grammar'. These cornerstones can be signalled by notions like language, grammar, rule, visual space versus Euclidean space, minima visibilia and colours. L. Wittgenstein s main interest takes the form of a research on language.". (shrink)

This volume gathers papers, which were read at the congress held at the University of Castilla-La Mancha in Toledo (Spain), in September 2007, under the general subject of phenomenology. The book is devoted to Wittgenstein’s thoughts on phenomenology. One of its aims is to consider and examine the lasting importance of phenomenology for philosophic discussion. For E. Husserl phenomenology was a discipline that endeavoured to describe how the world is constituted and experienced through a series of conscious acts. His fundamental (...) concept was that of intentional consciousness. What did drag Wittgenstein into working on phenomenology? In his 'middle period' work, Wittgenstein used the headline 'Phenomenology is Grammar'. These cornerstones can be signalled by notions like language, grammar, rule, visual space versus Euclidean space, minima visibilia and colours. L. Wittgenstein’s main interest takes the form of a research on language. (shrink)

This chapter contains sections titled: Setting the stage Leitmotifs External guidelines Necessary propositions and norms of representation Concerning the truth and falsehood of necessary propositions What necessary truths are about Illusions of correspondence: ideal objects, kinds of reality and ultra‐physics The psychology and epistemology of the a priori Propositions of logic and laws of thought Alternative forms of representation The arbitrariness of grammar A kinship to the non‐arbitrary Proof in mathematics Conventionalism.

This chapter contains sections titled: Following a rule Practices and techniques Doing the right thing and doing the same thing Privacy and the community view On not digging below bedrock.

This chapter contains sections titled: Initial compass bearings Accord and the harmony between language and reality Rules of inference and logical machinery Formulations and explanations of rules by examples Interpretations, fitting and grammar Further misunderstandings.

Review of G.P. Baker and P.M.S. Hacker's Wittgenstein: Rules, Grammar and Necessity, the second volume of their analytical commentary on Wittgenstein's Philosophical Investigations.

The paper explores Wittgenstein’s notion of grammar in the sense of a discipline or an activity, as opposed to the object sense of the term (grammar as a body of rules for the use of a language). I argue that the Wittgensteinian activity of grammar consists in giving expression to rules of our language use. It differs from the traditional grammarian’s activity not only in focusing on a different type of rules, but also in that it does (...) not aim at an explicit and exhaustive treatment of a specific domain of language. Instead, Wittgenstein conceives its goal as therapeutic: the dissolution of particular philosophical problems. Further, I attempt to reconcile his seemingly contradictory remarks on the character of grammatical statements, defining the senses in which they respectively can, and cannot, be considered descriptive assertions. I confront G. P. Baker’s and P. M. S. Hacker’s conceptions of the Wittgensteinian grammatical activity and I argue in favour of the former. Finally, I critically examine N. Garver’s claim that Wittgenstein, in his later conception of philosophy as grammar, succeeded in formulating a successfully self-referential criterion of philosophical critique. I also argue that grammatical activity, despite Wittgenstein’s overt commitment, is in fact not the only method of his later philosophy. (shrink)

When the NCAA adopted new rules allowing athletes to profit off their name, image, and likeness (NIL), few people took more interest than Reggie Bush who famously relinquished the Heisman trophy after being ruled retroactively ineligible for receiving "impermissible benefits." Bush has argued for his reinstatement and the "return" of his Heisman. In this paper, I argue that, while the NCAA never should have required players to be amateurs in the first place, Bush should not be reinstated or have (...) the Heisman "returned." Properly understood, Reggie Bush never won the Heisman Trophy in the first place, so "returning" it would be to rewrite history — almost literally, to move the goalposts. -/- Forthcoming in "College Sports and Ethics" by Lexington Books. (shrink)

In his final notebook, published posthumously as On Certainty , Wittgenstein offers a sustained and, at least apparently, fragmentary treatment of skeptical issues. Given the ambiguity and obscurity of some of its remarks, in the recent literature on the subject we can find a number of competing interpretations of OC, particularly of the elusive concept of ‘hinges’, central to Wittgenstein’s last work. In this paper, I will discuss the dominant interpretations of OC in order to show how they fail to (...) represent plausible renderings of his anti-skeptical thought. Finally, I will argue that the analogy between ‘hinges’ and ‘rules of grammar’, correctly understood and developed, can represent a plausible interpretation of Wittgenstein’s thought and, more importantly, a viable anti-skeptical strategy. (shrink)

The goal of this book is to develop a new approach to reading the rule-following sections guided by a simple idea. The simple idea is that Wittgenstein's remarks on rule-following are split between two distinct but complementary projects. The projects are marked not only by different guiding questions, but different presuppositions and methodologies. There is of course precedent for reading the rule-following remarks as comprising two parts. For example, there is the reading of (S. Kripke 1982) on which Wittgenstein first (...) presents a skeptical challenge before turning to its skeptical solution. And there are readings like (Minar 2011) and (Goldfarb 2012) that likewise treat the sections as roughly divided into two components or "chapters." But even these commentators do not explicitly associate their components with different questions and associated methodologies, and so fall short of attributing to Wittgenstein's text anything quite like the thoroughgoing bipartite structure I will be proposing here. (shrink)

P¯an.ini’s grammar includes several types of metarules which determine how its operational rules apply. Among them are “traffic rules” which constrain how rules interact with each other in grammatical derivations. These are typically formulated as designating a rule or class of rules asiddha “not effected” (or asiddhavat “as if not effected”) with respect to another rule or class of rules. For economy, the rules so designated are grouped into several sections, whose headings collectively declare (...) them to be asiddha(vat). The biggest such section, under the famous heading 8.2.1 p ¯urvatr¯asiddham, extends from 8.2.1 through the end of the grammar (8.4.68), and is hence called the Trip¯ad¯ı ‘Three Sections’. (shrink)

I examine the phenomenon of implicit learning, the process by which knowledge about the rule-governed complexities of the stimulus environment is acquired independently of conscious attempts to do so. Our research with the two seemingly disparate experimental paradigms of synthetic grammar learning and probability learning, is reviewed and integrated with other approaches to the general problem of unconscious cognition. The conclusions reached are as follows: Implicit learning produces a tacit knowledge base that is abstract and representative of the structure of (...) the environment; such knowledge is optimally acquired independently of conscious efforts to learn; and it can be used implicitly to solve problems and make accurate decisions about novel stimulus circumstances. Various epistemological issues and related problems such as intuition, neuroclinical disorders of learning and memory, and the relationship of evolutionary processes to cognitive science are also discussed. (shrink)

While rules and exemplars are usually viewed as opposites, this paper argues that they form end points of the same distribution. By representing both rules and exemplars as (partial) trees, we can take into account the fluid middle ground between the two extremes. This insight is the starting point for a new theory of language learning that is based on the following idea: If a language learner does not know which phrase‐structure trees should be assigned to initial sentences, (...) s/he allows (implicitly) for all possible trees and lets linguistic experience decide which is the “best” tree for each sentence. The best tree is obtained by maximizing “structural analogy” between a sentence and previous sentences, which is formalized by the most probable shortest combination of subtrees from all trees of previous sentences. Corpus‐based experiments with this model on the Penn Treebank and the Childes database indicate that it can learn both exemplar‐based and rule‐based aspects of language, ranging from phrasal verbs to auxiliary fronting. By having learned the syntactic structures of sentences, we have also learned the grammar implicit in these structures, which can in turn be used to produce new sentences. We show that our model mimicks children’s language development from item‐based constructions to abstract constructions, and that the model can simulate some of the errors made by children in producing complex questions. (shrink)

Geach’s rich paper ‘A Program for Syntax’ introduced many ideas into the arena of categorial grammar, not all of which have been given the attention they warrant in the thirty years since its first publication. Rather surprisingly, one of our findings (Section 3 below) is that the paper not only does not contain a statement of what has widely come to be known as “Geach’s Rule”, but in fact presents considerations which are inimical to the adoption of the rule in (...) question. With regard to at least some amongst the numerous other points extracted here from Geach’s discussion, we shall not be able to reach so definitive a conclusion, and content ourselves with giving the issues an airing. (shrink)

Lecture notes from Husserl's logic lectures published during the last 20 years offer a much better insight into his doctrine of the forms of meaning than does the fourth Logical Investigation or any other work published during Husserl's lifetime. This paper provides a detailed reconstruction, based on all the sources now available, of Husserl's system of logical grammar. After having explained the notion of meaning that Husserl assumes in his later logic lectures as well as the notion of form of (...) meaning as it features in ‘doctrine of the forms of meaning’, I present a system of rules that describes all the various forms of meaning that Husserl singles out in his lectures. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 10.2 (2003) 135-141 [Access article in PDF] On Delusions of Sense:A Response to Coetzee and Sass Rupert Read Keywords schizophrenia, Wittgenstein, Schreber, Faulkner, Benjy, grammar, madness, Cogito The great writings on and of severe mental affliction—those for instance of Schreber, 'Renee', Donna Williams, Artaud, Faulkner's The Sound and the Fury, Coetzee's In the Heart of the Country, Kafka's "Description of a struggle," and even (I (...) would add) key parts of The Lord of the Rings—present us with something deeply enigmatic. They have, we might say, a strong grammar, a grammar—a mode of hanging together, and (in this case) of linguistically seeming to make a sense that is not our sense and that we cannot make sense of—... they have a grammar all of their own, and all of its own, a grammar that resists and rejects interpretation even as it sometimes seems to offer interpretations. 1It was Wittgenstein's view that to make mental illness unpuzzling was a mistake, or (better, perhaps) a mythologically problematic move. Wittgenstein's aim in his philosophizing was to understand what was enigmatic when it could be understood without unwisely turning it into something altogether unpuzzling... and then to acknowledge that there are some things that may remain forever puzzling, without committing oneself to the metaphysically disastrous claim that the reason they are endlessly puzzling is that they lie outside the boundaries of human life, language, or reason, as though we could peek outside those alleged boundaries to see what was there, but never truly say anything about it.There simply may be places where our understanding—phenomenological understanding, understanding of what it is like—gives out, and not because it is (or we are) merely human. For instance, perhaps one cannot capture some mental illness by intellection alone, or even perhaps at all. Perhaps the best understanding one can have of mental illness is purely negative (in a sense at least as strong as that involved in negative theology, wherein God is only defined by what it is not).Louis Sass (2003) writes I am not sure whether or not Read would accept that a philosophical position can be understandable despite its containing deep, internal logical tensions (perhaps he would not). If such understanding is possible, however, it does seem to open the way for a similar understanding of conditions like that of Schreber. (p. 127) Unfortunately, however, it is indeed not possible—that is, this notion is merely a fantasy—by my lights. Philosophical positions, all of which turn out to contain such inexorable tensions, cannot be understood: they are mirages. Solipsism, as a position, is not any better (or worse) off [End Page 135] than (say) (Metaphysical) Realism. They are both mirages, just different aspects of the same nothing. 2 Considered as positions, they are in the end literally—when one understands their logic—one and the same nothing. Wittgenstein already made this quite clear in his Tractatus. Perhaps in contrast to Sass, there is then nothing special about solipsism: it is just one mode of presentation rather than another of the illusion that there can be philosophical positions.Sass claims that in saying this kind of thing I am committed to ruling out that some mental illness can be understood or can even exist, on abstract intellectual grounds. I am not. I am simply wishing to leave open the possibility that there may be things that some people utter or seem to experience which intellection gives us little or no assistance with. If Sass wishes to deny this, then I suggest that it is he and not I who is the absolutist intellectualist here, in insisting that rewriting something deeply strange in a weaker grammar is furthering our grasp of it.My suggestion concerning Faulkner's presentation of Benjy, in my original paper, was that it is a mythological mistake to think that Faulkner provides us with the tools for giving an unpuzzling rendition of what had perhaps appeared to be "another country," inaccessible to us. Rather, Faulkner's representation... (shrink)

The article suggests a reading of the 1st edition of H. L. A. Hart, The Concept of Law (1961) which involves bringing to the fore the elements of moral prescription which accompany the descriptive/explanatory intent which Hart himself explicitly espouses. Thus, the functionalist account of the emergence of secondary rules in complex societies draws on the importance of social benefits, relating to justice and efficiency, benefits which Hart endorses. Consideration is then given to how the posthumous Postscript in the (...) 2nd edition of The Concept of Law (1994) might be rewritten to take into account responses to Dworkin other than those adopted by Hart in the Postscript. The alternative of a tentative form of prescriptive ‘hard’ legal positivism would better preserve the prescriptive ingredients evident in the 1st edition and assist in the regeneration of contemporary analytic legal philosophy within the Hartian traditionResumen:Este artículo propone una lectura de la primera edición de El concepto de derecho (1961), de H. L. A. Hart, en la cual se recuperan los elementos de prescripción moral que acompañan a la tesis descriptiva/explicativa que Hart explícitamente defendió. De tal modo, la tesis funcionalista del surgimiento de normas secundarias en sociedades complejas depende de la importancia de los beneficios sociales, relacionados con la justicia y la eficiencia, los cuales Hart suscribe. Después se llama la atención sobre la forma en que el Post Scriptum a la segunda edición de El concepto de derecho (1994), publicado póstumamente, podría ser reescrito para tomar en cuenta respuestas a Dworkin distintas a las que ofreció el propio Hart en su Post Scriptum. La alternativa de una forma tentativa de un positivismo jurídico prescriptivo “duro” conservaría de mucho mejor manera los ingredientes prescriptivos que son tan evidentes en la 1a edición, y contribuiría a la regeneración de una filosofía del derecho analítica contemporánea dentro de la tradición hartiana. (shrink)

This paper explains how mathematical computation can be constructed from weaker recursive patterns typical of natural languages. A thought experiment is used to describe the formalization of computational rules, or arithmetical axioms, using only orally-based natural language capabilities, and motivated by two accomplishments of ancient Indian mathematics and linguistics. One accomplishment is the expression of positional value using versified Sanskrit number words in addition to orthodox inscribed numerals. The second is Pāṇini’s invention, around the fifth century BCE, of a (...) formal grammar for spoken Sanskrit, expressed in oral verse extending ordinary Sanskrit, and using recursive methods rediscovered in the twentieth century. The Sanskrit positional number compounds and Pāṇini’s formal system are construed as linguistic grammaticalizations relying on tacit cognitive models of symbolic form. The thought experiment shows that universal computation can be constructed from natural language structure and skills, and shows why intentional capabilities needed for language use play a role in computation across all media. The evolution of writing and positional number systems in Mesopotamia is used to transfer the thought experiment of “oral arithmetic” to inscribed computation. The thought experiment and historical evidence combine to show how and why mathematical computation is a cognitive technology extending generic symbolic skills associated with language structure, usage, and change. (shrink)

James Shaw has written an excellent book on Wittgenstein’s rule-following considerations. It manages to provide fresh perspectives on a topic on which it seemed.

The only obligatory temporal expression in English is tense, yet Hans Reichenbach (1947) has argued convincingly that the simplest sentence is understood in terms of three temporal notions. Additional possibilities for a simple sentence are limited: English sentences have one time adverbial each. It is not immediately clear how to resolve these matters, that is, how (if at all) Reichenbach's account can be reconciled with the facts of English. This paper attempts to show that they can be reconciled, and presents (...) an analysis of temporal specification that is based directly on Reichenbach's account.Part I is devoted to a study of the way the three times—speech time, reference time, event time—are realized and interpreted. The relevant syntactic structures and their interaction and interpretation are examined in detail. Part II discusses how a grammar should deal with time specification, and proposes a set of interpretive rules. The study offers an analysis of simple sentences, sentences with complements, and habitual sentences. It is shown that tense and adverbials function differently, depending on the structure in which they appear. The temporal system is relational: the orientation and values of temporal expressions are not fixed, but their relational values are consistent. This consistency allows the statement of principles of interpretation. (shrink)

This paper scrutinizes the topic of meaning manifestation and signification made known by the act of interpretation, which amounts to finding the organising principles of a text and rules of combination. The language of narrativity is a set of generational and transformational instances disguising textual content and initiating interpretation as such. The paper discusses the levels of description which assist in tackling the concept of change, or difference in degrees, as the result of both the artificial operation of rewriting (...) the text on various methodological grounds and real dynamics justified by the obvious layers of the plot. The elementary units of meaning articulation covered by this article run the gamut from Roland Barthes’ distributional and integrational narrative units to Algirdas Julien Greimas’ deep and surface structures with special stress on the narrative programs and narrative analysis, which is of primary interest here. The Greimassian narrative grammar successfully describes what exists and how to capture the process of signification. In the end, analytical instruments elaborated by Greimas are effectively adapted to interpret the case of Edgar Allan Poe’s story William Wilson. The story itself is retold and interpreted through narrative formalisation, which consists in speaking of the plot transformations in terms of utterances of states and utterances of doing. Within the framework of narrative structures that emerge on the way to the formalisation of William Wilson, small changes in narrative trajectories seem to be out of sight, not even readable. Contrariwise, Barthes’ touch on what the text is all about enables the reader to be engaged in looser textual interpretations, to gain more variable, highly detailed experience in noticing the slightest movement and tasting each nuance of the overall setting. This paper first and foremost takes on board the structural narrative analysis as it is set out in Greimas’ narrative grammar leading to a special kind of severe narrative absolutism; on top of that, the paper gives a clue of a couple of alternative views on the problem, views represented by Barthes. (shrink)

This is a critical response to Dr. Tamara Dobler's paper “What Is Wrong with Hacker's Wittgenstein? On Grammar, Context and Sense-Determination.” It demonstrates that Dr. Dobler has no idea of what Wittgenstein meant by “grammar” or “rule of grammar.” She does not know what Wittgenstein meant by “grammatical proposition,” nor does she know what a compositional account of meaning or a category mistake is. She labours under the illusion that to say, as Wittgenstein did, that a rule of grammar excludes (...) a form of words from use is incompatible with the claim that whether an utterance makes sense may be a context-dependent issue. Unlike Dr. Dobler, Wittgenstein did not. (shrink)

In Rules and Representations, first published in 1980, Noam Chomsky lays out many of the concepts that have made his approach to linguistics and human cognition so instrumental to our understanding of language.Chomsky arrives at his well-known position that there is a universal grammar, structured in the human mind and common to all human languages. Based on Chomsky's 1978 Woodbridge Lectures, this edition contains revised versions of the lectures and two new essays.

The primary concern of Stephen Mulhall's book is to investigate an interpretation of Wittgenstein's remarks on private language, associated paradigmatically with Norman Malcolm. On this reading, the grammar of our ordinary concepts of language, reference, meaning, rule, etc. is held to prohibit or exclude the idea of a private language. The attempt to give expression to the idea is held to result in a violation of the grammar of these concepts, which connects them essentially with the idea of public criteria (...) that are accepted as establishing that a word has been correctly used. Something that philosophers have been inclined to regard as a possibility is shown to be excluded: the attempt to describe it results in .. (shrink)

“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)

This note identifies and comments on selected crucial traits of Robert Aumann’s philosophy of game theory. In doing so, it aims at carving out and expressing some notions tacitly held by many working game theorists and ideally even at triggering subsequent reflection on the philosophy of game theory in general. According to my reconstruction of Aumann’s position, sophisticated, relatively precise rules of language—an ultra-refined grammar for interactions—constitute the heart of game theory. Consequently, the heart of game theory is devoid, (...) or almost devoid, of empirical content. The final section proposes a nomenclature for lucidly discussing the relationships between theories, different types of models, and “the real world” and employs it to explicate and slightly amend Aumann’s remarks on the role of truth and fruitfulness in game theory. (shrink)

Language sciences have long maintained a close and supposedly necessary coupling between the infinite productivity of the human language faculty and recursive grammars. Because of the formal equivalence between recursion and non-recursive iteration; recursion, in the technical sense, is never a necessary component of a generative grammar. Contrary to some assertions this equivalence extends to both center-embedded relative clauses and hierarchical parse trees. Inspection of language usage suggests that recursive rule components in fact contribute very little, and likely nothing significant, (...) to linguistic creativity. Further than this, if the productivity of human language is considered as not rigidly bound, but not infinite, then the need for any sort of iteration in generative grammars vanishes and can be replaced with a Non-iterative Explicit Alternatives Rule grammar. The knock-on effects of dispensing with recursive grammar components are: that language diversity can simply be based on rule and lexicon combinatorics with no potentially infinite dimensions derived from recursive or iterative components of rules; the oddity of a vast, multiply-infinite competence set of ‘grammatical but unacceptable’ productions is gone; and the development of a language faculty based on rules that eschew iterative rule components avoids any need for explaining ‘special’ mechanisms. On the broader front of searching for the mechanisms of mind, our analysis can be similarly applied to the proposals for a recursive basis for mind as an explanation for humanity’s great leap forward. (shrink)

In this paper, I present and criticize two highly influential anti-skeptical proposals inspired by Wittgenstein’s (1969) remarks on ‘hinges’, namely Pritchard’s ‘hinge commitment strategy’ (2012, forthcoming a, forthcoming b) and Moyal-Sharrock ‘non epistemic strategy’ (2004,2005). I argue that both these proposals fail to represent a valid response to skeptical worries. Furthermore, I argue that following Wittgenstein’s analogy between ‘hinges’ and ‘rules of grammar’ we should be able to get rid of Cartesian skeptical scenarios as nonsensical, even if apparently intelligible, (...) combination of signs excluded from our epistemic practices. (shrink)

This dissertation looks to make sense of the role 'grammar' plays in Wittgenstein's philosophy of mathematics during the middle period of his career. It constructs a formal model of Wittgenstein's notion of grammar as expressed in his writings of the early thirties, addresses the appropriateness of that model and then employs it to test Wittgenstein's claim that mathematical propositions are ultimately grammatical. ;In order to test Wittgenstein's claim that mathematical propositions are grammatical, the dissertation provides a formalized theory of grammatical (...) analysis and applies it to a portion of language big enough to contain numerical expressions. It attempts to prove that, if the object language contains the appropriate numerical expressions, the resulting grammar will include at least some rules with a natural mathematical interpretation. In particular, it tries to show that Wittgenstein's grammatical analysis of the ordinary use of numerical expressions yields familiar theorems of arithmetic. ;The dissertation also endeavors to extend these results to a coherent picture of Wittgenstein's peculiar view of mathematics as grammar. It fits the formal results into the larger picture of Wittgenstein's philosophy of mathematics during this period. A large portion of the dissertation explains how, by combining the notions of grammar and mathematics, Wittgenstein allowed himself to offer original answers to some central questions in the philosophy of mathematics. In this regard, the dissertation pays special attention to four main issues: Wittgenstein's use of the term 'grammar' in his philosophical writings of this period, in comparison with traditional understandings of 'grammar', mathematics as part of the syntax of language, his explanation of mathematical application, and Wittgenstein's account of mathematical necessity. Taking these points in consideration situates the dissertation's results in the larger philosophical discussion of these themes. (shrink)

Husserl’s Logical Grammar is intended to explain how complex expressions can be constructed out of simple ones so that their meaning turns out to be determined by the meanings of their constituent parts and the way they are put together. Meanings are thus understood as structured contents and classified into formal categories to the effect that the logical properties of expressions reflect their grammatical properties. As long as linguistic meaning reduces to the intentional content of pre-linguistic representations, however, it is (...) not trivial to account for how semantics relates to syntax in this context. In this paper, I analyze Husserl’s Logical Grammar as a system of recursive rules operating on representations and suggest that the syntactic form of representations contributes to their semantics because it carries information about semantic role. I further discuss Husserl’s syntactic account of the unity of propositions and argue that, on this account, logical form supervenes on syntactic form. In the last section I draw some implications for the phenomenology of thought and conjecture that the structural features it displays are likely to convey the syntactic structures of an underlying language-like representational system. (shrink)

This work investigates the possibility of iʿrâb (parsing) simple texts by utilizing simple grammar rules in the case of I’râb al-Awâmil al-mi’a. I'râb, which is the 'application of grammar on texts', is an operation of reinforcing the knowledge of grammar from one perspective, and an attempt to under-stand the text from another. Since I’râb al-Awâmil al-mi’a is the iʿrâb of al-Awâmil al-mi’a, which is a brief grammar treatise, it is clear that this iʿrâb is intended to reinforce grammar knowledge. (...) Moreover, it indicates the level of knowledge to be reinforced, as is evident from the text. We think that examining such iʿrâb works in terms of the applicability of iʿrâb, will provide data to the field of grammar teaching. The work is currently available in manuscript and consists of the object text and the author's iʿrâb. The object text of the work, al-Awâmil al-mi'a, is a textbook and the target reader of the book is beginners. It is naturally assumed that both al-Awâmil al-mi'a and its parsing were written for the same level. This work examines the validity of this assumption. In order to determine whether the rules used in the parsing are simple or complex, it was necessary to create a measure of simplicity. Three measures of simplicity have been created. If the grammatical rules used in parsing are: limited to the rules in al-Avâmil, or if they are within the frame of the main topics of grammar or if the author does not address controversial issues during the application of the iʿrâb, then they are simple rules. Moreover, in order to facilitate the achievement of our goal, we divided the sentences into simple and complex sentences, and selected our data mostly from complex sentences. We simplified complex sentences by dividing them into two types: sentences with missing elements and sentences that are semantically connected. We created a second dataset by adding some examples other than the complex sentence types. We selected the other sentence types that make up the second dataset from sentences that contain elements that al-Awāmil al-mi'a does not include. The mentioned datasets examined by utilizing the measure of simplicity that we created and the degree of complexity of the grammatical rules used in I'râb al-Awâmil al-mi'a, was revealed. (shrink)

This chapter contains sections titled: The continuation of the Early Draft into philosophy of mathematics Hidden isomorphism A common methodology The flatness of philosophical grammar.

I show that words with indefinite implicit complements occasion a dilemma for their model theory. There has been only two previous attempts to address this problem, one by Fodor and Fodor (1980) and one by Dowty (1981). Each requires that any word tolerating an implicit complement be treated as ambiguous between two different lexical entries and that a meaning postulate or lexical rule be given to constrain suitably the meanings of the various entries for the word. I show that the (...) positing of such an ambiguity runs counter to the facts and propose an alternative solution which does not appeal to ambiguity, meaning postulates or lexical rules. Indeed, I show that the dilemma posed by indefinite implicit complements is posed by all implicit complements and that a general solution to the problem of implicit complements follows from an independently motivated, single treatment of five other problems, that of subcategorization, that of phrasal projections of words, that of defining a model theoretic structure for phrase structure grammars, that of complement polyvalence and that of complement polyadicity. (shrink)

This chapter contains sections titled: From the Synoptic View to the Album “I don't know my way about” The Problem of Grammar Essential Complexity The Practice of Language Hyper ‐ complexity.

Grammar is a word associated with the rules that govern language and its related pedagogy for articulating types of declarative knowledge. It can also refer to the organizational structure of practices and their related forms of knowledge, as described here by Ludwig Wittgenstein: “Essence is expressed in grammar.... Grammar tells us what kind of object anything is. (Theology as grammar.)”1 Wittgenstein’s remark about theology can be generalized to visual art, and, by extension, to the grammatical structure of art education. (...) The grammar of creative practices is described by George Steiner as the “articulate organisation of perception, reflection and experience, the nerve structure of consciousness when.. (shrink)