The study of the interdependence of grammar and logic at the beginning of the twelfth century is a difficult subject and progress here has been slow. With the recent publication of the Notae Dunelmenses, however, we are now able to see rather more clearly how closely the two disciplines were bound to one another. The following article draws upon this newly published material and on unpublished material from contemporary commentaries on Aristotle’s Categories to investigate how the grammarians’ account (...) of number was reconciled with that given by Aristotle. It considers in particular the problem of the meaning of numerical terms such as ‘pair’ and of collective names such as ‘people’ and how attempting to solve it shaped thinking about the metaphysics of number. (shrink)

Boethius, the Roman philosopher, was executed for treason and pilloried by modern scholars for misinterpreting Aristotle to the West. This book examines his semantics and logic, attempting to clear his name and lend him new credence.

Taki Suto, Boethius on Mind, Grammar and Logic: A Study of Boethius’ Commentaries on Peri Hermeneias. Leiden: Brill Publishers, 2012. xiii + 296 pp. $148. €108. ISBN 9789004214187.Reviewed by Marga...

Wittgenstein's conception of logical and conceptual truth is often thought to rival that of the logical positivists. This paper argues that there are important respects in which these conceptions complement each other. Analyticity, in the positivists' sense, coincides, not with Wittgenstein's notion of a grammatical proposition, but rather with his notion of a tautology. Grammatical propositions can usually be construed as analyticity postulates in Carnap's sense of the term. This account of grammatical and analytic propositions will be illustrated by appeal (...) to logical, conceptual and arithmetic truths. Its consequences for an analysis of corresponding modal notions will be indicated. (shrink)

As an undergraduate I was taught to multiply two numbers with the help of log tables, using the formulaHaving graduated to teach calculus to Engineers, I learned that log tables were to be replaced by slide rules. It was then that Imade the fateful decision that there was no need for me to learn how to use this tedious device, as I could always rely on the students to perform the necessary computations. In the course of time, slide rules were (...) replaced by pocket calculators and personal computers, but I stuck to my original decision.My computer phobia did not prevent me from taking an interest in the theoretical question: what is a computation? This question goes back to Hilbert's 10th problem, which asks for an algorithm, or computational procedure, to decide whether any given polynomial equation is solvable in integers. It quickly leads to the related question: which numerical functionsf: ℕn→ ℕ are computable?While Hilbert's 10th problem was only resolved in 1970, this related question had some earlier answers, of which I shall single out the following three:f is recursive,f is computable on an abstract machine,f is definable in the untyped λ-calculus.These tentative answers were shown to be equivalent by Church [1936] and Turing [1936–7].I shall discuss here some of the more recent developments of these notions of computability and their relevance to linguistics and logic. I hope to be forgiven for dwelling on some of the work I have been involved with personally, with greater emphasis than is justified by its historical significance. (shrink)

This book argues that a basic grasp of philosophy and logic can produce written and spoken material that is both grammatically correct and powerful. The author analyses errors in grammar, word choice, phrasing and sentences that even the finest writers can fail to notice; concentrating on subtle missteps and errors that can make the difference between good and excellent prose. Each chapter addresses how common words and long-established grammatical rules are often misused or ignored altogether – including such (...) common words as ‘interesting’, ‘possible’, and ‘apparent’. By tackling language in this way, the author provides an illuminating and practical stylistic guide that will interest students and scholars of grammar and philosophy, as well as readers looking to improve their technical writing skills. (shrink)

P.F. Strawson's essay traces some formal characteristics of logic and grammar to their roots in general features of thought and experience.

“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)

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)

Could a computer be programmed to make moral judgments about cases of intentional harm and unreasonable risk that match those judgments people already make intuitively? If the human moral sense is an unconscious computational mechanism of some sort, as many cognitive scientists have suggested, then the answer should be yes. So too if the search for reflective equilibrium is a sound enterprise, since achieving this state of affairs requires demarcating a set of considered judgments, stating them as explanandum sentences, and (...) formulating a set of algorithms from which they can be derived. The same is true for theories that emphasize the role of emotions or heuristics in moral cognition, since they ultimately depend on intuitive appraisals of the stimulus that accomplish essentially the same tasks. Drawing on deontic logic, action theory, moral philosophy, and the common law of tort, particularly Terry's five-variable calculus of risk, I outline a formal model of moral grammar and intuitive jurisprudence along the foregoing lines, which defines the abstract properties of the relevant mapping and demonstrates their descriptive adequacy with respect to a range of common moral intuitions, which experimental studies have suggested may be universal or nearly so. Framing effects, protected values, and implications for the neuroscience of moral intuition are also discussed. (shrink)

1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement (...) of mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. The 1st is the 1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. https://www.academia.edu/s/fb081b1886?source=link . (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Gilbert Ryle never pursued research under Edmund Husserl. However, Ryle was indeed Husserl’s student in a broader sense, as much of his own work was deeply influenced by his studies of Husserl’s pre-World War I writings. While Ryle is the thinker whose name typically comes to mind in connection with the concern over category mistakes I argue that (1) Husserl deserves to be known for precisely this concern as well, and (2) the similarity between them is no accident. Developing this (...) reading of Ryle’s Husserlian pedigree forces a broader reevaluation of each of their roles in twentieth-century thought. (shrink)

This paper introduces λ-grammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, λ-grammars are multi-dimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of linear combinators, closed pure λ-terms in which each abstractor binds exactly one variable. Mathematically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the method seems more practicable.

Although the two volumes of _Logic, Language, and Meaning _can be used independently of one another, together they provide a comprehensive overview of modern logic as it is used as a tool in the analysis of natural language. Both volumes provide exercises and their solutions.

Gilbert Ryle never pursued research under Edmund Husserl. However, Ryle was indeed Husserl’s student in a broader sense, as much of his own work was deeply influenced by his studies of Husserl’s pre-World War I writings. While Ryle is the thinker whose name typically comes to mind in connection with the concern over category mistakes I argue that (1) Husserl deserves to be known for precisely this concern as well, and (2) the similarity between them is no accident. Developing this (...) reading of Ryle’s Husserlian pedigree forces a broader reevaluation of each of their roles in twentieth-century thought. (shrink)

i How can one talk about 'understanding' and 'not understanding' a proposition? Surely it is not a proposition until it's understood ? ...

Analytical commentary -- Fruits upon one tree -- The continuation of the early draft into philosophy of mathematics -- Hidden isomorphism -- A common methodology -- The flatness of philosophical grammar -- Following a rule 185-242 -- Introduction to the exegesis -- Rules and grammar -- 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 (...) -- Exegesis 185-8 -- Accord with a rule -- 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 -- Exegesis 189-202 -- Following rules, mastery of techniques and practices -- 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 -- Private linguists and private linguists : Robinson Crusoe sails again -- Is a language necessarily shared with a community of speakers? -- Innate knowledge of a language -- Robinson Crusoe sails again -- Solitary cavemen and monolinguists -- Private languages and private languages -- Exegesis 203-37 -- Agreement in definitions, judgements, and form of life -- The scaffolding of facts -- The role of our nature -- Forms of life -- Agreement : consensus of human beings and their actions -- Exegesis 238-42 -- Grammar and necessity -- 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 of the A priori -- Knowledge -- Belief -- Certainty -- Surprise -- Discoveries and conjectures -- Compulsion -- 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. (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)

This paper takes distances from two influential images of Wittgenstein's philosophy: the image of a primarily ethical philosopher defended by the so-called «resolute» interpreters and that of an ascetically "analytical" philosopher transmitted by the standard interpretation. Instead of contrasting images (that of Wittgenstein as an "aesthetic" philosopher and that of the "ethical" Wittgenstein), this paper focuses on the analysis of the fractures and tensions characterizing not only the relationship between Wittgenstein's philosophy and aesthetics, but also the very style of Wittgenstein's (...) thought. Addressing a specific issue from a conceptual and textual standpoint (the unity of Ethics and Aesthetics in the Tractatus) seems to the author to be a fruitful strategy that allows us not only to understand whether and how determinant and central the aesthetic problem is for Wittgenstein, but also to see how aesthetics itself can be radically reshaped through the filter offered by his thought. In the first place, then, it is clarified what the Tractatus claim that ethics and aesthetics «sind eins» might entail. Secondly, it is checked if and how the conceptual consistency of the «being one» of ethics and aesthetics is transformed during the 1930s, to the point that it requires a different configuration: the metamor¬phosis of the logical unity between the two conceptual fields into an analogical affinity. Analyzing this conceptual metamorphosis the paper considers also the idea of an asymmetry of the aesthetic over the ethical as already evident, despite appearances, in the 1929 Lecture on Ethics. This asymmetry is then developed focusing the image of grammatical mechanism with its degrees of freedom of which Wittgensteins writes from 1930. In connection with this image the author outlines finally the idea of an aesthetic mechanism arising from primitive reactions and strictly related with the genesis of language games. (shrink)

ABSTRACT This part of the series has a dual purpose. In the first place we will discuss two kinds of theories of proof. The first kind will be called a theory of linear proof. The second has been called a theory of suppositional proof. The term "natural deduction" has often and correctly been used to refer to the second kind of theory, but I shall not do so here because many of the theories so-called are not of the second kind--they (...) must be thought of either as disguised linear theories or theories of a third kind (see postscript below). The second purpose of this part is 25 to develop some of the main ideas needed in constructing a comprehensive theory of proof. The reason for choosing the linear and suppositional theories for this purpose is because the linear theory includes only rules of a very simple nature, and the suppositional theory can be seen as the result of making the linear theory more comprehensive. CORRECTION: At the time these articles were written the word ‘proof’ especially in the phrase ‘proof from hypotheses’ was widely used to refer to what were earlier and are now called deductions. I ask your forgiveness. I have forgiven Church and Henkin who misled me. (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 paper explains (in Part A) Wittegnstein’s understanding of the ‘grammar’ of our (or any) language, tracing its origins in the Tractatus’s concept of logical syntax, and then examining the senses in which Wittegnstein, in his later work, viewed grammar as being ‘arbitrary’. Then, armed with this understanding, it moves on (in Part B) to the task of examining how, within the framework of a Wittegnsteinian view of language, we should understand the inescapable ‘compellingness’ of logical necessity – (...) what Wittegnstein calls the “hardness of the logical must”. Whereas it is often thought that Wittegnstein’s views on the nature of the ‘grammar’ of our concpets leads him towards a vitiatingly conventionalist or anti‐realist understanding of necessity, in which its logical ‘superhardness’ becomes problematic, this paper will argue that there is actually no such tension in Wittegnstein’s thought. In fact, it will be argued, an understanding of the ways in which our conceptual grammar is arbitrary casts a great deal of light on how it is that our concepts can nevertheless support a logically superhard, and normatively commanding, notion of necessity. In support of this view, I distinguish Wittegnstein’s views on necessity from the ‘classical’ conventionalism of the Vienna Circle, and from the radical conventionalism of Michael Dummett, and defend Wittegnstein’s view from a powerful recent attack from Quassim Cassam. (shrink)

There are in fact two questions to be discussed. Is the importance of a philosophical thesis relative to language? Is the validity of a philosophical thesis relative to language? The answer to both questions is "yes." It can be shown that two well-known philosophical theses--the logical distinction between numerals and adjectives drawn by Frege, and the distinction between tasks and achievements drawn by Ryle--are true but trivial when stated in Chinese. This is the program for the first part of this (...) paper. In the second part, I will try to show that Mrs. Daitz's statement "a sentence states, but its parts cannot state" and Strawson's grammatical criterion for the subject-predicate distinction are both invalidated by Chinese grammar. (shrink)

Available from UMI in association with The British Library. Requires signed TDF. ;Even among Wittgenstein's admirers, his conception of philosophy as a therapy for conceptual confusion is generally considered to be the weakest part of his later work. It seems to consist of slogans, which are unsupported by argument and belied by his own 'theory construction'. It may even be self-refuting--a philosophical theory that denies the possibility of philosophical theory. ;Unless these objections can be met, current attempts to apply Wittgenstein's (...) philosophical methods remain problematic. I have therefore sought to provide a systematic account and defense of Wittgenstein's position. I have identified the common ground between Wittgenstein and the philosophical tradition, distinguished five basic features of his position and shown how they are linked. ;I go on to argue that his conception of philosophy is consistent and appears untenable only because of a failure to appreciate its connection with other aspects of his work, notably his understanding of logical necessity and of language. I have not examined Wittgenstein's conception of philosophy in isolation, for only by pitting it against its major rivals can its cogency be properly assessed. Throughout, I have examined the links between methodological issues and problems in philosophical logic, epistemology and metaphysics. ;The principle threat to Wittgenstein's conception of philosophy is Quine's assimilation of philosophy and science. His rejection of the distinctions between the a priori and the empirical and between analytic and synthetic propositions represent a profoundly important challenge to Wittgenstein's normative conception of language. However, I argue the challenge can be met by reducing Quine's behaviorist model of language to absurdity. (shrink)

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)

This paper explains Wittegnstein’s understanding of the ‘grammar’ of our language, tracing its origins in the Tractatus’s concept of logical syntax, and then examining the senses in which Wittegnstein, in his later work, viewed grammar as being ‘arbitrary’. Then, armed with this understanding, it moves on to the task of examining how, within the framework of a Wittegnsteinian view of language, we should understand the inescapable ‘compellingness’ of logical necessity – what Wittegnstein calls the “hardness of the logical (...) must”. Whereas it is often thought that Wittegnstein’s views on the nature of the ‘grammar’ of our concpets leads him towards a vitiatingly conventionalist or anti‐realist understanding of necessity, in which its logical ‘superhardness’ becomes problematic, this paper will argue that there is actually no such tension in Wittegnstein’s thought. In fact, it will be argued, an understanding of the ways in which our conceptual grammar is arbitrary casts a great deal of light on how it is that our concepts can nevertheless support a logically superhard, and normatively commanding, notion of necessity. In support of this view, I distinguish Wittegnstein’s views on necessity from the ‘classical’ conventionalism of the Vienna Circle, and from the radical conventionalism of Michael Dummett, and defend Wittegnstein’s view from a powerful recent attack from Quassim Cassam. (shrink)

Unification grammars are known to be Turing-equivalent; given a grammar G and a word w, it is undecidable whether w L(G). In order to ensure decidability, several constraints on grammars, commonly known as off-line parsability (OLP), were suggested, such that the recognition problem is decidable for grammars which satisfy OLP. An open question is whether it is decidable if a given grammar satisfies OLP. In this paper we investigate various definitions of OLP and discuss their interrelations, proving that (...) some of the OLP variants are indeed undecidable. We then present a novel, decidable OLP constraint which is more liberal than the existing decidable ones. (shrink)