History of Logic, Misc

David Miller claims that every valid deductive argument begs the question. Other philosophers and logicians have made similar claims. I show that the claim is false. Its appeal depends on the existence of logical terminology, particularly concerning what a proposition 'contains' or its 'logical content,' that is best understood as metaphoric and that, given its aptness to mislead, would be better eschewed. I show how the terminology appears to derive from early modern theories of the nature of mind, ideas and (...) reasoning that have since been rejected. (shrink)

This paper offers an expressivist account of logical form, arguing that in order to fully understand it one must examine what valid arguments make us do (or: what Achilles does and the Tortoise doesn’t, in Carroll’s famed fable). It introduces Charles Peirce’s distinction between symbols, indices and icons as three different kinds of signification whereby the sign picks out its object by learned convention, by unmediated indication, and by resemblance respectively. It is then argued that logical form is represented by (...) the third, iconic, kind of sign. It is noted that icons uniquely enjoy partial identity between sign and object, and argued that this holds the key to Carroll’s puzzle. Finally, from this examination of sign-types metaphysical morals are drawn: that the traditional foes metaphysical realism and conventionalism constitute a false dichotomy, and that reality contains intriguingly inference-binding structures. (shrink)

ABSTRACTThis paper seeks to clarify Husserl’s critical remarks about Kant’s view of logic by comparing their respective views of logic. In his Formal and Transcendental Logic Husserl c...

In 1907 Einstein had the insight that bodies in free fall do not “feel” their own weight. This has been formalized in what is called “the principle of equivalence.” The principle motivated a critical analysis of the Newtonian and special-relativistic concepts of inertia, and it was indispensable to Einstein’s development of his theory of gravitation. A great deal has been written about the principle. Nearly all of this work has focused on the content of the principle and whether it has (...) any content in Einsteinian gravitation, but more remains to be said about its methodological role in the development of the theory. I argue that the principle should be understood as a kind of foundational principle known as a criterion of identity. This work extends and substantiates a recent account of the notion of a criterion of identity by William Demopoulos. Demopoulos argues that the notion can be employed more widely than in the foundations of arithmetic and that we see this in the development of physical theories, in particular space–time theories. This new account forms the basis of a general framework for applying a number of mathematical theories and for distinguishing between applied mathematical theories that are and are not empirically constrained. (shrink)

We argue that, if taken seriously, Kripke's view that a language for science can dispense with a negation operator is to be rejected. Part of the argument is a proof that positive logic, i.e., classical propositional logic without negation, is not categorical.

We take an argument of Gödel's from his ground‐breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: "the sentence G says about itself that it is not provable, and G is indeed not provable; therefore, G is true".

This paper challenges a standard interpretation according to which Frege’s conception of logic (early and late) is at odds with the contemporary one, because on the latter’s view logic is formal, while on Frege’s view it is not, given that logic’s subject matter is reality’s most general features. I argue that Frege – in Begriffsschrift – retained the idea that logic is formal; Frege sees logic as providing the ‘logical cement’ that ties up together the contentful concepts of specific sciences, (...) not the most general truths. Finally, I discuss how Frege conceives of the application of Begriffsschrift, and of its status as a ‘lingua characteristica’. (shrink)

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two di erent ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of (...) inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. (shrink)

In this paper, I identify an assumption at play in anti-semantic interpretative approaches to Frege: the notion that translatability to Frege’s concept script functions as a criterion for deciding whether a thought is expressed in a sentence or utterance. I question the viability of this assumption by pointing to Frege’s accounts of the aim and character of his logical language and scientific discourse more generally, and by looking at his remarks on poetic forms of language, literature and fiction (Dichtung). Since (...) it seems clear that the sentences used in poetic and literary forms of language that Frege discusses, have Sinn and are possible to understand, in his view, I argue that the translatability criterion for thoughts is flawed. A discussion of Frege’s appeal to an approach of willingness to understand in a reader, and the relation between Frege’s use of elucidatory discourse and his conception of Dichtung is central to my exposition in this paper. (shrink)

The paper discusses Husserl’s notion of definiteness as presented in his Göttingen Mathematical Society Double Lecture of 1901 as a defense of two, in many cases incompatible, ideals, namely full characterizability of the domain, i.e., categoricity, and its syntactic completeness. These two ideals are manifest already in Husserl’s discussion of pure logic in the Prolegomena: The full characterizability is related to Husserl’s attempt to capture the interconnection of things, whereas syntactic completeness relates to the interconnection of truths. In the Prolegomena (...) Husserl argues that an ideally complete theory gives an independent norm for objectivity for logic and experiential sciences, hence the notion is central to his argument against psychologism. In the Double Lecture the former is captured by non-extendibility, that is, categoricity of the domain, from which, so Husserl assumes, syntactic completeness is thought to follow. In the so-called ‘mathematical manifolds’ the expressions of the theory are equations that are reducible to equations between elements of the theory. With such an equational reduction structure individual elements of the domain are given criteria of identity and hence they are fully determined. (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)

We claim that Kant's doctrine of the "transcendental ideal of pure reason" contains, in an anticipatory sense, a second-order theory of reality (as a second-order property) and of the highest being. Such a theory, as reconstructed in this paper, is a transformation of Kant's metatheoretical regulative and heuristic presuppositions of empirical theories into a hypothetical ontotheology. We show that this metaphysical theory, in distinction to Descartes' and Leibniz's ontotheology, in many aspects resembles Gödel's theoretical conception of the possibility of a (...) supreme being. The proposed second-order modal formalization of the Kantian doctrine of the "transcendental ideal" of the supreme being includes some specific features of Kantian general logic like "categorical", "hypothetical", and exclusive disjunctive propositions as basic forms. (shrink)

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by (...) his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School – a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical studies (...) and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines.The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School (among others, Ajdukiewicz, Lesniewski, Łukasiewicz,Tarski), accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name – Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period – the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of followers are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz (...) and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz (...) and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

The paper examines Husserl’s interactions with logicians in the 1930s in order to assess Husserl’s awareness of Gödel’s incompleteness theorems. While there is no mention about the results in Husserl’s known exchanges with Hilbert, Weyl, or Zermelo, the most likely source about them for Husserl is Felix Kaufmann (1895–1949). Husserl’s interactions with Kaufmann show that Husserl may have learned about the results from him, but not necessarily so. Ultimately Husserl’s reading marks on Friedrich Waismann’s Einführung in das mathematische Denken: die (...) Begriffsbildung der modernen Mathematik, 1936, show that he knew about them before his death in 1938. (shrink)

Essays on Aristotle's Sea-Battle, Lazy Argument, Argument Reaper, Diodorus' Master Argument.

Socrates’ expert-analogies is frequent both in Plato’s dialogues and in the Socratic writings of Xenophon, and is also ascribed to Socrates by Aristotle and Aeschines. Socrates makes an analogy from a non-controversial expert (or an expertise) like the cobbler or ship-captain, to another (often controversial) expert (or expertise) like the statesman. This paper defends an interpretation of the expert-analogy as valid deductions. It infers from one type of expert (such as the ship-captain) to another type of expert (such as the (...) statesman), and the attribute inferred (for instance ‘should not be selected by lot’) belongs to these types of experts because they are experts. The general logical form, which makes no mention of the expert and which is valid in virtue of its form, infers from one or more species to another species of the same genus, and the attribute inferred is presupposed to belong as such to the genus and only accidentally to the species. (shrink)

The “sign of consequence” is a notation for propositional logic that Peirce invented in 1886 and used at least until 1894. It substituted the “copula of inclusion” which he had been using since 1870.

The aim of Kant’s Sources in Translation is to retrieve the rich intellectual world that influenced Kant’s philosophical development. In its first stage, the series makes available the most important textbooks Kant used throughout his long teaching career. Many of these textbooks are in Latin or in German and remain inaccessible to Anglophone readers. Lacking this material, however, it is difficult to appreciate Kant’s originality and process of philosophical maturation, for readers are unable to understand what prompted Kant to introduce (...) a distinction, offer a qualification, attack a position, or develop a new thesis. This background is essential to understand the genesis of Kant’s thought. -/- This volume provides a translation of Georg Friedrich Meier’s Auszugaus der Vernunftlehre, the textbook that serves not only as the basis for Kant’s lectures on logic and related Reflexionen but is also crucial for studying the so-called Jäsche Logic of 1800, which is redacted from the marginal and interleave notes found in Kant’s personal copy of Meier’s book. Given the recent growth in scholarship on Kant’s logic, normative epistemology, and the psychology of belief, this volume makes a major contribution to contemporary debates in the field. (shrink)

This article assesses the role of the laws of the French logician and educational reformer Petrus Ramus in the writings of Francis Bacon. The laws of Ramus derive from Aristotle’s grounds for necessary propositions. Necessary propositions, according to Aristotle, Ramus, and Bacon, are required for the premises of scientific syllogisms. It is argued that in Bacon’s Advancement of Learning and De augmentis scientiarum the only role for these laws is in the transmission of knowledge that has already been acquired. However, (...) in the early Valerius Terminus Bacon also considered them to be relevant to the interpretation of nature. Interestingly, this, in turn, sheds light on the two precepts for the discovery of forms in Aphorism 4 of Book Two of the Novum organum that appear to derive from Valerius Terminus. All of this bears importantly on Bacon’s views on the problem of gaining epistemic access to the inner natures or forms of things. (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians.

In the popular imagination, the relevance of Turing's theoretical ideas to people producing actual machines was significant and appreciated by everybody involved in computing from the moment he published his 1936 paper ‘On Computable Numbers’. Careful historians are aware that this popular conception is deeply misleading. We know from previous work by Campbell-Kelly, Aspray, Akera, Olley, Priestley, Daylight, Mounier-Kuhn, Haigh, and others that several computing pioneers, including Aiken, Eckert, Mauchly, and Zuse, did not depend on Turing's 1936 universal-machine concept. Furthermore, (...) it is not clear whether any substance in von Neumann's celebrated 1945 ‘First Draft Report on the EDVAC’ is influenced in any identifiable way by Turing's work. This raises the questions: When does Turing enter the field? Why did the Association for Computing Machinery honor Turing by associating his name to ACM's most prestigious award, the Turing Award? Previous authors have.. (shrink)

Panorama of the concept of logic century-by-century since Aristotle.

This two-part paper reviews a scholarly debate on an alleged tension in Frege ’s philosophy of logic. In Section 1 of Part I, I discuss Frege ’s view that logic is concerned with establishing norms for correct thinking and is therefore a normative science. In Section 2, I explore a different understanding of the role of logic that Frege seems to advance: logic is constitutive of the very possibility of thought, because it sets forth necessary conditions for thought. Hence, the (...) tension the view according to which logic is normative for thought seems to be incompatible with the idea that abiding by the laws of logic forms a precondition for thought. In Section 1 of Part II, I survey a number of interpretations of Frege ’s conception of logic that deal with this question. I show that they are for the most part either normative readings or constitutive readings. Finally, in Section 2, I adjudicate the debate and aim at reconciling the normative and the constitutive strands in Frege ’s conception of logic. (shrink)

It is commonly claimed that the conclusion of a valid deductive argument is contained in its premises and says nothing new. In 'Deduction and Novelty,' in The Reasoner 5 (4), pp. 56-57, I refuted that claim. In The Reasoner, 8 (3), pp. 24-25, David McBride criticised my refutation. I show that McBride’s arguments are unsound.

This chapter situates Mill’s System of Logic (1843/1872) in the context of some of the meta-logical themes and disputes characteristic of the 19th century as well as Mill’s empiricism. Particularly, by placing the Logic in relation to Whately’s (1827) Elements of Logic and Mill’s response to the “great paradox” of the informativeness of syllogistic reasoning, the chapter explores the development of Mill’s views on the foundation, function, and the relation between ratiocination and induction. It provides a survey of the Mill-Whewell (...) debate on the nature of induction, Mill’s account of putatively a priori disciplines such as the science of number, and Frege’s criticisms of the Logic as psychologistic. (shrink)

I saggi che compongono questo libro intendono presentare in maniera organica e interdisciplinare, anche se da una prospettiva fondazionale logico-filosofica, il ruolo che il concetto di prova svolge in differenti ambiti del sapere. L’elemento innovativo e caratterizzante del volume è quello di stabilire e formulare quali sono le condizioni di adeguatezza materiale e formale per una corretta esplicazione del concetto di prova nelle sue differenti applicazioni. Si cercherà, inoltre, di cogliere cosa ha qualificato storicamente e qualifica tuttora il concetto di (...) prova tentando di individuarne, da un lato, le caratteristiche persistenti o invarianti e, dall’altro, le differenze specifiche che esso assume in molteplici settori della ricerca. Vengono così presentati sette saggi che spaziano sui seguenti temi: dalle procedure logiche di prova medievali (Alfredo Di Giorgio), si va alla nozione di prova da Frege a Wittgenstein (Carlo Penco), al problema delle verità delle proposizioni gödeliane (Sergio Galvan), alla fondazione delle leggi logiche (Ciro De Florio), ai fondamenti del linguaggio pragmatico (Antonio Negro e Davide Sergio), alla relazione tra conoscibilità e prova (Massimiliano Carrara e Daniele Chiffi), fino all’analisi del paradosso del buon samaritano (Andrea Favaro e Silvia Gaio). (shrink)

This paper compares the trajectories and defining moments in the evolution of dialectic in Islamic and European civilizations in the early modern intellectual history. The paper begins with a discussion of the concept of contradiction in the history of dialectic, and traces the shifts in the meaning of contradiction in the pre- and post-Hegel periods by focusing on the works of Immanuel Kant, G.W.F. Hegel and Karl Marx to illuminate how Islamic intellectuals understood dialectic at the very moment in which (...) a new form was being constructed in Europe. This paper aims to demonstrate that in Islamic civilization, dialectic was used as a tool for finding the truth whereas in Europe, especially in the post-Hegel period, dialectic was a tool for understanding the contradiction in nature and the world. (shrink)

Frege’s system of first-order logic is presented in a contemporary framework. The system described is distinguished by economy of expression and an unusual syntax.

With material on his early philosophical views, his contributions to set theory and his work on nominalism and higher-order quantification, this book offers a uniquely expansive critical commentary on one of analytical philosophy’s great ...

The goal of this paper is to provide an extensive account of Robert Leslie Ellisʼs largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellisʼs (...) reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge. (shrink)

History and Philosophy of Logic, Volume 33, Issue 3, Page 293-296, August 2012.

International Studies in the Philosophy of Science, Volume 26, Issue 3, Page 345-348, September 2012.

Here I revisit Bolzano's criticisms of Kant on the nature of logic. I argue that while Bolzano is correct in taking Kant to conceive of the traditional logic as a science of the activity of thinking rather than the content of thought, he is wrong to charge Kant with a failure to identify and examine this content itself within logic as such. This neglects Kant's own insistence that traditional logic does not exhaust logic as such, since it must be supplemented (...) by a transcendental logic that will in fact study nothing other than thought's content. Once this feature of Kant's views is brought to light, a much deeper accord emerges between the two thinkers than has hitherto been appreciated, on both the nature of the content that is at issue in logic and the sense of logic's generality and formality. (shrink)

Throughout his career, Bolzano presents his account of knowledge and science as an alternative to 'the Critical philosophy' of Kant and his followers. The aim of this essay is to evaluate the success of Bolzano's own account—and especially, its heavy emphasis on the objectivity of cognitive content—in enabling him to escape what he takes to be the chief shortcomings of the 'subjective idealist philosophy'. I argue that, because Bolzano's own position can be seen to be beset by problems that are (...) both recognizably similar to, and possibly even worse than, those that he takes to afflict Kant's account of the elements of our knowledge, Bolzano's attempt to fully overcome the alleged vices of Kant's idealism by 'extruding' semantic content from the mind must be judged to be less than satisfactory. (shrink)

A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness and (...) completeness -- Why is first order logic "First Order"? (shrink)

[ES] El principio de contexto que Frege establece en Fundamentos de la Aritmética no reaparece en sus escritos posteriores, hecho que podría sugerir que a consecuencia de la sustitución de la noción de contenido juzgable por las de sentido y significado Frege lo rechazara después de 1890. Dummett arguye en 1995 que el principio de contexto continúa formando una parte central de la filosofía de lenguaje de Frege, pero que las teorías establecidas a partir del tomo I de Leyes fundamentales (...) de la aritmética ya no son compatibles con la noción de la perspectiva internalista atribuida a Frege por van Heijenoort e Hintikka. El presente trabajo arguye que Frege no sólo jamás abandonó el principio de contexto, sino que también mantuvo toda su vida la fe en el lenguaje como medio universal con la consiguiente imposibilidad de establecer, vía una teoría semántica, externamente las referencias de nuestros términos. [EN] Frege´s context principle, as put forward in Foundations of Arithmetic, is not mentioned in his later writings which might suggest that as a consequence of substituting the notion of judgeable content by sense and meaning, Frege would have abandoned the context principle after 1890. Dummett argues in 1993 that the context principle continues to form a central part of Frege’s language philosophy; theories put forth from volume I of Fundamental Laws of Arithmetic, however, are no longer compatible with the notion of an internalist perspective attributed to Frege by van Heijenoort and Hintikka. I argue currently that Frege not only never abandoned the context principle but all his life maintained his faith en language as universal medium making and therefore held it impossible to establish reference for our terms externally by means of a theory of meaning. (shrink)

This article deals with a set-theoretic interpretation of the Talmudic rules of General and Specific, known as Klal and Prat (KP), Prat and Klal (PK), Klal and Prat and Klal (KPK) and Prat and Klal and Prat (PKP).

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

Strawson offers three accounts of singular predication: a grammatical, a category and a mediating account. I argue that the grammatical and mediating accounts are refuted by a host of counter-examples and that the latter is worse than useless. In later works Strawson defends only the category account. This account entails that singular terms cannot be predicates; it excludes non-denoting singular terms from being logical subjects, except by means of an ad hoc analogy; it depends upon a notion of identification that (...) is too vague; and it is unnecessarily complicated, relying on analogies where a more uniform explanation should be possible. But I show how the account can be corrected to avoid all these difficulties and to provide an accurate account of singular predication. (shrink)

This edited volume presents a comprehensive history of modern logic from the Middle Ages through the end of the twentieth century.

This dissertation is an analysis of the development of dialectic and argumentation theory in post-classical Islamic intellectual history. The central concerns of the thesis are; treatises on the theoretical understanding of the concept of dialectic and argumentation theory, and how, in practice, the concept of dialectic, as expressed in the Greek classical tradition, was received and used by five communities in the Islamic intellectual camp. It shows how dialectic as an argumentative discourse diffused into five communities (theologicians, poets, grammarians, philosophers (...) and jurists) and how these local dialectics that the individual communities developed fused into a single system to form a general argumentation theory (adab al-bahth) applicable to all fields. I evaluate a treatise by Shams al-Din Samarqandi (d.702/1302), the founder of this general theory, and the treatises that were written after him as a result of his work. I concentrate specifically on work by 'Ad}ud al-Din al-Iji (d.756/1355), Sayyid Sharif al-Jurjani (d.816/1413), Taşköprüzâde (d.968/1561), Saçaklızâde (d.1150/1737) and Gelenbevî (d.1205/1791) and analyze how each writer (from Samarqandi to Gelenbevî) altered the shape of argumentative discourse and how later intellectuals in the post-classical Islamic world responded to that discourse bequeathed by their predecessors. What is striking about the period that this dissertation investigates (from 1300-1800) is the persistence of what could be called the linguistic turn in argumentation theory. After a centuries-long run, the jadal-based dialectic of the classical period was displaced by a new argumentation theory, which was dominantly linguistic in character. This linguistic turn in argumentation dates from the final quarter of the fourteenth century in Iji's impressively prescient work on 'ilm al-wad'. This idea, which finally surfaced in the post-classical period, that argumentation is about definition and that, therefore, defining is the business of language—even perhaps, that language is the only available medium for understanding and being understood—affected the way that argumentation theory was processed throughout most of the period in question.The argumentative discourse that started with Ibn al-Rawandi in the third/ninth century left a permanent imprint on Islamic intellectual history, which was then full of concepts, terminology and objectives from this discourse up until the late nineteenth century. From this perspective, Islamic intellectual history can be read as the tension between two languages: the "language of dialectic" (jadal) and the "language of demonstration" (burhan), each of which refer not only to a significant feature of that history, but also to a feature that could dramatically alter the interpretation of that history. (shrink)

Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell's reduction of functions to relations over Frege's reduction of relations to functions. There is an interesting (...) system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an over-simplification: one can't assimilate predication to functional application.<br>. (shrink)