History of Logic, Misc

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)

La lógica matemática es matemática en cuanto que usa herramientas matemáticas. En este sentido, la lógica matemática es matemática en el mismo sentido que lo es, digamos, la mecánica newtoniana. En ambos casos, el método es matemático, pero las ciencias mismas no lo son, pues su objeto de estudio pertenece a una realidad objetiva e independiente. En particular, las herramientas matemáticas que usa la lógica simbólica contemporánea —tanto en su simbolismo como en su cálculo— se crearon originalmente para el desarrollo (...) algebraico de la geometría, y luego fueron adaptadas al resto de las matemáticas y la lógica. A estas herramientas se les llama formales, pues permiten el cálculo con formas generales. (shrink)

The influence of Cajetan’s De Nominum Analogia is due largely to its first three chapters, which introduce Cajetan’s three modes of analogy: analogy of inequality, analogy of attribution, and analogy of proportionality. Interpreters typically ignore the final eight chapters, which describe further features of analogy of proportionality. This article explains this neglect as a symptom of a failure to appreciate Cajetan’s particular semantic concerns, taken independently from the question of systematizing the thought of Aquinas. After an exegesis of the neglected (...) chapters, which describe the semantics of analogy through the three levels of cognition, the article concludes with observations about the relationship between Cajetan and Aquinas and the philosophical and historical signifi cance of Cajetan’s approach to the semantics of analogy. (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)

This interesting and imaginative monograph is based on the author’s PhD dissertation supervised by Saul Kripke. It is dedicated to Timothy Smiley, whose interpretation of PRIOR ANALYTICS informs its approach. As suggested by its title, this short work demonstrates conclusively that Aristotle’s syllogistic is a suitable vehicle for fruitful discussion of contemporary issues in logical theory. Aristotle’s syllogistic is represented by Corcoran’s 1972 reconstruction. The review studies Lear’s treatment of Aristotle’s logic, his appreciation of the Corcoran-Smiley paradigm, and his understanding (...) of modern logical theory. In the process Corcoran and Scanlan present new, previously unpublished results. Corcoran regards this review as an important contribution to contemporary study of PRIOR ANALYTICS: both the book and the review deserve to be better known. (shrink)

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)

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)

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.

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)

This article is a translation of the paper in Polish (Alfred Tarski - człowiek, który zdefiniował prawdę) published in Ruch Filozoficzny 4 (4) (2007). It is a personal Alfred Tarski memories based on my stay in Berkeley and visit the Alfred Tarski house for the invitation of Janusz Tarski.

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)

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".

Regarding the linguistic turn in History of Ideas, there are logical topics to be considered in order to increase the accuracy of reconstruccions oriented to describe and explain conceptual formations of the past. This paper propose an overview of these topics and a methodological approach to deal with its difficulties.

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)

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)

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 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)

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

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)

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)

Chapin reviewed this 1972 ZEITSCHRIFT paper that proves the completeness theorem for the logic of variable-binding-term operators created by Corcoran and his student John Herring in the 1971 LOGIQUE ET ANALYSE paper in which the theorem was conjectured. This leveraging proof extends completeness of ordinary first-order logic to the extension with vbtos. Newton da Costa independently proved the same theorem about the same time using a Henkin-type proof. This 1972 paper builds on the 1971 “Notes on a Semantic Analysis of (...) Variable Binding Term Operators” (Co-author John Herring), Logique et Analyse 55, 646–57. MR0307874 (46 #6989). A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. Kalish-Montague 1964 proposed using vbtos to formalize definite descriptions “the x: x+x=2”, set abstracts {x: F}, minimization in recursive function theory “the least x: x+x>2”, etc. However, they gave no semantics for vbtos. Hatcher 1968 gave a semantics but one that has flaws described in the 1971 paper and admitted by Hatcher. In 1971 we give a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture, proved in this paper with Hatcher’s help, was proved independently about the same time by Newton da Costa. (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)

Os grafos existenciais (GE) são uma notação lógica de caráter topológico e estão entre as mais originais invenções de C.S. Peirce (1839-1914). Trata-se de um sistema gráfico de diagramas lógicos por meio do qual, segundo Peirce, “qualquer desenvolvimento do pensamento pode ser representado com precisão” (CP 4.530). Eles se dividem em três subsistemas – alfa, beta e gama –, aproximadamente equivalentes ao cálculo proposicional clássico, ao cálculo de predicados clássico de primeira ordem, e a um tipo de lógica modal. Nosso (...) propósito é apresentar o sistema beta. O que apresentaremos se restringe ao cálculo de predicados monádicos, com ênfase na silogística categórica de Aristóteles. (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.

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)

Universal logic is to logic what universal algebra is to algebra. It is not a specific system of logic that would apply to everything but a general theory of all existing and possible logics. This new field has been slowly emerging through the new directions of research in logic of the past decades and the name was coined 15 years ago. In the Spring of 2005 was organized in Montreux, Switzerland, the First World Congress on Universal Logic. This exciting event (...) gathered more than 200 people from about 40 different countries. Many famous logicians were present there, such as Dov Gabbay, Krister Segerberg, Saul Kripke, Michael Dunn, Jouko Väännänen. Most of the papers in this book arose from talks - and discussions that follow - presented during this event. They are concerned with philosophical and historical approaches discussing the basic concepts of logic and reflecting the meaning of a universal approach to logic, the development and elaboration of general tools in order to construct frameworks useful for studying classes of logics, applications of logics to concrete problems through the universal perspective. (shrink)

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

[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)

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately (...) for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’. (shrink)

DEFINING OUR TERMS A “paradox" is an argumentation that appears to deduce a conclusion believed to be false from premises believed to be true. An “inconsistency proof for a theory" is an argumentation that actually deduces a negation of a theorem of the theory from premises that are all theorems of the theory. An “indirect proof of the negation of a hypothesis" is an argumentation that actually deduces a conclusion known to be false from the hypothesis alone or, more commonly, (...) from the hypothesis augmented by a set of premises known to be true. A “direct proof of a hypothesis" is an argumentation that actually deduces the hypothesis itself from premises known to be true. Since `appears', `believes' and `knows' all make elliptical reference to a participant, it is clear that `paradox', `indirect proof' and `direct proof' are all participant-relative. PARTICIPANT RELATIVITY In normal mathematical writing the participant is presumed to be “the community of mathematicians" or some more or less well-defined subcommunity and, therefore, omission of explicit reference to the participant is often warranted. However, in historical, critical, or philosophical writing focused on emerging branches of mathematics such omission often invites confusion. One and the same argumentation has been a paradox for one mathematician, an inconsistency proof for another, and an indirect proof to a third. One and the same argumentation-text can appear to one mathematician to express an indirect proof while appearing to another mathematician to express a direct proof. WHAT IS A PARADOX’S SOLUTION? Of the above four sorts of argumentation only the paradox invites “solution" or “resolution", and ordinarily this is to be accomplished either by discovering a logical fallacy in the “reasoning" of the argumentation or by discovering that the conclusion is not really false or by discovering that one of the premises is not really true. Resolution of a paradox by a participant amounts to reclassifying a formerly paradoxical argumentation either as a “fallacy", as a direct proof of its conclusion, as an indirect proof of the negation of one of its premises, as an inconsistency proof, or as something else depending on the participant's state of knowledge or belief. This illustrates why an argumentation which is a paradox to a given mathematician at a given time may well not be a paradox to the same mathematician at a later time. -/- The present article considers several set-theoretic argumentations that appeared in the period 1903-1908. The year 1903 saw the publication of B. Russell's Principles of mathematics, [Cambridge Univ. Press, Cambridge, 1903; Jbuch 34, 62]. The year 1908 saw the publication of Russell's article on type theory as well as Ernst Zermelo's two watershed articles on the axiom of choice and the foundations of set theory. The argumentations discussed concern “the largest cardinal", “the largest ordinal", the well-ordering principle, “the well-ordering of the continuum", denumerability of ordinals and denumerability of reals. The article shows that these argumentations were variously classified by various mathematicians and that the surrounding atmosphere was one of confusion and misunderstanding, partly as a result of failure to make or to heed distinctions similar to those made above. The article implies that historians have made the situation worse by not observing or not analysing the nature of the confusion. -/- RECOMMENDATION This well-written and well-documented article exemplifies the fact that clarification of history can be achieved through articulation of distinctions that had not been articulated (or were not being heeded) at the time. The article presupposes extensive knowledge of the history of mathematics, of mathematics itself (especially set theory) and of philosophy. It is therefore not to be recommended for casual reading. AFTERWORD: This review was written at the same time Corcoran was writing his signature “Argumentations and logic”[249] that covers much of the same ground in much more detail. https://www.academia.edu/14089432/Argumentations_and_Logic . (shrink)

An evaluation of the historiography of logic is attempted.

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)

This self-contained one page paper produces one valid two-premise premise-conclusion argument that is a counterexample to the entire three traditional rules of distribution. These three rules were previously thought to be generally applicable criteria for invalidity of premise-conclusion arguments. No longer can a three-term argument be dismissed as invalid simply on the ground that its middle is undistributed, for example. The following question seems never to have been raised: how does having an undistributed middle show that an argument's conclusion does (...) not follow from its premises? This result does nothing to vitiate the theories of distribution developed over the period beginning in medieval times. What it does vitiate is many if not all attempts to use distribution in tests of invalidity outside of the standard two-premise categorical arguments—where they were verified on a case-by-case basis without further theoretical grounding. In addition it shows that there was no theoretical basis for many if not all claims of fundamental status of rules of distribution. These results are further support for approaching historical texts using mathematical archeology. (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)

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)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

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)