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)
Cantor's abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor's proposal based upon the set theoretic framework of Bourbaki - called BK - which is a First-order set theory extended with Hilbert's ε-operator. Moreover, it is argued that the BK system and the ε-operator provide a faithful reconstruction of Cantor's insights on cardinal numbers. (...) I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the ε-operator. Then, after presenting Cantor's abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo-von Neumann and Frege-Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege's objections to Cantor's proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the ε-operator in the BK definition of cardinal numbers. (shrink)
This chapter explores the metaphysical views about higher-order logic held by two individuals responsible for introducing it to philosophy: Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970). Frege understood a function at first as the remainder of the content of a proposition when one component was taken out or seen as replaceable by others, and later as a mapping between objects. His logic employed second-order quantifiers ranging over such functions, and he saw a deep division in nature between objects and functions. (...) Russell understood propositional functions as what is obtained when constituents of propositions are replaced by variables, but eventually denied that they were entities in their own right. Both encountered contradictions when supposing there to exist as many objects as functions, and both adopted views about the meaningfulness of higher-order discourse that were difficult to state from within their own strictures. (shrink)
This chapter clarifies that it was the works Giuseppe Peano and his school that first led Russell to embrace symbolic logic as a tool for understanding the foundations of mathematics, not those of Frege, who undertook a similar project starting earlier on. It also discusses Russell’s reaction to Peano’s logic and its influence on his own. However, the chapter also seeks to clarify how and in what ways Frege was influential on Russell’s views regarding such topics as classes, functions, meaning (...) and denotation, etc., and summarizes the correspondence between Frege and Russell and the light it sheds on the philosophical logic of both. (shrink)
To believe that logic has no history might at first seem peculiar today. But since the early 20th century, this position has been repeatedly conflated with logical monism of Kantian provenance. This logical monism asserts that only one logic is authoritative, thereby rendering all other research in the field marginal and negating the possibility of acknowledging a history of logic. In this paper, I will show how this and many related issues have developed, and that they are founded on only (...) one prominent statement by Kant. I will argue, however, that this statement takes on a very different meaning in a broader context of the history and philosophy of science, and that Kant and his supporters never advocated the logical monism that they are still said to hold today. (shrink)
Boethius identifies God both with esse ipsum and esse suum. This paper explains Boethius's general semantic use of "esse" and the application of this use to God. It questions the helpfulness of attributing to Boethius "existence" words and argues for a more robust role in Boethius’s thought for Hilary of Poitiers’s and Augustine’s exegeses of Exodus 3:14-15 than has been acknowledged in recent scholarship.
Frege and Peano started in 1896 a debate where they contrasted the respective conceptions on the theory and practice of mathematical definitions. Which was (if any) the influence of the Frege-Peano debate on the conceptions by the two authors on the theme of defining in mathematics and which was the role played by this debate in the broader context of their scientific interaction?
John Cook Wilson is increasingly recognised as an important predecessor of ordinary language philosophy. He emphasizes the authority of ordinary language in philosophical theorizing. At the same time, however, he circumscribes the limits of that authority and identifies cases in which it threatens to mislead us. My aim is to consider in detail one case where, according to Cook Wilson, ordinary language has misled philosophical theorizing. Judgement was one of the core notions of the logic, epistemology, and philosophy of mind (...) of Cook Wilson’s time. Cook Wilson rejects this notion, in the form developed by his contemporaries, in part because it is based on a problematic analogy between ordinary language and the thoughts expressed in that language. Cook Wilson’s discussion of judgement also highlights the extent to which Cook Wilson was critical of, but also responsive to, his contemporaries. In addition, variants of the language-thought analogy Cook Wilson opposes continue to feature in 21st century epistemology and philosophy of mind. Cook Wilson’s criticism of the analogy thus raises questions about recent work as well as the theories of his contemporaries. (shrink)
The purpose of this paper is to analyse and compare two concepts which tend to be treated as synonymous, and to show the difference between them: these are critical thinking and logical culture. Firstly, we try to show that these cannot be considered identical or strictly equivalent: i.e. that the concept of logical culture includes more than just critical thinking skills. Secondly, we try to show that Christian philosophers, when arguing about philosophical matters and teaching philosophy to students, should not (...) focus only on critical thinking skills, but rather also consider logical culture. This, as we argue, may help to improve debate both within and outside of Christian philosophy. (shrink)
The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)
Logic diagrams have been increasingly studied and applied for a few decades, not only in logic, but also in many other fields of science. The history of logic diagrams is an important subject, as many current systems and applications of logic diagrams are based on historical predecessors. While traditional histories of logic diagrams cite pioneers such as Leibniz, Euler, Venn, and Peirce, it is not widely known that Kant and the early Kantians in Germany and England played a crucial role (...) in popularising Euler(-type) diagrams. In this paper, the role of the Kantians in the late eighteenth and early nineteenth centuries will be analysed in more detail. It shows that diagrams (or intuition in general) were a highly contentious topic that depend on the philosophical attitude and went beyond logic to touch on issues of physics, metaphysics, linguistics and, above all, mathematics. (shrink)
In the Laws of Thought, Boole establishes a theory of secondary propositions based upon the notion of time. This temporal interpretation of secondary propositions has historically been met with wide disapproval and is usually dismissed in the modern literature as a philosophical non-starter. What was Boole thinking? This paper attempts to give an answer to this question. Specifically, it provides an account according to which Boole’s temporal interpretation follows from his psychologistic conception of logic, in addition to certain background assumptions (...) regarding the psychological necessity of the concept of time. Once Boole’s psychological premises are laid bare, it becomes clearer how he might have viewed the temporal interpretation to be an essential feature of his theory of secondary propositions. (shrink)
Kierkegaard, like Plato, though using different methods and conclusions, sought to ground knowledge in the ineffability of subjectivity. For Plato, knowledge comes subjectively (internally); for Kierkegaard, it comes by God's grace through faith. Socrates becomes the facilitator for the slave in the /Meno/, as does God for the man of faith. Again, Kierkegaard is also concerned with passion. "...the paradox is the passion of thought, and the thinker without the paradox is like the lover without passion; a mediocre fellow" (p. (...) 37). The paradox is necessitated by the metaphysical nature of the inquiry. Only knowledge through faith can approach the paradox since it is by definition beyond our knowledge. Passion must accompany the leap of faith, since knowledge acquisition for the man of faith is guided by God. (shrink)
In his Doppelvortrag, Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the history of completeness. Some commentators defended that the meaning of these notions should be understood as categoricity, while other scholars believed that it is closer to syntactic completeness. A detailed study of the early twentieth-century axiomatics and Husserl’s Doppelvortrag shows, however, that many concepts of completeness were conflated as equivalent. Although “absolute definiteness” was principally an attempt to characterize (...) non-extendible manifolds and axiom systems, an absolutely definite theory has a unique model and, thus, it is non-forkable and semantically complete. Non-forkability and decidability were formally delimited by Fraenkel and Carnap almost three decades later and, in fact, they mentioned Husserl as precursor of the latter. Therefore, this paper contributes to a reassessment of Husserl’s place in the history of logic. (shrink)
Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s \-operator. Moreover, it is argued that the BK system and the \-operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first (...) the axiomatic setting of BK and the definition of cardinal numbers by means of the \-operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the \-operator in the BK definition of cardinal numbers. (shrink)
Since Peirce defined the first operators for three-valued logic, it is usually assumed that he rejected the principle of bivalence. However, I argue that, because bivalence is a principle, the strategy used by Peirce to defend logical principles can be used to defend bivalence. Construing logic as the study of substitutions of equivalent representations, Peirce showed that some patterns of substitution get realized in the very act of questioning them. While I recognize that we can devise non-classical notations, I argue (...) that, when we make claims about those notations, we inevitably get saddled with bivalent commitments. I present several simple inferences to show this. The argument that results from those examples is ‘pragmatic’, because the inevitability of the principle is revealed in use (not mention); and it is ‘semiotic’, because this revelation happens in the use of signs. (shrink)
In contemporary historical studies Peano is usually linked to the logical tradition pioneered by Frege. In this paper I question this association. Specifically, I claim that Frege and Peano developed significantly different conceptions of a logical calculus. First, I clam that while Frege put the systematisation of the notion of inference at the forefront of his construction of an axiomatic logical system, Peano modelled his early logical systems as mathematical calculi and did not really attempt to justify reasoning. Second, I (...) argue that in later works on logic Peano advanced towards a deductive approach that was closer to Frege’s standpoint. (shrink)
After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain (...) Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. -/- In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme. (shrink)
In this article I develop an interpretation of the opening passages of Hegel's essay ‘With what must the beginning of science be made?’ I suggest firstly that Hegel is engaging there with a distinctive problem, the overcoming of which he understands to be necessary in order to guarantee the scientific character of the derivation of the fundamental categories of thought which he undertakes in the Science of Logic. I refer to this as ‘the problem of beginning’. I proceed to clarify (...) the nature of the problem, which I understand to be motivated by a concern to avoid arbitrariness, and then to detail the nature of Hegel's proposed solution, which turns on understanding how the concept of ‘pure being’, understood in a specific sense to be both mediated and immediate, avoids the concerns about arbitrariness which accompany attempts to begin merely with something mediated, or merely with something immediate. On this basis, I offer a number of criticisms of alternative approaches to the beginning of Hegel's Logic. (shrink)
Lewis Carroll published a system of logic in the symbolic tradition that developed in his time. Carroll’s readers may be puzzled by his system. On the one hand, it introduced innovations, such as his logic notation, his diagrams and his method of trees, that secure Carroll’s place on the path that shaped modern logic. On the other hand, Carroll maintained the existential import of universal affirmative Propositions, a feature that is rather characteristic of traditional logic. The object of this paper (...) is to untangle this dilemma by exploring Carroll’s guidelines in the design of his logic, and in particular his theory of existential import. It will be argued that Carroll’s view reflected his belief in the social utility of symbolic logic. (shrink)
In Begriffsschrift, Frege presented a formal system and used it to formulate logical definitions of arithmetical notions and to deduce some noteworthy theorems by means of logical axioms and inference rules. From a contemporary perspective, Begriffsschrift’s deductions are, in general, straightforward; it is assumed that all of them can be reproduced in a second-order formal system. Some deductions in this work present—according to this perspective—oddities that have led many scholars to consider it to be Frege’s inaccuracies which should be amended. (...) In this paper, we continue with the analysis of Begriffsschrift’s logic undertaken in an earlier work and argue that its deductive system must not be reconstructed as a second-order calculus. This leads us to argue that Begriffsschrift’s deductions do not need any correction but, on the contrary, can be explained in coherence with a global reading of this work and, in particular, with its fundamental distinction between function and argument. (shrink)
This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language.
I critically discuss Dale Jacquette’s Frege: A Philosophical Biography. First, I provide a short overview of Jacquette’s book. Second, I evaluate Jacquette’s interpretation of Frege’s three major works, Begriffsschrift, Grundlagen der Arithmetik and Grundgesetze der Arithmetik; and conclude that the author does not faithfully represent their content. Finally, I offer some technical and general remarks.
By the beginning of the 19th century Hegel's dialectic turn contradiction (conceived as unity of a concept with its determined negation) into distinguished inference. In the course of 20th century a family of systems known as "paraconsistent" formalized dialectical logic according to the contemporary paradigm of inference, oriented to truth-preserving, and not powered anymore solely by contradiction. In this way, nevertheless, Hegel's idea of logic as unfolding of concepts ordered by degree of "determination" reached at every step of the process, (...) was put aside. We attempt to recover the importance of that idea for the hegelian concept of "logical necessity", pointing out in addition that it may well play the role of an order relation for dialectical inference. (shrink)
In his Berlin Lectures of the 1820s, the German philosopher Arthur Schopenhauer (1788–1860) used spatial logic diagrams for philosophy of language. These logic diagrams were applied to many areas of semantics and pragmatics, such as theories of concept formation, concept development, translation theory, clarification of conceptual disputes, etc. In this paper we first introduce the basic principles of Schopenhauer’s philosophy of language and his diagrammatic method. Since Schopenhauer often gives little information about how the individual diagrams are to be understood, (...) we then make the attempt to reconstruct, specify and further develop one diagram type for the field of conceptual analysis. (shrink)
North-American philosophy was bolstered with the doctrines of the Jesuits. The penetration of the Coimbra Jesuits in the United States of America can be examined through the paradigmatic case of Charles Sanders Peirce. The extent to which Peirce was affected by the Coimbra Jesuits has not yet been researched. However, it is known that Peirce was acquainted with the Coimbra Jesuit Aristotelian Course.
This paper concerns Bertrand Russell’s changing views on negative judgement. ‘Negative judgement’ is considered in the context of three theories of judgement that Russell put forth at different times: a dual relation theory ; a multiple relation theory ; a psychological theory of judgement. Four issues are singled out for a more detailed discussion: quality dualism versus quality monism, that is, the question whether judgement comes in two kinds, acceptance and rejection, or whether there is only one judgement-quality ; the (...) structure of negative judging; the problem of truth-making for negative facts; the different roles of ‘fact’ in Russell’s theories of truth. What emerges from the discussion is a rough chronology of Russell’s views on negative judgement during the period from 1903 to 1948. (shrink)
The question of naturalness in logic is widely discussed in today’s research literature. On the one hand, naturalness in the systems of natural deduction is intensively discussed on the basis of Aristotelian syllogistics. On the other hand, research on “natural logic” is concerned with the implicitly existing logical laws of natural language, and is therefore also interested in the naturalness of syllogistics. In both research areas, the question arises what naturalness exactly means, in logic as well as in language. We (...) show, however, that this question is not entirely new: In his Berlin Lectures of the 1820s, Arthur Schopenhauer already discussed in depths what is natural and unnatural in logic. In particular, he anticipates two criteria for the naturalness of deduction that meet current trends in research: (1) Naturalness is what corresponds to the actual practice of argumentation in everyday language or scientific proof; (2) Naturalness of deduction is particularly evident in the form of Euler-type diagrams. (shrink)
Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice of references (...) and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas. (shrink)
In this paper, we would show how the logical object “square of opposition”, viewed as semiotic object, has been historically transformed since its appearance in Aristotle’s texts until the works of Vasiliev. These transformations were accompanied each time with a new understanding and interpretation of Aristotle’s original text and, in the last case, with a transformation of its geometric configuration. The initial textual codification of the theory of opposition in Aristotle’s works is transformed into a diagrammatic one, based on a (...) new “reading” of the Aristotelian text by the medieval scholars that altered the semantics of the O form. Further, based on the medieval “Neo-Aristotelian” reading, the logicians of the nineteenth century suggest new diagrammatic representations, based on new interpretations of quantification of judgements within the algebraic and the functional logical traditions. In all these interpretations, the original square configuration remains invariant. However, Nikolai A. Vasiliev marks a turning point in history. He explicitly attacks the established logical tradition and suggests a new alternation of semantics of the O form, based on Aristotelian concepts that were neglected in the Aristotelian tradition of logic, notably the concept of indefinite judgement. This leads to a configurational transformation of the “square” of opposition into a “triangle”, where the points standing for the O and I forms are contracted into one point, the M form that now stands for particular judgement with altered semantics. The new transformation goes beyond the Aristotelian logic paradigm to a new “Non-Aristotelian” logic, i.e. to paraconsistent logic, although the argumentation used in support of it is phrased in Aristotelian style and the context of discovery is foundational. It establishes a bifurcation point in the development of logic. No unique logic is recognized, but different logics concerning different domains. One branch of logic remains to be the “Neo-Aristotelian” one, while the new logic is “Non-Aristotelian”. (shrink)
During 1881, the British logicians John Venn and Hugh MacColl engaged in a brief dispute in Nature about ‘symbolical logic’. The letters to the editor shed interesting light on the early reception of MacColl’s contributions to logic and his position in the logical community of the Victorian era. Drawing on the correspondence between Venn and William Stanley Jevons, this paper analyzes the background and context of these letters, adding to the recent interest in the social dimensions of the development of (...) British algebraic logic in the confusing intermediary period between the 1870s and 1890s. (shrink)
The translation of both ‘bedeuten’ and ‘Bedeutung’ in Frege's works remains sufficiently problematic that some contemporary authors prefer to leave these words untranslated. Here a case is made for returning to Russell's initial choice of ‘to indicate’ and ‘indication’ as better alternatives than the more usual ‘meaning’, ‘reference’, or ‘denotation’. It is argued that this choice has the philosophical payoff that Frege's controversial doctrines concerning the semantic values of sentences and predicative expressions are rendered far more comprehensible by it, and (...) that this translational strategy fulfills the desiderata of offering a translation which is acceptable both before and after Frege introduced the distinction between sense and reference or, as this paper would have it, between the sense of an expression and what it indicates. (shrink)
Analytic philosophy has been perhaps the most successful philosophical movement of the twentieth century. While there is no one doctrine that defines it, one of the most salient features of analytic philosophy is its reliance on contemporary logic, the logic that had its origin in the works of George Boole and Gottlob Frege and others in the mid‐to‐late nineteenth century. Boolean algebra, the heart of Boole's contributions to logic, has also come to represent a cornerstone of modern computing. Frege had (...) broad philosophical interests, and his writings on the nature of logical form, meaning and truth remain the subject of intense theoretical discussion, especially in the analytic tradition. Frege's works, and the powerful new logical calculi developed at the end of the nineteenth century, influenced many of its most seminal figures, such as Bertrand Russell, Ludwig Wittgenstein and Rudolf Carnap. (shrink)
This volume collects nine essays that investigate the work of Gottlob Frege. The contributors address Frege’s work in relation to literature and fiction (Dichtung), the humanities (Geisteswissenschaften), and science (Wissenschaft). Overall, the essays consider internal connections between different aspects of Frege’s work while acknowledging the importance of its philosophical context. There are also further common strands between the papers, such as the relation between Frege’s and Wittgenstein’s approaches to philosophical investigations, the relation between Frege and Kant, and the place of (...) Frege’s work in the philosophical landscape more generally. The volume is therefore of direct relevance to several current debates in philosophy in general, in addition to Frege and Wittgenstein research in particular. Even though Frege’s great significance for contemporary philosophy is not disputed, the question of how we are to understand the character and aims of his project is debated. The debate has a starting point in Frege’s specific conception of logic. The volume elucidates this conception as well as the relation between natural language and the Begriffsschrift. It will help philosophers, researchers, and students better understand the nuances of this great thinker. By extension, it will also help readers seeking to understand Wittgenstein’s approach to philosophical difficulties and his struggle to find an apt form of presentation for his philosophical investigations. (shrink)
This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand (...) Tarski’s and Gödel’s work, explaining why the problems they discussed are still unsolved. Finally, the book reports on some of the most influential positions in contemporary philosophy of mathematics, i.e., Maddy’s mathematical naturalism and Shapiro’s mathematical structuralism. Last but not least, the book introduces Biancani’s Aristotelian philosophy of mathematics as this is considered important to understand current philosophical issue in the applications of mathematics. One of the main purposes of the book is to stimulate readers to reconsider the Aristotelian position, which disappeared almost completely from the scene in logic and mathematics in the early twentieth century. (shrink)
George Boole emerged from the British tradition of the “New Analytic”, known for the view that the laws of logic are laws of thought. Logicians in the New Analytic tradition were influenced by the work of Immanuel Kant, and by the German logicians Wilhelm Traugott Krug and Wilhelm Esser, among others. In his 1854 work An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Boole argues that the laws of thought acquire (...) normative force when constrained to mathematical reasoning. Boole’s motivation is, first, to address issues in the foundations of mathematics, including the relationship between arithmetic and algebra, and the study and application of differential equations (Durand-Richard, van Evra, Panteki). Second, Boole intended to derive the laws of logic from the laws of the operation of the human mind, and to show that these laws were valid of algebra and of logic both, when applied to a restricted domain. Boole’s thorough and flexible work in these areas influenced the development of model theory (see Hodges, forthcoming), and has much in common with contemporary inferentialist approaches to logic (found in, e.g., Peregrin and Resnik). (shrink)
This text aims to contrast the metaphysical beginnings of the philosophies of Hegel and Kierkegaard. For this task, the notion of Being Pure of the Hegel of Logic will be used in relation with the concept of Irony that Kierkegaard expresses in his Concept of irony. The need for this "contrast of beginnings" seeks to clarify, from a “metaphysical awakening”, the evident theoretical courtship that has so far distanced by the dominant historiographical traditions of continental philosophy.
We discuss the many aspects and qualities of the number one: the different ways it can be represented, the different things it may represent. We discuss the ordinal and cardinal natures of the one, its algebraic behaviour as a neutral element and finally its role as a truth-value in logic.
Working within the broad lines of general consensus that mark out the core features of John Stuart Mill’s (1806–1873) logic, as set forth in his A System of Logic (1843–1872), this chapter provides an introduction to Mill’s logical theory by reviewing his position on the relationship between induction and deduction, and the role of general premises and principles in reasoning. Locating induction, understood as a kind of analogical reasoning from particulars to particulars, as the basic form of inference that is (...) both free-standing and the sole load-bearing structure in Mill’s logic, the foundations of Mill’s logical system are briefly inspected. Several naturalistic features are identified, including its subject matter, human reasoning, its empiricism, which requires that only particular, experiential claims can function as basic reasons, and its ultimate foundations in ‘spontaneous’ inference. The chapter concludes by comparing Mill’s naturalized logic to Russell’s (1907) regressive method for identifying the premises of mathematics. (shrink)
It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is (...) concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete. (shrink)
Not all truths are on a par. The realm of truths is structured: some propositions are only true because others are. The relation that endows the realm of truths with this structure is often called grounding. Grounding has achieved much attention in 21st century metaphysics, but the topic is arguably as old as philosophy itself. -/- This becomes apparent when investigating the works of the 19th-century philosopher Bernard Bolzano, who developed what is perhaps the first comprehensive theory of grounding, drawing (...) on a rich tradition that goes back to Aristotle’s Posterior Analytics. Roski’s book provides, for the first time, a comprehensive study of Bolzano’s theory of grounding in its entirety, paying more attention than previous studies to the interaction between grounding and the consequence-relation of deducibility. -/- . (shrink)