Logic, Mathematics and Philosophy

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today.

Gottlob Frege (1848–1925) was unquestionably one of the most important philosophers of all time. He trained as a mathematician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of language and philosophical logic. Frege is generally seen (along with Russell and Wittgenstein) as one of the fathers of the analytic method, which dominated philosophy in English-speaking countries for most of the twentieth century. His work is studied today not just for its historical importance but also because many of his ideas are still seen as relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research.

This chapter analyzes how the pursuit of truth in logic had begun with Gottlob Frege and was continued by Bertrand Russell and the rest, so that axiomatic logic had become the norm by the 1920s. The axioms were supposed to express the most basic logical truths, and just two rules of proof expressed the passage from instances of axioms to the theorems of logic. The latter were, supposedly, the less basic truths, but sometimes theorems were simpler than axioms. There was another aspect to axiomatic logic, namely a fundamental relativity in the choice of the basic notions. Russell's axiomatization of propositional logic, from the Principia Mathematica, uses disjunction and negation as the primitive connectives and became a standard.

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of "revenge Liar" arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's "useful four-valued logic", one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a "bi-and-a-half" lattice which determines seven-valued logics different from Priest's Logic of Paradox.

It is a consequence of Quine’s confirmation holism that the logical laws are in principle revisable. Some have worried this is at odds with another dictum in Quine, viz., that any translation which construes speakers as systematically illogical is ipso facto inadequate. In this paper, I try to formulate exactly what the problem is here, and offer a solution to it by (1) disambiguating the term ‘logic,’ and (2) appealing to a Quinean understanding of ‘necessity.’ The result is that the different theses in Quine’s philosophy of logic are to be situated within different contexts of inquiry.

Modernism and the Specter of Psychologism Martin Jay (bio) No genealogist of the complex and heterogeneous cultural field we have come to call aesthetic modernism can fail to acknowledge its multiple intersections with that other richly articulated field known as modern psychology. In both cases, an unprecedented preoccupation with the interior landscape of the subject, a no longer self-confident self functioning with increased difficulty in the larger world outside its threatened and vulnerable boundaries, led to voyages of scientific and artistic discovery whose endpoints have yet been reached. Whether it be the empiricist tradition of Ernst Mach, Franz Brentano and William James recently foregrounded by Judith Ryan in The Vanishing Subject, 1 Jean-Martin Charcot’s investigations of hysteria inspirational for such modernist movements as Surrealism, or Freudian psychoanalysis, whose importance for modernism is evident in such accounts as Carl Schorske’s Fin-de-Siècle Vienna, 2 the story of aesthetic innovation has seemed impossible to narrate without reference to the no less radical developments in psychology happening at virtually the same time. What Ryan calls the “complex intertextuality” 3 between modernism and psychology has perhaps reached its zenith in Louis Sass’s recent Madness and Modernism, which boldly seeks to understand such familiar aspects of the modernist temper as perspectivism, dehumanization, alienation, ironic detachment, hyperreflexivity, and spatialized form precisely by finding affinities with the workings of the schizophrenic mind, which was itself first labeled in the 1890s and quickly became, he tells us, “the quintessential form of madness in our time.” 4 [End Page 93] Cultural fields, however, not only intersect and parallel each other, producing reinforcements, homologies, and elective affinities, but also sometimes interact through repulsions and negations, achieving their fragile definitions by means of abjecting the other. In the case of modernism and psychology, I want to argue, such a process of negative exclusion did, in fact, occur, a process which overlapped and sometimes interfered with the more positive interaction that has been explored by other scholars. In particular, I want to show that what the philosophers of this period called “psychologism” and which many sought mightily to banish from their search for truth also emerged as a source of anxiety at certain key moments in the genesis of aesthetic modernism as well. 5 My evidence will come from both literary and visual instances of this anxiety. 6 Although I do not want to be understood as saying modernism was essentially a repudiation of psychologism, let alone unrelated to developments in the psychological sciences per se, no understanding of its emergence can afford to ignore the haunting presence of this stigmatized “-ism” as a specter that refused to be effortlessly exorcised. The philosophical critique of psychologism can be traced at least as far back as Kant’s claim that: in logic we do not want to know how understanding is and thinks and how it hitherto has proceeded in thinking, but how it ought to proceed in thinking. Logic must teach us the correct use of the understanding, i.e. that in which it is in agreement with itself. 7 His transcendental deduction was thus explicitly distinguished between the questio facti of the “physiological derivation” of a priori ideas and the questio juris of their validity. The most elaborate and influential development of the critique, however, appeared after a period in which psychologism gained ground among philosophers and the upstart discipline of experimental psychology had emerged to challenge philosophy as the royal road to unlocking the mysteries of the mind. 8 Although there were rumblings throughout the nineteenth century, only at its end did psychologism emerge as a primary target of a philosophical counterattack. In 1884, Gottlob Frege published The Foundations of Arithmetic, which vigorously criticized attempts by philosophers such as Friedrich Beneke, Jakob Fries, John Stuart Mill, and Franz Brentano to reduce the mind, in particular its logical function, to the psyche. 9 More radical psychologizers like Nietzsche were beneath Frege’s consideration. 10 Husserl, although himself initially indebted to Brentano, was putatively won over to Frege’s position by a devastating review the latter wrote of Husserl’s early work The Foundations of Arithmetic in 1894. 11 By his Logical Investigations of 1900...

During his long career, Rudolf Carnap held various different views about the concept of truth and its philosophical significance. As a good logical empiricist, he insisted on the distinction between logical and factual statements, and employed his technical powers to give rigorous characterizations of the notions of logical, analytic, and factual truth. The development of Carnap’s views reflected his ability to quickly absorb new influences and his broad interests ranging from logic to epistemology, philosophy of language, and philosophy of science (Sections 1 and 2). Carnap was the co-founder of logical semantics with Alfred Tarski (Section 3), and therefore it is especially interesting to see how Carnap’s work was related to Tarski’s early definition of truth and to the later Tarskian model theory (Section 4). In Section 5, we discuss some difficulties in Carnap’s liberally empiricist treatment of scientific theories. The final section 6 makes some remarks on Carnap’s contemporary relevance.

Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of papers. The first aim of this book, then, is to consider the philosophical uses of model theory, focusing on the central topics of reference, realism, and doxology. Its second aim is to address important questions in the philosophy of model theory, such as: sameness of theories and structure, the boundaries of logic, and the classification of mathematical structures. Philosophy and Model Theory will be accessible to anyone who has completed an introductory logic course. It does not assume that readers have encountered model theory before, but starts right at the beginning, discussing philosophical issues that arise even with conceptually basic model theory. Moreover, the book is largely self-contained: model-theoretic notions are defined as and when they are needed for the philosophical discussion, and many of the most philosophically significant results are given accessible proofs.

This study investigates how collaboration between design research and technological philosophy can benefit both fields. More emphasis must be placed on the quality and quantity of philosophical education and its application within the framework of "workmanship philosophy," given the available tools for artistic expression. In expressions instruction, several tactics and approaches should be utilized, and students should be capable of analyzing and comprehending works of art using a variety of approaches. Is it required to utilize philosophical terminology in the illustration of handicraft philosophy? Is it feasible, on the other hand, that philosophical terminology provides a vulnerability for these problems to be examined and multiple answers to be found? Nonetheless, we must be interested in Plato's discourse and the significance of Aristotle's poetics to create even more appealing statements. Instead of emphasizing a complete education in philosophy, art students must learn and comprehend the properties of knowledge that form the foundation of philosophy. This assists in determining the placement and significance of philosophy in speech. Based on three contextual studies employing technology philosophy hypotheses and methodologies in a planning setting, we will explain how these businesses profited from a more reflective viewpoint. Aside from contemplating and debating, one may argue that the Philosophy of Technology is still functioning. As well as technology philosophy with the mind, there is also technology philosophy with the hands. About the prior experimental shift in Philosophy of Technology, we refer to this partnership as the "Useful Approach in Philosophy of Technology." Based on three contextual studies using technology philosophy hypotheses and methods in a planning environment, we will explain how these companies benefited from a more reflective perspective. Other than thinking and discussing, one may claim that the Philosophy of Technology is still active. Technology philosophy with the head, as well as technology philosophy with the hands As a result, we refer to this collaboration as the 'Useful Approach in Philosophy of Technology,' in reference to the preceding experimental turn in Philosophy of Technology.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

This introduction discusses the development of Quine’s system over time and the centrality of structure to it. It explains the contributions made in this volume to our understanding of Quine’s thought on structure and ontology, especially with respect to philosophical logic, philosophy of language, history of philosophy, mathematics, philosophy of time, and set theory. Chapters by Michael Resnik, Frederique Janssen-Lauret and Fraser MacBride, John Collins, Jaroslav Peregrin, and Paul Gregory explore whether Quine’s structuralism is epistemological, language-based, or ontological. Greg Frost-Arnold, Robert Sinclair, and Gary Kemp and Andrew Lugg explore Quine’s views on structure from a historical point of view. Nathan Salmón, Gila Sher, Marianna Antonutti Marfori, and Natalja Deng consider Quine’s views on the structure of logic, language, and theories in relation to contemporary philosophy, specifically ontology, the philosophy of logic and mathematics, philosophy of set theory, and philosophy of time.

<i>The Parmenidean Ascent</i>

An attempt is made to show that Wittgenstein's later philosophy of logic is not the kind of conventionalism which is often ascribed to him. On the contrary, Wittgenstein gives expression to a “mixed” theory which is not only interesting but tends to resolve the perplexities usually associated with the question of the a priori character of logical truth. I try to show that Wittgenstein is better understood not as denying that there are such things as “logical rules” nor as denying that the results of applying such rules are “logically necessary,” but as trying to understand what it is to appeal to a logical rule and what it means to say that the results of applying such a rule are “necessary.” He is not so much overthrowing standard accounts of logical necessity as discovering the limits of the concept.

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of the arithmetization is still regarded as the standard one. It is highly compressed, however, and very difficult to read. My goals in this paper are expository: to present the basics of Kleene's arithmetization in a less compressed, more easily readable form, in a setting that highlights its relevance to issues in the philosophy of logic, especially to Quine's substitutional definition of logical truth, and to formulate the Hilbert–Bernays Theorem in a way that incorporates Kreisel's, Mostowski's, and Putnam's refinements of Kleene's result.

Preface. Acknowlegements. Chapter 1 Introduction: Fregea s Life and Work, Biography, The Origin and Development of Fregea s Philosophy, Fregea s Contributions to Philosophy. Chapter 2: Logic,The Purpose of Conceptional Notation, Logic Before Frege, Fregean Logic. Chapter 3: Number, Aims of the Foundations of Arithmetic, Rebuttal of Earlier Attempts, The Development of Fregea s Own Position. Chapter 4: Philosophical Logic, On Function and Concept, On Concept and Object. Chapter 5: Theory of Meaning, The Distinction between Sense and Reference, Indirect Reference, The Objectivity of Sense, Challenges to Sense. Appendix. Bibliography. Index