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)

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom (...) Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

A framework for the mind-matter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive non-Boolean description of a world without an a priori mind-matter distinction. Such a description in terms of a locally Boolean but globally non-Boolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are (...) no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely non-Boolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain.In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given a priori we have to expect correlations between these two domains. In physics it corresponds to Newton 's separation of universal laws of nature and contingent initial conditions Both descriptions have a non-Boolean structure and can be encompassed into a single non-Boolean description. Tensed and tenseless time can be synchronized by holistic correlations. (shrink)

Mi objetivo es discutir las principales dificultades que Frank P. Ramsey encontró en Principia Mathematica y la solución que, vía el Tractatus Logico-Philosophicus, propuso al respecto. Sostengo que las principales dificultades que Ramsey encontró en Principia Mathematica están, todas, relacionadas con que Russell y Whitehead desatendieron la forma lógica de las proposiciones matemáticas, las cuales, según Ramsey, deben ser tautológicas.

We attempt here to trace the evolution of Frege’s thought about truth. What most frames the way we approach the problem is a recognition that hardly any of Frege’s most familiar claims about truth appear in his earliest work. We argue that Frege’s mature views about truth emerge from a fundamental re-thinking of the nature of logic instigated, in large part, by a sustained engagement with the work of George Boole and his followers, after the publication of Begriffsschrift and the (...) appearance of critical reviews by members of the Boolean school. (shrink)

Logical form has always been a prime concern for philosophers belonging to the analytic tradition. For at least one century, the study of logical form has been widely adopted as a method of investigation, relying on its capacity to reveal the structure of thoughts or the constitution of facts. This book focuses on the very idea of logical form, which is directly relevant to any principled reflection on that method. Its central thesis is that there is no such thing as (...) a correct answer to the question of what is logical form: two significantly different notions of logical form are needed to fulfil two major theoretical roles that pertain respectively to logic and to semantics. This thesis has a negative and a positive side. The negative side is that a deeply rooted presumption about logical form turns out to be overly optimistic: there is no unique notion of logical form that can play both roles. The positive side is that the distinction between two notions of logical form, once properly spelled out, sheds light on some fundamental issues concerning the relation between logic and language. (shrink)

One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject.

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

Original source :「委員会の論理」『世界文化』[Culture du monde], n° 13, janvier 1936, 2–17 ; n° 14, février 1936, 16–33 ; n° 15, mars 1936, 12–25. Repris dans nmz 1 : 46–108. La version des œuvres complètes présente un certain nombre de variantes par rapport à la première publication en 1936. « La logique des comités » a été traduit en espagnol par Agustín J. Zavala. Je remercie vivement Saitō Takako pour sa relecture et ses précieuses remarques.

Prior Analytics by the Greek philosopher Aristotle and Laws of Thought by the English mathematician George Boole are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle's system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss many other historically and philosophically important aspects (...) of Boole's book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole's contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology.… using mathematical methods … has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to … when Boole's masterpiece was published.Paul Rosenbloom 1950. (shrink)

An investigation of what Frege means by his doctrine that functions (and so concepts) are 'unsaturated'. We argue that this doctrine is far less peculiar than it is usually taken to be. What makes it hard to understand, oddly enough, is the fact that it is so deeply embedded in our contemporary understanding of logic and language. To see this, we look at how it emerges out of Frege's confrontation with the Booleans and how it expresses a fundamental difference between (...) Frege's approach to logic and theirs. (shrink)

Natural languages can express some logical propositions that humans are able to understand. We illustrate this fact with a famous text that Conan Doyle attributed to Holmes: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth”. This is a subtle logical statement usually felt as an evident true. The problem we are trying to solve is the cognitive reason for such a feeling. We postulate here that we (...) accept Holmes’ maxim as true because our adult brains are equipped with neural modules that perform naturally modal logical computations. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated in this paper (...) have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims atproviding an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic hasfewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic PPC3, with two kinds of falsity and two negations. It introduces the notion of -space for a sentence A as the basis of logical model theory, and the notion of P A /, functioning as a `private' subuniverse for A/A. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. PPC3 and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of PPC3 as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: 1a for the conjoined presuppositions of a, ã for the complement of a within 1a, and â for the complement of 1a within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted byPC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to presentPCas a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, inPC. The (...) transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection betweenPCand the alpha system. (shrink)

C. I. Lewis (I883-I964) was the first major figure in history and philosophy of logic—-a field that has come to be recognized as a separate specialty after years of work by Ivor Grattan-Guinness and others (Dawson 2003, 257).Lewis was among the earliest to accept the challenges offered by this field; he was the first who had the philosophical and mathematical talent, the philosophical, logical, and historical background, and the patience and dedication to objectivity needed to excel. He was blessed with (...) many fortunate circumstances, not least of which was entering the field when mathematical logic, after only six decades of toil, had just reaped one of its most important harvests with publication of the monumental Principia Mathematica. It was a time of joyful optimism which demanded an historical account and a sober philosophical critique. Lewis was one of the first to apply to mathematical logic the Aristotelian dictum that we do not understand a living institution until we see it growing from its birth. (shrink)

Julius König is famous for his mistaken attempt to demonstrate that the continuum hypothesis was false. It is also known that the only positive result that could have survived from his proof is the paradox which bears his name. Less famous is his 1914 book Neue Grundlagen der Logik, Arithmetik und Mengenlehre. Still, it contains original contributions to logic, like the concept of metatheory and the solution of paradoxes based on the refusal of the law of bivalence. We are going (...) to discover them by analysing the content of the book. (shrink)

This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order-Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of (...) modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplares (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which Firs-Order-Logic matured will be seen to have played a primary role in its shaping. (shrink)

In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aiin of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world (...) is made up of things, or properties, or facts, may be no more than a matter of how we look at it. (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)

This article examines the traditional and modern doctrines of categorical propositions and argues that both doctrines have serious problems. While the doctrines disagree about existential imports...

This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)

Most recent discussions of John Stuart Mill’s System of Logic (1843) neglect the fifth book concerned with logical fallacies. Mill not only follows the revival of interest in the traditional Aristotelian doctrine of fallacies in Richard Whately and Augustus De Morgan, but he also develops new categories and an original analysis which enhance the study of fallacies within the context of what he calls ‘the philosophy of error’. After an exploration of this approach, the essay relates the philosophy of error (...) to the discussion of truth and error in chapter two of On Liberty (1859) concerned with freedom of thought and discussion. Drawing on Socratic and Baconian perspectives, Mill defends both the traditional study of logic against Jevons, Boole, De Morgan, and others, as well as the study of fallacies as the key to maintaining truth and its dissemination in numerous fields, such as science, morality, politics, and religion. In Mill’s view the study of fallacies also liberates ordinary people to explore the truth and falsity of ideas and, as such, to participate in society and politics and develop themselves as progressive beings. (shrink)

It is a commonplace that in the development of modern logic towards its actual shape at least two directions or traditions have to be distinguished. These traditions may be called, following the mo...

Para Frege las relaciones lógicas se dan entre contenidos judicables, entre pensamientos. Aquellas relaciones son inferenciales. Los pensamientos se definen a través de sus relaciones inferenciales con otros. De acuerdo con esto es discutible afirmar, como lo hizo Schröder, que el proyecto lógico de Frege es como el proyecto lógico de Boole. También es cuestionable afirmar, como lo hizo Dummett, que la relación inferencial no es siempre central en el proyecto fregeano. En este texto defenderé una lectura del proyecto lógico (...) de Frege, en contra de las posturas de Schröder y Dummett. (shrink)

Quine’s philosophy comprises a bewildering set of views whose integrating principle is his “confirmed extensionalism”. The paper offers a historical as well as an intellectual reconstruction of extensionalism. Traditional extensionalism (Boole) freed logic from Aristotelian essentialism that had inhibited the development of logic. Quine’s confirmed extensionalism is the acceptance, as a matter of course, of the validity of Frege’s criticism of [Boole’s] extensionalism. His confirmed extensionalism is a generalized version of the philosophy of science known as conventionalism. As such, it (...) places the advancement of science outside the province of science proper. It is, thus, at odds with Quine’s repeated expressions of alliance with the Popperian (hypothetico-deductive) model of science. (shrink)

This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence (...) of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort). (shrink)

By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the other hand, (...) is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems. (shrink)

George Boole collected ideas for the improvement of his Mathematical analysis of logic(1847) on interleaved copies of that work. Some of the notes on the interleaves are merely minor changes in explanation. Others amount to considerable extension of method in his mathematical approach to logic. In particular, he developed his technique in solving simultaneous elective equations and handling hypotheticals and elective functions. These notes and extensions provided a source for his later book Laws of thought(1854).

This paper provides equational semantics for Dung's argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the ?extensions? of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, and much more. The equational approach has its conceptual roots in the nineteenth (...) century following the algebraic equational approach to logic by George Boole, Louis Couturat, and Ernst Schroeder. (shrink)

This paper discusses George Boole’s two distinct approaches to the explanatory relationship between logical and psychological theory. It is argued that, whereas in his first book he attributes a substantive role to psychology in the foundation of logical theory, in his second work he abandons that position in favour of a linguistically conceived foundation. The early Boole espoused a type of psychologism and later came to adopt a type of anti-psychologism. To appreciate this invites a far-reaching reassessment of his philosophy (...) of logic. (shrink)

In IV, xvii, 19–22 of his Essay, Locke employs Latin labels for four kinds of argument, of which one (ad hominem) was already in circulation and one (ad judicium) has never had much currency. The present proposal seeks to locate and clarify what Locke was aiming to describe, and to contrast what he says with some subsequent uses that have been made of these labels as if they named fallacies. Though three of the four kinds of argument that Locke picks (...) out are often less than decisive, he casts no aspersion on the legitimacy of their use in debate. (shrink)