Relations between logic and mathematics in the work of Benjamin and Charles S. Peirce.

Charles Peirce (1839-1914) was one of the most important logicians of the nineteenth century. This thesis traces the development of his algebraic logic from his early papers, with especial attention paid to the mathematical aspects. There are three main sources to consider. 1) Benjamin Peirce (1809-1880), Charles's father and also a leading American mathematician of his day, was an inspiration. His memoir Linear Associative Algebra (1870) is summarised and for the first time the algebraic structures behind its 169 algebras are analysed in depth. 2) Peirce's early papers on algebraic logic from the late 1860s were largely an attempt to expand and adapt George Boole's calculus, using a part/whole theory of classes and algebraic analogies concerning symbols, operations and equations to produce a method of deducing consequences from premises. 3) One of Peirce's main achievements was his work on the theory of relations, following in the pioneering footsteps of Augustus De Morgan. By linking the theory of relations to his post-Boolean algebraic logic, he solved many of the limitations that beset Boole's calculus. Peirce's seminal paper `Description of a Notation for the Logic of Relatives' (1870) is analysed in detail, with a new interpretation suggested for his mysterious process of logical differentiation. Charles Peirce's later work up to the mid 1880s is then surveyed, both for its extended algebraic character and for its novel theory of quantification. The contributions of two of his students at the Johns Hopkins University, Oscar Mitchell and Christine Ladd-Franklin are traced, specifically with an analysis of their problem solving methods. The work of Peirce's successor Ernst Schröder is also reviewed, contrasting the differences and similarities between their logics. During the 1890s and later, Charles Peirce turned to a diagrammatic representation and extension of his algebraic logic. The basic concepts of this topological twist are introduced. Although Peirce's work in logic has been studied by previous scholars, this thesis stresses to a new extent the mathematical aspects of his logic - in particular the algebraic background and methods, not only of Peirce but also of several of his contemporaries

Recapture, Transparency, Negation and a Logic for the Catuṣkoṭi

The recent literature on Nāgārjuna’s catuṣkoṭi centres around Jay Garfield’s (2009) and Graham Priest’s (2010) interpretation. It is an open discussion to what extent their interpretation is an adequate model of the logic for the catuskoti, and the Mūla-madhyamaka-kārikā. Priest and Garfield try to make sense of the contradictions within the catuskoti by appeal to a series of lattices – orderings of truth-values, supposed to model the path to enlightenment. They use Anderson & Belnaps\u27s (1975) framework of First Degree Entailment. Cotnoir (2015) has argued that the lattices of Priest and Garfield cannot ground the logic of the catuskoti. The concern is simple: on the one hand, FDE brings with it the failure of classical principles such as modus ponens. On the other hand, we frequently encounter Nāgārjuna using classical principles in other arguments in the MMK. There is a problem of validity. If FDE is Nāgārjuna’s logic of choice, he is facing what is commonly called the classical recapture problem: how to make sense of cases where classical principles like modus pones are valid? One cannot just add principles like modus pones as assumptions, because in the background paraconsistent logic this does not rule out their negations. In this essay, I shall explore and critically evaluate Cotnoir’s proposal. In detail, I shall reveal that his framework suffers collapse of the kotis. Taking Cotnoir’s concerns seriously, I shall suggest a formulation of the catuskoti in classical Boolean Algebra, extended by the notion of an external negation as an illocutionary act. I will focus on purely formal considerations, leaving doctrinal matters to the scholarly discourse – as far as this is possible

Recapture, Transparency, Negation and a Logic for the Catuskoti

The recent literature on Nāgārjuna’s catuṣkoṭi centres around Jay Garfield’s (2009) and Graham Priest’s (2010) interpretation. It is an open discussion to what extent their interpretation is an adequate model of the logic for the catuskoti, and the Mūla-madhyamaka-kārikā. Priest and Garfield try to make sense of the contradictions within the catuskoti by appeal to a series of lattices – orderings of truth-values, supposed to model the path to enlightenment. They use Anderson & Belnaps's (1975) framework of First Degree Entailment. Cotnoir (2015) has argued that the lattices of Priest and Garfield cannot ground the logic of the catuskoti. The concern is simple: on the one hand, FDE brings with it the failure of classical principles such as modus ponens. On the other hand, we frequently encounter Nāgārjuna using classical principles in other arguments in the MMK. There is a problem of validity. If FDE is Nāgārjuna’s logic of choice, he is facing what is commonly called the classical recapture problem: how to make sense of cases where classical principles like modus pones are valid? One cannot just add principles like modus ponens as assumptions, because in the background paraconsistent logic this does not rule out their negations. In this essay, I shall explore and critically evaluate Cotnoir’s proposal. In detail, I shall reveal that his framework suffers collapse of the kotis. Furthermore, I shall argue that the Collapse Argument has been misguided from the outset. The last chapter suggests a formulation of the catuskoti in classical Boolean Algebra, extended by the notion of an external negation as an illocutionary act. I will focus on purely formal considerations, leaving doctrinal matters to the scholarly discourse – as far as this is possible

On relationships between the logic of law, legal positivism and semiotics of law

The issue of reciprocal relationships between the logic of law, positivistic theory of the logic of law, and legal semiotics is among the most important questions of the modern theoretical jurisprudence. This paper has not attempted to provide any comprehensive account of the modern jurisprudence (and legal logic). Instead, the emphasis has been laid on those aspects of positivist legal theories, logical studies of law and legal semiotics that allow tracing the common points or the differences between these paradigms of legal research. One of the theses of the present work is that, at the comparative methodological level, the limits of legal semiotics and its object of inquiry could only be defined in relation to legal posi tivism and logical studies of law. This paper also argues for a proper position for legal semiotics in between legal positivism and legal logic. The differences between legal positivism, legal logic and legal semiotics are best captured in the issue of referent

On Logic in the Law: Something, but not All

In 1880, when Oliver Wendell Holmes (later to be a Justice of the U.S. Supreme Court) criticized the logical theology of law articulated by Christopher Columbus Langdell (the first Dean of Harvard Law School), neither Holmes nor Langdell was aware of the revolution in logic that had begun, the year before, with Frege\u27s Begriffsschrift. But there is an important element of truth in Holmes\u27s insistence that a legal system cannot be adequately understood as a system of axioms and corollaries ; and this element of truth is not obviated by the more powerful logical techniques that are now available

Axiomatic Structure and the Method of Analysis: Shifting Styles in the History of Mathematics

This article surveys the different views of mathematical methodology that occurred from ancient Greek times through the early modern period up until its codification around 1900. After summarizing the axiomatic approach advocated by Aristotle and implemented in mathematics by Euclid, the talk explores the character of analysis in ancient Greek times, its development into a symbolic algebra by Viete and Descartes, and its expansion into a calculus of fluxions and differentials by Newton and Leibniz. The article concludes by touching on the recovery and transformation of the deductive ideal for mathematics by Pasch, Peano, and Hilbert during the late nineteenth and early twentieth centuries

Working memory and human reasoning : an individual differences approach

Experiments 1-3 investigated the relationship between working memory and syllogistic and five-ten-n series spatial inference. A secondary aim was to replicate the findings of Shah and Miyake (1996) who suggested the use of separate central resources of working memory for spatial and verbal ability. The correlational analysis showed that the complex verbal and spatial working memory span tasks were associated together and consistently predicted reasoning performance in both verbal and visual modalities. The confirmatory factor analysis showed that three factors best accounted for the data -a verbal, a spatial, and a general resource. All the span tasks and most of the reasoning tasks significantly and consistently loaded the general factor. Experiments 4-6 investigated the relationship between working memory and a range of reasoning tasks - identified as either propositional. spatial, or quantifiable tasks. These experiments were based on the work of Stanovich and West (1998) who found that a range of reasoning tasks were predicted by cognitive ability and a reasoner's thinking style. The correlational anaylsis showed that the complex verbal and spatial working memory span tasks were associated together and consistently predicted reasoning perforinance. Two clusters of reasoning task emerged from the correlational analysis - one cluster related to the propositional and simple spatial reasoning tasks, whilst the other related to the quantifiable and complex spatial reasoning tasks. The confin-natory factor analysis showed that four factors best accounted for the data -a verbal, a spatial, a general, and a thinking style resource. All the span tasks and the reasoning tasks loaded the general factor, and most of the reasoning tasks further loaded the thinking disposition factor. These results are discussed in light of models of workino memory, theories of reasoning, and how to best characterise factor 3 (executive function) and factor 4 (thinking style) from tile factor analysis.Economic and Social Research Counci