123 research outputs found

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

    Get PDF
    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

    A proof-theoretic analysis of the classical propositional matrix method

    Get PDF
    The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory. © 2012 The Author, 2012. Published by Oxford University Press

    The Algebra of Logic Tradition

    Get PDF
    The algebra of logic, as an explicit algebraic system showing the underlying mathematical structure of logic, was introduced by George Boole (1815-1864) in his book The Mathematical Analysis of Logic (1847). The methodology initiated by Boole was successfully continued in the 19th century in the work of William Stanley Jevons (1835-1882), Charles Sanders Peirce (1839-1914), Ernst Schröder (1841-1902), among many others, thereby establishing a tradition in (mathematical) logic. From Boole's first book until the influence after WWI of the monumental work Principia Mathematica (1910 1913) by Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970), versions of thealgebra of logic were the most developed form of mathematical above allthrough Schröder's three volumes Vorlesungen über die Algebra der Logik(1890-1905). Furthermore, this tradition motivated the investigations of Leopold Löwenheim (1878-1957) that eventually gave rise to model theory. Inaddition, in 1941, Alfred Tarski (1901-1983) in his paper On the calculus of relations returned to Peirce's relation algebra as presented in Schröder's Algebra der Logik. The tradition of the algebra of logic played a key role in thenotion of Logic as Calculus as opposed to the notion of Logic as Universal Language . Beyond Tarski's algebra of relations, the influence of the algebraic tradition in logic can be found in other mathematical theories, such as category theory. However this influence lies outside the scope of this entry, which is divided into 10 sections.Fil: Burris, Stanley. University of Waterloo; CanadáFil: Legris, Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Interdisciplinario de Economía Politica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Económicas. Instituto Interdisciplinario de Economía Politica de Buenos Aires; Argentin

    Two new gestures. On Peirce's continuum and the existential graphs

    Get PDF
    The article presents two gestures corresponding to two profound new understandings of Peirce's Continuum (Vargas 2015) and Peirce's Existential Graphs (Oostra 2010). Vargas and Oostra have revolutionized Peirce's mathematical studies, thanks to a first complete model for Peirce's continuum provided by Vargas, and thanks to the emergence of intuitionistic existential graphs provided by Oostra. The article aims at showing how these careful mathematical constructions can be encrypted in very simple gestures

    On Logic in the Law: Something, but not All

    Get PDF
    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

    Kripke Semantics for a Logical Framework

    Get PDF
    We present a semantics (using Kripke lambda models) for a logical framework (minimal implicational predicate logic with quantification over all higher types). We apply the semantics to obtain straightforward adequacy proofs for encodings of logics in the framework. 1 Introduction There has been much recent interest in the development and use of logical frameworks. A logical framework is a formal system within which many different logics can be easily represented. It is hoped that such frameworks will facilitate the rapid development of proof assistants for the wide variety of different logics used in computer science and other fields. In this paper we give a semantic analysis (using Kripke lambda models) of the use of minimal implicational predicate logic (with quantification over all higher types) as a logical framework. We choose this framework because it is relatively straightforward to give it a useful semantics. The use of such a logic as a framework is not new. Similar logics ha..

    From mathematics in logic to logic in mathematics : Boole and Frege

    Get PDF
    This project proceeds from the premise that the historical and logical value of Boole's logical calculus and its connection with Frege's logic remain to be recognised. It begins by discussing Gillies' application of Kuhn's concepts to the history oflogic and proposing the use of the concept of research programme as a methodological tool in the historiography oflogic. Then it analyses'the development of mathematical logic from Boole to Frege in terms of overlapping research programmes whilst discussing especially Boole's logical calculus. Two streams of development run through the project: 1. A discussion and appraisal of Boole's research programme in the context of logical debates and the emergence of symbolical algebra in Britain in the nineteenth century, including the improvements which Venn brings to logic as algebra, and the axiomatisation of 'Boolean algebras', which is due to Huntington and Sheffer. 2. An investigation of the particularity of the Fregean research programme, including an analysis ofthe extent to which certain elements of Begriffsschrift are new; and an account of Frege's discussion of Boole which focuses on the domain common to the two formal languages and shows the logical connection between Boole's logical calculus and Frege's. As a result, it is shown that the progress made in mathematical logic stemmed from two continuous and overlapping research programmes: Boole's introduction ofmathematics in logic and Frege's introduction oflogic in mathematics. In particular, Boole is regarded as the grandfather of metamathematics, and Lowenheim's theorem ofl915 is seen as a revival of his research programme
    • …
    corecore