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

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 a phaneroscopy beyond human consciousness: Building a phenomenology of multiple realities

This essay wants to rescue the concept of phaneroscopy, created by Charles Sanders Peirce, to adapt it in a phenomenological condition of multiple realities. Therefore, in addition to review the reflection of Peirce, we visited the approach of phenomenology of multiple realities proposed by Alfred Schutz in his reading of William James. The idea is to seek a phenomenology that goes beyond the human consciousness to other research subjects

Valental aspects of Peircean algebraic logic

AbstractThis paper describes a system of logic that has both an algebraic syntax and a graphical syntax and that may be regarded as a kind of semantic net. The paper analyses certain “valental” characteristics of terms and graphs of this logical system: characteristics defined in terms of the “valence”, or number of argument-places, in the terms and graphs. The most famous valental result is the so-called “Reduction Thesis” of Charles Sanders Peirce. The paper briefly explicates the author's proof, which is to be published in a forthcoming book, of this Reduction Thesis. Several additional valental results are proved, and the potential of PAL as a bridge between logic and topological graph theory is suggested

A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine

Combining insights from the study of type refinement systems and of monoidal closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of presheaves using a full and faithful embedding into a monoidal closed bifibration living now over the compact closed category of small categories and distributors. Besides revealing dualities which are not immediately apparent in the traditional presentation of the presheaf hyperdoctrine, this reconstruction leads us to an axiomatic treatment of directed equality predicates (modelled by hom presheaves), realizing a vision initially set out by Lawvere (1970). It also leads to a simple calculus of string diagrams (representing presheaves) that is highly reminiscent of C. S. Peirce's existential graphs for predicate logic, refining an earlier interpretation of existential graphs in terms of Boolean hyperdoctrines by Brady and Trimble. Finally, we illustrate how this work extends to a bifibrational setting a number of fundamental ideas of linear logic.Comment: Identical to the final version of the paper as appears in proceedings of LICS 2016, formatted for on-screen readin

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

Rhemata

The article offers an analysis of Peirce's notion of “rhema.” It examines and explains Peirce's definition of the rhema; it identifies and solves two problems that are direct consequences of the definition. The first problem is that proper names, while classified as rhemata, do not satisfy Peirce's definition of the rhema. The second problem is that Peirce also calls “rhemata” the results of propositional analysis that however do not satisfy his own definition of the rhema. Peirce himself solves the first problem by generalizing the notion of rhema into that of “seme.” I argue that we can solve the second problem if, following M. Dummett, we distinguish propositional analysis from propositional decomposition

Semiotics and Methods of Legal Inquiry: Interpretation and Discovery in Law from the Perspective of Peirce\u27s Speculative Rhetoric

Symposium: Semiotics, Dialectic, and the Law Held at Indiana University School of Law - Oct. 13, 198