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

Proceedings of the Seventh International Symposium on Methodologies for Intelligent Systems (Poster Session)

This report contains the following papers: Implications in vivid logic; a self-learning bayesian expert system; a natural language generation system for a heterogeneous distributed database system; competence-switching'' managed by intelligent systems; strategy acquisition by an artificial neural network: Experiments in learning to play a stochastic game; viewpoints and selective inheritance in object-oriented modeling; multivariate discretization of continuous attributes for machine learning; utilization of the case-based reasoning method to resolve dynamic problems; formalization of an ontology of ceramic science in CLASSIC; linguistic tools for intelligent systems; an application of rough sets in knowledge synthesis; and a relational model for imprecise queries. These papers have been indexed separately

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

Proceedings of the Joint Automated Reasoning Workshop and Deduktionstreffen: As part of the Vienna Summer of Logic – IJCAR 23-24 July 2014

Preface For many years the British and the German automated reasoning communities have successfully run independent series of workshops for anybody working in the area of automated reasoning. Although open to the general public they addressed in the past primarily the British and the German communities, respectively. At the occasion of the Vienna Summer of Logic the two series have a joint event in Vienna as an IJCAR workshop. In the spirit of the two series there will be only informal proceedings with abstracts of the works presented. These are collected in this document. We have tried to maintain the informal open atmosphere of the two series and have welcomed in particular research students to present their work. We have solicited for all work related to automated reasoning and its applications with a particular interest in work-in-progress and the presentation of half-baked ideas. As in the previous years, we have aimed to bring together researchers from all areas of automated reasoning in order to foster links among researchers from various disciplines; among theoreticians, implementers and users alike, and among international communities, this year not just the British and German communities

What is truth?

I defend the correspondence theory of truth, according to which a statement’s truth consists in a relation of correspondence with extralinguistic fact. There are well-known objections to this view, which I consider and rebut, and also important rival accounts, principal among which are so-called deflationist theories and epistemic theories. Epistemic theories relate the concept of truth to our state of knowledge, but fail, I argue, to respect the crucial distinction between a criterion of truth and the meaning of truth: the view that one cannot do semantics, or metaphysics, without addressing epistemic issues is rejected by this work. Against epistemic theories, I illustrate how truth is independent of epistemic considerations. Deflationism is the more popular of the rival accounts and has gained considerable momentum over the past two decades. It is therefore dealt with in greater detail by this work. Deflationist theories exploit the paradigmatic ‘“Snow is white” is true iff snow is white’ biconditional to argue for an insubstantialist account, according to which truth is conservative with respect to non-semantical facts. On this view, truth’s raison d’être is merely to perform the useful expressive function of generalising over possibly infinite sets of assertions. Against deflationist theories, I claim that the work done by Jeffrey Ketland and Stewart Shapiro conclusively demonstrates how truth is informationally additive over non-semantic facts, while deflationism itself is also an excessively impoverishing theory, inadequate to the tasks it purports to accomplish. This work also defends the thesis that Alfred Tarski’s well-known theory of truth is an authentic correspondence theory. To say this is to say that the clauses of a Tarskian truth-definition can be interpreted in terms of a relation of correspondence that holds between true sentences and the states of affairs they describe. I provide a precise account of what the correspondence in question consists in, claiming that true sentences are homomorphic images of facts, i.e. a true sentence represents, in a form-preserving manner, the truth-making facts in it. This gives precise expression to Wittgenstein’s thesis that true sentences picture the world

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic