In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, (...) there was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist view (...) is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance. (shrink)

Applied Logic presents logic as a relevant tool for interpreting arguments, not simply as an abstract discipline such as mathematics or physics. Through English-language arguments that are debated today—health care, economic problems, AIDS, and so on—the author strives to give students an appreciation of logic's applicability. An appendix provides answers to selected exercises. Applied Logic and the Instructor's Manual will be of great value to those teaching elementary logic.

Kant identifies knowledge [Wissen], belief [Glaube], and opinion [Meinung] as our three primary modes of “holding-to-be-true” [Fürwahrhalten]. He also identifies opinion as making up the greatest part of our cognition. After a preliminary sketch of Kant’s system of propositional attitudes, this paper will explore what he says about the norms governing opinion and empirical hypotheses. The final section will turn to what, in the Critique of Pure Reason and elsewhere, Kant refers to as “General Applied Logic”. It concerns (...) the “contingent conditions of the subject, which can hinder or promote” good inquiry; and, though rarely mentioned in the secondary literature, it offers Kant’s methodological alternative to the traditional epistemological goal of finding “a sufficient and at the same time general criterion of truth”. (shrink)

This instructor's manual, to accompany Applied Logic, contains answers to the exercises found in the text and a workbook useful for preparing students for exams. The text itself presents logic as a relevant tool for interpreting arguments, not simply as an abstract discipline such as mathematics or physics. Through English-language arguments that are debated today—health care, economic problems, AIDS, and so on—the author strives to give students an appreciation of logic's applicability. This book will be of (...) interest to those teaching elementary logic. (shrink)

Contemporary logicians have expanded upon the old notions of psychologism in logic and proposed new, weakened versions of it. Those weakened versions postulate that psychologistic logic does not have to inform about the ontology or metaphysics of reasoning. Instead, logic applied in cognitive science could serve as one of many paradigms for making empirical predictions about the observable process of human reasoning. The purpose of this article is to entertain this notion and answer the question: what (...) properties should a logical system formally representing actual human reasoning have? Based on the existing evidence from cognitive science and neuroscience we identified three potential candidates: context-sensitivity (satisfied for example by adaptive logics), content-sensitivity (satisfied by non-Fregean logics) and probabilism (satisfied for example by fuzzy logics). (shrink)

This Is A New Release Of The Original 1883 Edition.

This special issue of the Journal of Applied Logics deals with the logical aspects of the concept of God. It contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions (...) to the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago. (shrink)

This classic text, written by philosopher and mathematician William Thomson, presents a systematic exposition of the laws of thought and their role in science, logic, and philosophy. The book is still widely used in philosophy and mathematics courses today. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. (...) Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Twenty-four U.S. states have enacted HIV exposure laws that prohibit HIV-positive persons from engaging in sexual activities with partners to whom they have not disclosed their HIV status. There is little standardization among existing HIV exposure laws, which vary substantially with respect to the sexual activities that are prohibited without prior serostatus disclosure. Logical analysis and mathematical modeling were used to explore the HIV prevention effectiveness of two types of HIV exposure laws: “strict” laws that require HIV-positive persons to disclose (...) their serostatus to prospective partners prior to any sexual activity and “flexible” laws that require seropositive status disclosure only prior to high-risk sex . These laws were compared relative to each other and to a no-law alternative. The results of these analyses indicate that, under most circumstances, both strict and flexible exposure laws can be expected to reduce HIV transmission risk relative to the no-law alternative, with flexible exposure laws producing the greater reduction in risk. This study demonstrates how logical analysis and mathematical modeling techniques can make an important contribution to the construction of a rational basis for decisions about a highly contested public health policy issue. (shrink)