Diversity Reading List

Summary: This book is an introductory textbook on mathematical logic. It covers Propositional Logic and Predicate Logic. For each of these formalisms it presents its syntax and formal semantics as well as a tableaux-style method of consistency-checking and a natural deduction-style deductive calculus. Moreover, it discusses the metatheory of both logics.

Abstract: Gottlob Frege's work in logic and the foundations of mathemat- ics centers on claims of logical entailment; most important among these is the claim that arithmetical truths are entailed by purely logical principles. Occupying a less central but nonetheless important role in Frege's work are claims about failures of entailment. Here, the clearest examples are his theses that the truths of geometry are not entailed by the truths of logic or of arithmetic, and that some of them are not entailed by each other. As he, and we, would put it: the truths of Eluclidean geometry are independent of the truths of logic, and some of them are independent of one another.' Frege's talk of independence and related notions sounds familiar to a modern ear: a proposition is independent of a collection of propositions just in case it is not a consequence of that collection, and a proposition or collection of propositions is consistent just in case no contradiction is a consequence of it. But some of Frege's views and procedures are decidedly tinmodern. Despite developing an extremely sophisticated apparattus for demonstrating that one claim is a consequience of others, Frege offers not a single demon- stration that one claim is not a conseqtuence of others. Thus, in par- tictular, he gives no proofs of independence or of consistency. This is no accident. Despite his firm commitment to the independence and consistency claims just mentioned, Frege holds that independence and consistency cannot systematically be demonstrated.2 Frege's view here is particularly striking in light of the fact that his contemporaries had a fruitful and systematic method for proving consistency and independence, a method which was well known to him. One of the clearest applications of this method in Frege's day came in David Hilbert's 1899 Foundations of Geometry,3 in which he es- tablishes via essentially our own modern method the consistency and independence of various axioms and axiom systems for Euclidean geometry. Frege's reaction to Hilbert's work was that it was simply a failure: that its central methods were incapable of demonstrating consistency and independence, and that its usefulness in the founda- tions of mathematics was highly questionable.4 Regarding the general usefulness of the method, it is clear that Frege was wrong; the last one hundred years of work in logic and mathemat- ics gives ample evidence of the fruitfulness of those techniques which grow directly from the Hilbert-style approach. The standard view today is that Frege was also wrong in his claim that Hilbert's methods fail to demonstrate consistency and independence. The view would seem to be that Frege largely missed Hilbert's point, and that a better under- standing of Hilbert's techniques would have revealed to Frege their success. Despite Frege's historic role as the founder of the methods we now use to demonstrate positive consequence-results, he simply failed, on this account, to understand the ways in which Hilbert's methods could be used to demonstrate negative consequence-results. The purpose of this paper is to question this account of the Frege- Hilbert disagreement. By 1899, Frege had a well-developed view of log- ical consequence, consistency, and independence, a view which was central to his foundational work in arithmetic and to the epistemologi- cal significance of that work. Given this understanding of the logical relations, I shall argue, Hilbert's demonstrations do fail. Successful as they were in demonstrating significant metatheoretic results, Hilbert's proofs do not establish the consistency and independence, in Frege's sense, of geometrical axioms. This point is important, I think, both for an understanding of the basis of Frege's epistemological claims about mathematics, and for an understanding of just how different Frege's conception of logic is from the modern model-theoretic conception that has grown out of the Hilbert-style approach to consistency.

Publisher's Note: In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic. She argues that the fruitfulness of Frege's conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege's logicist project. Blanchette argues that despite a number of difficulties, Frege's use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege's intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. In the second part of the book, Blanchette explores the resulting conception of logic itself, and some of the straightforward ways in which Frege's conception differs from its now-familiar descendants. In particular, Blanchette argues that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege's position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.

Publisher’s Note:
A native of St. Thomas, West Indies, Edward Wilmot Blyden (1832-1912) lived most of his life on the African continent. He was an accomplished educator, linguist, writer, and world traveler, who strongly defended the unique character of Africa and its people. Christianity, Islam and the Negro Race is an essential collection of his writings on race, culture, and the African personality.

Summary: A comprehensive introduction to ancient (western) logic from the 5th century BCE to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Topics include pre-Aristotelian logic, Aristotelian logic, Peripatetic logic, Stoic Logic and a note on Epicureans and their views on logic.

Abstract: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out.

Abstract: Lady Mary Shepherd (1777-1847) argued for distinctive accounts of causation, perception, and knowledge of an external world and God. However, her work, engaging with Berkeley and Hume but written after Kant, does not fit the standard periodisation of early modern philosophy presupposed by many philosophy courses, textbooks, and conferences. This paper argues that Shepherd should be added to the canon as a Scottish philosopher. The practical reason for doing so is that it would give Shepherd a disciplinary home, opening up additional possibilities for research and teaching. The philosophical reason is that her views share certain features characteristic of canonical Scottish philosophers.

Publisher's Note: Appearance and Reality: An Introduction to the Philosophy of Physics addresses quantum mechanics and relativity and their philosophical implications, focusing on whether these theories of modern physics can help us know nature as it really is, or only as it appears to us. The author clearly explains the foundational concepts and principles of both quantum mechanics and relativity and then uses them to argue that we can know more than mere appearances, and that we can know to some extent the way things really are. He argues that modern physics gives us reason to believe that we can know some things about the objective, real world, but he also acknowledges that we cannot know everything, which results in a position he calls "realistic realism." This book is not a survey of possible philosophical interpretations of modern physics, nor does it leap from a caricature of the physics to some wildly alarming metaphysics. Instead, it is careful with the physics and true to the evidence in arriving at its own realistic conclusions. It presents the physics without mathematics, and makes extensive use of diagrams and analogies to explain important ideas. Engaging and accessible, Appearance and Reality serves as an ideal introduction for anyone interested in the intersection of philosophy and physics, including students in philosophy of physics and philosophy of science courses.

Publisher's Note: In this rich and detailed study of early modern women's thought, Jacqueline Broad explores the complexity of women's responses to Cartesian philosophy and its intellectual legacy in England and Europe. She examines the work of thinkers such as Mary Astell, Elisabeth of Bohemia, Margaret Cavendish, Anne Conway and Damaris Masham, who were active participants in the intellectual life of their time and were also the respected colleagues of philosophers such as Descartes, Leibniz and Locke. She also illuminates the continuities between early modern women's thought and the anti-dualism of more recent feminist thinkers. The result is a more gender-balanced account of early modern thought than has hitherto been available. Broad's clear and accessible exploration of this still-unfamiliar area will have a strong appeal to both students and scholars in the history of philosophy, women's studies and the history of ideas.

Publisher's Note: This ground-breaking book surveys the history of women's political thought in Europe from the late medieval period to the early modern era. The authors examine women's ideas about topics such as the basis of political authority, the best form of political organisation, justifications of obedience and resistance, and concepts of liberty, toleration, sociability, equality, and self-preservation. Women's ideas concerning relations between the sexes are discussed in tandem with their broader political outlooks; and the authors demonstrate that the development of a distinctively sexual politics is reflected in women's critiques of marriage, the double standard, and women's exclusion from government. Women writers are also shown to be indebted to the ancient idea of political virtue, and to be acutely aware of being part of a long tradition of female political commentary. This work will be of tremendous interest to political philosophers, historians of ideas, and feminist scholars alike.