Blanchette, Patricia. Frege and Hilbert on Consistency
1996, Journal of Philosophy 93 (7):317
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Added by: Clotilde Torregrossa, Contributed by: Alex Yates
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.Blanchette, Patricia. Frege’s Conception of Logic2012, New York: Oxford University Press.-
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Added by: Clotilde Torregrossa, Contributed by: Alex Yates
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.Comment: This book would be a suitable resource for independent study, or for a historically oriented course on philosophy of logic, of math, or on early analytic philosophy, especially one which looks at philosophical approaches to axiomatic systems.
Blyden, Edward Wilmot. Christianity, Islam, and the Negro Race1887, Black Classic Press-
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, Contributed by: Quentin PharrPublisher’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.Comment: This collection of essays is seminal in the intellectual foundations of Pan-Africanism, African Islamism, African Anti-colonialism, the Back-to-Africa Movement, and the educational revival in Liberia/West Africa. The essays are great for courses on African thought, or African anti-colonialism/postcolonialism. They would also be excellent companion texts for reading Marcus Garvey or Kwame Nkrumah, or vice versa.
Bobzien, Susanne. Ancient Logic2016, The Stanford Encyclopedia of Philosophy-
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Added by: Berta Grimau, Contributed by: Giada Fratantonio
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.Comment: This paper would be ideal as an introductory overview for a course on ancient logic. Alternatively, it could serve as an overview for a module on ancient logic within a more general course on the history of logic. No prior knowledge of logic is required; formalisms are for the most part avoided in the paper. Note that this is a SEP entry, so it's completely accessible to students.
Bobzien, Susanne. Stoic Syllogistic1996, Oxford Studies in Ancient Philosophy 14: 133-92.-
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Added by: Berta Grimau, Contributed by: Giada Fratantonio
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.Comment: This paper can be used as specialised/further reading for an advanced undergrad or postgraduate course on ancient logic or as a primary reading in an advanced undergrad or postgraduate course on Stoic logic. Alternatively, given that the text argues that there are important parallels between Stoic logic and Relevance logic, it could be used in a course on Relevance logic as well. It requires prior knowledge of logic (in particular, proof theory).
Boyle, Deborah. Expanding the Canon of Scottish Philosophy: The Case for Adding Lady Mary Shepherd2017, Journal of Scottish Philosophy, 15(3), pp.275-293.-
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Added by: Benny Goldberg
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.Comment: A good paper for any classes on how to teach philosophy, on early modern philosophy, the philosophy of history, or feminism
Brading, Katherine, Elena Castellani. Symmetry and Symmetry Breaking2013, The Standford Encyclopedia of Philosophy-
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Added by: Laura Jimenez
Introduction: Symmetry considerations dominate modern fundamental physics, both in quantum theory and in relativity. Philosophers are now beginning to devote increasing attention to such issues as the significance of gauge symmetry, quantum particle identity in the light of permutation symmetry, how to make sense of parity violation, the role of symmetry breaking, the empirical status of symmetry principles, and so forth. These issues relate directly to traditional problems in the philosophy of science, including the status of the laws of nature, the relationships between mathematics, physical theory, and the world, and the extent to which mathematics suggests new physics. This entry begins with a brief description of the historical roots and emergence of the concept of symmetry that is at work in modern science. It then turns to the application of this concept to physics, distinguishing between two different uses of symmetry: symmetry principles versus symmetry arguments. It mentions the different varieties of physical symmetries, outlining the ways in which they were introduced into physics. Then, stepping back from the details of the various symmetries, it makes some remarks of a general nature concerning the status and significance of symmetries in physics.Comment: This article offers a good introduction to the topic of symmetries. The entry begins with a brief description of the historical roots and emergence of the concept of symmetry that could serve as a reading for undergraduates. It then turns to the application of this concept to physics and merges the discussion with issues in relativity and quantum mechanics. This second part of the article is thus more suitable to postgraduate courses in philosophy of science, specially, philosophy of physics. It could serve as a secondary reading for those researching the laws of nature.
Brake, Elizabeth. Rawls and Feminism: What Should Feminists Make of Liberal Neutrality?2004, Journal of Moral Philosophy 1 (3):293-309.-
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Added by: Carl Fox
Abstract: I argue that Rawls’s liberalism is compatible with feminist goals. I focus primarily on the issue of liberal neutrality, a topic suggested by the work of Catharine MacKinnon. I discuss two kinds of neutrality: neutrality at the level of justifying liberalism itself, and state neutrality in political decision-making. Both kinds are contentious within liberal theory. Rawls’s argument for justice as fairness has been criticized for non-neutrality at the justificatory level, a problem noted by Rawls himself in Political Liberalism. I will defend a qualified account of neutrality at the justificatory level, taking an epistemic approach to argue for the exclusion of certain doctrines from the justificatory process. I then argue that the justification process I describe offers a justificatory stance supportive of the feminist rejection of state-sponsored gender hierarchy. Further, I argue that liberal neutrality at the level of political decision-making will have surprising implications for gender equality. Once the extent of the state’s involvement in the apparently private spheres of family and civil society is recognized, and the disproportionate influence of a sexist conception of the good on those structures—and concomitant promotion of that ideal—is seen, state neutrality implies substantive change. While—as Susan Moller Okin avowed—Rawls himself may have remained ambiguous on how to address gender inequality, his theory implies that the state must seek to create substantive, not merely formal, equality. I suggest that those substantive changes will not conflict with liberal neutrality but instead be required by it.Comment: A sympathetic treatment of Rawls's Political Liberalism from a feminist perspective. It introduces the alleged clash between liberalism and feminism in a clear way and goes on to argue that (political) liberalism's commitment to substantive rather than merely formal equality makes it compatible with core feminist concerns.
Broad, Jacqueline, Karen Detlefsen. (eds.) Women and Liberty, 1600-1800: Philosophical Essays2017, Oxford University Press.-
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Added by: Francesca Bruno
Publisher's Note: This book addresses the theme of liberty as it is found in the writing of women philosophers of the seventeenth and eighteenth centuries, or as it is theorized with respect to women and their lives. It covers both theoretical and practical philosophy, with chapters grappling with problems in the metaphysics of free will (both human and God's), the liberty (or lack thereof) of women in their moral, personal lives as well as their social-political, public lives, and the interactions between the metaphysical and normative issues. The chapters draw upon writing of both women and men, and notably, upon a wide range of genres, including more standard philosophical treatises as well as polemical texts, poetry, plays, and other forms of fiction. As such, this book alerts the reader to the wide range of conceptions of what counts as a philosophical text in the early modern period. Several chapters also grapple with the relation between early modern and contemporary ways of thinking about the theme of women and liberty, thus urging the reader to appreciate the continuing importance of these earlier philosophers in the history of philosophy and of feminism. Ultimately, the chapters in this text show how crucial it is to recover the too-long forgotten views of female and women-friendly male philosophers of the seventeenth and eighteenth centuries, for in the process of recovering these voices, our understanding of philosophy in the early modern period is not only expanded, but also significantly altered toward a more accurate history of our discipline.Comment: This volume covers ethical, political, metaphysical, and religious notions of liberty, including chapters on women's ideas about the metaphysics of free will and chapters examining the topic of women's freedom (or lack thereof) in their moral and personal lives. Some of the papers in this collection could be assigned individually in an undergraduate early modern survey course; or it could be one of the main texts for a more advanced (undergraduate or graduate) course on the topic of liberty/freedom, from a variety of philosophical perspectives (ethical, political, metaphysical, and religious).
Broad, Jacqueline. Women Philosophers of the Seventeenth Century2002, Cambridge University Press.-
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Added by: Alison Stone, Contributed by: Karen Green
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.Comment: The book is organised around six authors: Elisabeth of Bohemia, Margaret Cavendish, Anne Conway, Mary Astell, Damaris Masham and Catherine Trotter Cockburn. The book focuses on their relations to Cartesianism and this means the book can be readily used on a history of modern philosophy course. It can be treated as introducing the ideas of all the women philosophers just mentioned and, e.g., a chapter could be further reading each week accompanying primary texts by the women philosophers in question.
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Comment: Good for a historically-based course on philosophy of logic or mathematics.