Diversity Reading List

Abstract:

In this paper, we provide a recipe that not only captures the common structure of semantic paradoxes but also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first discuss a well-known schema introduced by Graham Priest, namely,the Inclosure Schema. Without rehashing previous arguments against the Inclosure Schema, we contribute different arguments for the same concern that the Inclosure Schema bundles together the wrong paradoxes. That is, we will provide further arguments on why the Inclosure Schema is both too narrow and too broad. We then spell out our recipe. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry’s paradox, Validity Curry, Provability Liar, Provability Curry, Knower’s paradox, Knower’s Curry, Grelling-Nelson’s paradox, Russell’s paradox in terms of extensions, alternative Liar and alternative Curry, and hitherto unexplored paradoxes.We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions the recipe suggests if we want to adhere to the Principle of Uniform Solution.

Abstract: Recent criticism of feminist philosophy poses a dilemma. Feminism is taken to be a substantive set of empirical claims and political commitments, whereas philosophy is taken to be a discipline of thought organized by the pursuit of truth, but uncommitted to any particular truth. This paper responds to this dilemma, and defends the project of feminist philosophy.The first task toward understanding the feminist critique of reason, Alcoff argues, is to historically situate it within the rather long tradition of critiquing reason that has existed within the mainstream of philosophy itself.

Abstract: What sort of claims do scientific models make and how do these claims then underwrite empirical successes such as explanations and reliable policy interventions? In this paper I propose answers to these questions for the class of models used throughout the social and biological sciences, namely idealized deductive ones with a causal interpretation. I argue that the two main existing accounts misrepresent how these models are actually used, and propose a new account.

Abstract: Many solutions have been proposed for solving the problem of macroscopic superpositions of wave function ontology. A possible solution is to assume that, while the wave function provides the complete description of the system, its temporal evolution is not given by the Schroedinger equation. The usual Schroedinger evolution is interrupted by random and sudden "collapses". The most promising theory of this kind is the GRW theory, named after the scientists that developed it: Gian Carlo Ghirardi, Alberto Rimini and Tullio Weber. It seems tempting to think that in GRW we can take the wave function ontologically seriously and avoid the problem of macroscopic superpositions just allowing for quantum jumps. In this paper it is argued that such "bare" wave function ontology is not possible, neither for GRW nor for any other quantum theory: quantum mechanics cannot be about the wave function simpliciter. All quantum theories should be regarded as theories in which physical objects are constituted by a primitive ontology. The primitive ontology is mathematically represented in the theory by a mathematical entity in three-dimensional space, or space-time.

Abstract:

We present a case study of how mathematicians write for mathematicians. We have conducted interviews with two research mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research paper they wrote together. Over the course of 2 years, Adam and Thomas revised Adam’s very detailed first draft. At the beginning of this collaboration, Adam was very knowledgeable about the subject of the paper and had good presentational skills but, as a new PhD student, did not yet have experience writing research papers for mathematicians. Thus, one main purpose of revising the paper was to make it take into account the intended audience. For this reason, the changes made to the initial draft and the authors’ purpose in making them provide a window for viewing how mathematicians write for mathematicians. We examined how their paper attracts the interest of the reader and prepares their proofs for validation by the reader. Among other findings, we found that their paper prepares the proofs for two types of validation that the reader can easily switch between.

Abstract: Mathematicians appear to have quite high standards for when they will rely on testimony. Many mathematicians require that a number of experts testify that they have checked the proof of a result p before they will rely on p in their own proofs without checking the proof of p. We examine why this is. We argue that for each expert who testifies that she has checked the proof of p and found no errors, the likelihood that the proof contains no substantial errors increases because different experts will validate the proof in different ways depending on their background knowledge and individual preferences. If this is correct, there is much to be gained for a mathematician from requiring that a number of experts have checked the proof of p before she will rely on p in her own proofs without checking the proof of p. In this way a mathematician can protect her own work and the work of others from errors. Our argument thus provides an explanation for mathematicians’ attitude towards relying on testimony.

Abstract:

Is it possible for one and the same person to be a feminist and a logician, or does this entail a psychic rift of such proportions that one is plunged into an endless cycle of self-contradiction? Andrea Nye's book, Words of Power (1990), is an eloquent affirmation of the psychic rift position. In what follows, I shall discuss Nye's proscription of logic as well as her perceived alternatives of a woman's language and reading. This will be followed by a discussion more sharply focused on Nye's feminist response to logic, namely, her claim that feminism and logic are incompatible. I will end by offering a sketch of a class in the life of a feminist teaching logic, a sketch which is both a response to Nye (in Nye's sense of the word) and a counter-example to her thesis that logic is necessarily destructive to any genuine feminist enterprise.

Abstract: This chapter is based on the talk that I gave in August 2018 at the ICM in Rio de Janeiro at the panel on The Gender Gap in Mathematical and Natural Sciences from a Historical Perspective. It provides some examples of the challenges and prejudices faced by women mathematicians during last two hundred and fifty years. I make no claim for completeness but hope that the examples will help to shed light on some of the problems many women mathematicians still face today.

Publisher's note: This volume is an accessible introduction to the subject of many-valued and fuzzy logic suitable for use in relevant advanced undergraduate and graduate courses. The text opens with a discussion of the philosophical issues that give rise to fuzzy logic - problems arising from vague language - and returns to those issues as logical systems are presented. For historical and pedagogical reasons, three valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic. The major fuzzy logical systems - Lukasiewicz, Godel, and product logics - are then presented as generalizations of three-valued systems that successfully address the problems of vagueness. Semantic and axiomatic systems for three-valued and fuzzy logics are examined along with an introduction to the algebras characteristic of those systems. A clear presentation of technical concepts, this book includes exercises throughout the text that pose straightforward problems, ask students to continue proofs begun in the text, and engage them in the comparison of logical systems.

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: This paper argues that the prominent accounts of logical knowledge have the consequence that they conflict with ordinary reasoning. On these accounts knowing a logical principle, for instance, is having a disposition to infer according to it. These accounts in particular conflict with so-called 'reasoned change in view', where someone does not infer according to a logical principle but revise their views instead. The paper also outlines a propositional account of logical knowledge which does not conflict with ordinary reasoning.

Publisher's Note: Although sequent calculi constitute an important category of proof systems, they are not as well known as axiomatic and natural deduction systems. Addressing this deficiency, Proof Theory: Sequent Calculi and Related Formalisms presents a comprehensive treatment of sequent calculi, including a wide range of variations. It focuses on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, linear logic, and modal logic. In the first chapters, the author emphasizes classical logic and a variety of different sequent calculi for classical and intuitionistic logics. She then presents other non-classical logics and meta-logical results, including decidability results obtained specifically using sequent calculus formalizations of 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.

Abstract: Description: This article is a short overview of philosophical and formal issues in the treatment and analysis of logical consequence. The purpose of the paper is to provide a brief introduction to the central issues surrounding two questions: (1) that of the nature of logical consequence and (2) that of the extension of logical consequence. It puts special emphasis in the role played by formal systems in the investigation of logical consequence.