Diversity Reading List

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.

Abstract: The chapter addresses the philosophical issues raised by the use of hypothetical modeling in the social sciences. Hypothetical modeling involves the construction and analysis of simple hypothetical systems to represent complex social phenomena for the purpose of understanding those social phenomena. To highlight its main features hypothetical modeling is compared both to laboratory experimentation and to computer simulation. In analogy with laboratory experiments, hypothetical models can be conceived of as scientific representations that attempt to isolate, theoretically, the working of causal mechanisms or capacities from disturbing factors. However, unlike experiments, hypothetical models need to deal with the epistemic uncertainty due to the inevitable presence of unrealistic assumptions introduced for purposes of analytical tractability. Computer simulations have been claimed to be able to overcome some of the strictures of analytical tractability. Still they differ from hypothetical models in how they derive conclusions and in the kind of understanding they provide. The inevitable presence of unrealistic assumptions makes the legitimacy of the use of hypothetical modeling to learn about the world a particularly pressing problem in the social sciences. A review of the contemporary philosophical debate shows that there is still little agreement on what social scientific models are and what they are for. This suggests that there might not be a single answer to the question of what is the epistemic value of hypothetical models in the social sciences.

Summary: In this paper Beebee argues that the problem of induction, which she describes as a genuine sceptical problem, is the same for Humeans than for Necessitarians. Neither scientific essentialists nor Armstrong can solve the problem of induction by appealing to IBE (Inference to the Best Explanation), for both arguments take an illicit inductive step.

Abstract: In recent years, protesters around the world have been calling for the removal of commemorations honouring those who are, by contemporary standards, generally regarded as seriously morally compromised by their racism. According to one line of thought, leaving racist memorials in place is profoundly disrespectful, and doing so tacitly condones, and perhaps even celebrates, the racism of those honoured and memorialized. The best response is to remove the monuments altogether. In this article, I first argue against a prominent offense-based account of the wrong of simply leaving memorials in place, unaltered, before offering my own account of this wrong. In at least some cases, these memorials wrong insofar as they express and exemplify a morally objectionable attitude of race-based contempt. I go on to argue that the best way of answering this disrespect is through a process of expressively “dehonouring” the subject. Removal of these commemorations is ultimately misguided, in many cases, because removal, by itself, cannot adequately dishonour, and simple removal does not fully answer the ways in which these memorials wrong. I defend a more nuanced approach to answering the wrong posed by these monuments, and I argue that public expressions of contempt through defacement have an ineliminable role to play in an apt dishonouring process.

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: In this article, I discuss the commemoration of public figures such as Nelson Mandela and Yitzhak Rabin. In many cases, our commemoration of such figures is based on the admiration we feel for them. However, closer inspection reveals that most (if not all) of those we currently honour do not qualify as fitting objects of admiration. Yet, we may still have the strong intuition that we ought to continue commemorating them in this way. I highlight two problems that arise here: the problem that the expressed admiration does not seem appropriate with respect to the object and the problem that continued commemorative practices lead to rationality issues. In response to these issues, I suggest taking a fictionalist position with respect to commemoration. This crucially involves sharply distinguishing between commemorative and other discourses, as well as understanding the objects of our commemorative practices as fictional objects.

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: In The Grammar of Society, first published in 2006, Cristina Bicchieri examines social norms, such as fairness, cooperation, and reciprocity, in an effort to understand their nature and dynamics, the expectations that they generate, and how they evolve and change. Drawing on several intellectual traditions and methods, including those of social psychology, experimental economics and evolutionary game theory, Bicchieri provides an integrated account of how social norms emerge, why and when we follow them, and the situations where we are most likely to focus on relevant norms. Examining the existence and survival of inefficient norms, she demonstrates how norms evolve in ways that depend upon the psychological dispositions of the individual and how such dispositions may impair social efficiency. By contrast, she also shows how certain psychological propensities may naturally lead individuals to evolve fairness norms that closely resemble those we follow in most modern societies.

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.

Abstract: This paper examines the connection between model-theoretic truth and necessary truth. It is argued that though the model-theoretic truths of some standard languages are demonstrably "necessary" (in a precise sense), the widespread view of model-theoretic truth as providing a general guarantee of necessity is mistaken. Several arguments to the contrary are criticized.