Diversity Reading List

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.

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.

Publisher's note: We are frequently confronted with arguments. Arguments are attempts to persuade us - to influence our beliefs and actions - by giving us reasons to believe this or that. Critical Thinking: A Concise Guide will equip students with the concepts and techniques used in the identification, analysis and assessment of arguments. Through precise and accessible discussion, this book provides the tools to become a successful critical thinker, one who can act and believe in accordance with good reasons, and who can articulate and make explicit those reasons. Key topics discussed include:

• Core concepts in argumentation.
• How language can serve to obscure or conceal the real content of arguments; how to distinguish argumentation from rhetoric.
• How to avoid common confusions surrounding words such as 'truth', 'knowledge' and 'opinion'.
• How to identify and evaluate the most common types of argument.
• How to distinguish good reasoning from bad in terms of deductive validly and induction.

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.

Abstract: This article surveys several interrelated issues in the metaphysics of chance. First, what is the relationship between the probabilities associated with types of trials (for instance, the chance that a twenty?eight?year old develops diabetes before age thirty) and the probabilities associated with individual token trials (for instance, the chance that I develop diabetes before age thirty)? Second, which features of the the world fix the chances: are there objective chances at all, and if so, are there non?chancy facts on which they supervene? Third, can chance be reconciled with determinism, and if so, how?

Publisher’s Note: Plato's Sun-Like Good is a revolutionary discussion of the Republic's philosopher-rulers, their dialectic, and their relation to the form of the good. With detailed arguments Sarah Broadie explains how, if we think of the form of the good as 'interrogative', we can re-conceive those central reference-points of Platonism in down-to-earth terms without loss to our sense of Plato's philosophical greatness. The book's main aims are: first, to show how for Plato the form of the good is of practical value in a way that we can understand; secondly, to make sense of the connection he draws between dialectic and the form of the good; and thirdly, to make sense of the relationship between the form of the good and other forms while respecting the contours of the sun-good analogy and remaining faithful to the text of the Republic itself.

Abstract:

This chapter begins with a discussion of humanist criticisms of scholastic logic. It then discusses the evolution of the scholastic tradition and the influence of Renaissance Aristotelianism, Descartes and his influence, the Port-Royal Logic, the emergence of a logic of cognitive faculties, logic and mathematics in the late 17th century, Gottfried Wilhelm Leibniz's role in the history of formal logic, and Kant's influence on logic.

Publisher's Note: This paper examines the methodology used by Kepler to discover a quantitative law of refraction. The aim is to argue that this methodology follows a heuristic method based on the following two Pythagorean principles: (1) sameness is made known by sameness, and (2) harmony arises from establishing a limit to what is unlimited. We will analyse some of the author's proposed analogies to find the aforementioned law and argue that the investigation's heuristic pursues such principles.

Abstract: This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept formation as well as representations of proofs. In addition we note that 'visualization' is used in two different ways. In the first sense 'visualization' denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense 'visualization' denotes a diagram or representation of something.

Publisher's Note: This teaching book is designed to help its readers to reason systematically, reliably, and to some extent self-consciously, in the course of their ordinary pursuits-primarily in inquiry and in decision making. The principles and techniques recommended are explained and justified - not just stated; the aim is to teach orderly thinking, not the manipulation of symbols. The structure of material follows that of Quine's Methods of Logic, and may be used as an introduction to that work, with sections on truth-functional logic, predicate logic, relational logic, and identity and description. Exercises are based on problems designed by authors including Quine, John Cooley, Richard Jeffrey, and Lewis Carroll.

Abstract: In this paper, I raise the following problem: How does Avicenna define modalities? What oppositional relations are there between modal propositions, whether quantified or not? After giving Avicenna's definitions of possibility, necessity and impossibility, I analyze the modal oppositions as they are stated by him. This leads to the following results: 1. The relations between the singular modal propositions may be represented by means of a hexagon. Those between the quantified propositions may be represented by means of two hexagons that one could relate to each other. 2. This is so because the exact negation of the bilateral possible, i.e. 'necessary or impossible' is given and applied to the quantified possible propositions. 3. Avicenna distinguishes between the scopes of modality which can be either external (de dicto) or internal (de re). His formulations are external unlike al-F̄ar̄ab̄;’s ones. However his treatment of modal oppositions remains incomplete because not all the relations between the modal propositions are stated explicitly. A complete analysis is provided in this paper that fills the gaps of the theory and represents the relations by means of a complex figure containing 12 vertices and several squares and hexagons.

Abstract: In this article, I analyze Quine's conception of science, which is a radical defence of extensionalism on the grounds that first?order logic is the most adequate logic for science. I examine some criticisms addressed to it, which show the role of modalities and probabilities in science and argue that Quine's treatment of probability minimizes the intensional character of scientific language and methods by considering that probability is extensionalizable. But this extensionalizing leads to untenable results in some cases and is not consistent with the fact that Quine himself admits confirmation which includes probability. Quine's extensionalism does not account for this fact and then seems unrealistic, even if science ought to be extensional in so far as it is descriptive and mathematically expressible.

Abstract:

A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practice seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue ‘morality’, but the term is not of my coining: there are mathematicians across the world who use the word ‘morally’ to great effect in private, and I propose that there should be a public theory of what they mean by this. The issue arises because proofs, despite being revered as the backbone of mathematical truth, often contribute very little to a mathematician’s understanding. ‘Moral’ considerations, however, contribute a great deal. I will first describe what these ‘moral’ considerations might be, and why mathematicians have appropriated the word ‘morality’ for this notion. However, not all mathematicians are concerned with such notions, and I will give a characterisation of ‘moralist’ mathematics and ‘moralist’ mathematicians, and discuss the development of ‘morality’ in individuals and in mathematics as a whole. Finally, I will propose a theory for standardising or universalising a system of mathematical morality, and discuss how this might help in the development of good mathematics.

Publisher's Note: Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings. A Structural Account of Mathematics will be required reading for anyone working in this field. generally put forward, which he maintains have led to serious misunderstandings.