Diversity Reading List

Summary: This article provides the basics of a typical logic, sometimes called 'classical elementary logic' or 'classical first-order logic', in a rigorous yet accessible manner. Section 2 develops a formal language, with a syntax and grammar. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. Section 4 provides a model-theoretic semantics. Section 5 turns to the relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. The authors show that an argument is derivable only if it is valid (soundness). Then they establish a converse: that an argument is valid only if it is derivable (completeness). They also briefly indicate other features of the logic, some of which are corollaries to soundness and completeness. The final section, Section 6, is devoted to a brief examination of the philosophical position that classical logic is 'the one right logic'.

Abstract:

This paper presents a new view of logical pluralism. This pluralism takes into account how the logical connectives shift, depending on the context in which they occur. Using the Question-Under-Discussion Framework as formulated by Craige Roberts, I identify the contextual factor that is responsible for this shift. I then provide an account of the meanings of the logical connectives which can accommodate this factor. Finally, I suggest that this new pluralism has a certain Carnapian flavour. Questions about the meanings of the connectives or the best logic outside of a specified context are not legitimate questions.

Abstract:

Logical pluralism is the view that there is more than one right logic. A particular version of the view, what is sometimes called domain-specific logical pluralism, has it that the right logic and connectives depend somehow on the domain of use, or context of use, or the linguistic framework. This type of view has a problem with cross-framework communication, though: it seems that all such communication turns into merely verbal disputes. If two people approach the same domain with different logics as their guide, then they may be using different connectives, and hence talking past each other. In this situation, if we think we are having a conversation about “ ¬ A”, but are using different “ ¬ ”s, then we are not really talking about the same thing. The communication problem prevents legitimate disagreements about logic, which is a bad result. In this paper I articulate a possible solution to this problem, without giving up pluralism, which requires adopting a notion of metalinguistic negotiation, and allows people to communicate and disagree across domains/contexts/frameworks.

Introduction: In increasing numbers, philosophers are coming to read Sellars' "Empiricism and the Philosophy of Mind" (1997, hereafter EPM) as having dealt the definitive death blow to the idea that inner states with epistemic authority could have this authority immediately. EPM purportedly proves that instead, such states necessarily show up already embedded within a web of inferentially articulated conceptual knowledge, and that in order for this to be possible,  the epistemic subject must be a negotiator of a normative space in which standards of justification and correctness are already recognized. [...] In this paper I will attempt to show that Sellars' mythical explanations in EPM employ a very specific and rhetorically complex methodology, and likewise that we will not be in a position to critically assess the paper's arguments unless we give careful attention to its overall textual structure and to the nature of the mythical explanations it employs.

Abstract: Several accounts of representation in cognitive systems have recently been proposed. These look for a theory that will establish how a representation comes to have a certain content, and how these representations are used by cognitive systems. Covariation accounts are unsatisfactory, as they make intelligent reasoning and cognition impossible. Cummins' interpretation-based account cannot explain the distinction between cognitive and non-cognitive systems, nor how certain cognitive representations appear to have intrinsic meaning. Cognitive systems can be defined as model-constructers, or systems that use information from interpreted models as arguments in the functions they execute. An account based on this definition solves many of the problems raised by the earlier proposals

Abstract: In this paper I examine some research on how to diminish or eliminate stereotype threat in mathematics. Some of the successful strategies include: informing our students about stereotype threat, challenging the idea that logical intelligence is an 'innate' ability, making students In threatened groups feel welcomed, and introducing counter-stereotypical role models. The purpose of this paper is to take these strategies that have proven successful and come up with specific ways to incorporate them into introductory logic classes. For example, the possible benefit of presenting logic to our undergraduate students by concentrating on aspects of logic that do not result in a clash of schemas.

Publisher's Note: Mary Leng offers a defense of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focussed on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. Leng, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized.

Summary: Surveys the opposition between views of mathematics which take mathematics to represent a independent mathematical reality and views which take mathematical axioms to define or circumscribe their subject matter; and defends the latter view against influential objections.

Summary: Defends an account of mathematical knowledge in which mathematical knowledge is a kind of modal knowledge. Leng argues that nominalists should take mathematical knowledge to consist in knowledge of the consistency of mathematical axiomatic systems, and knowledge of what necessarily follows from those axioms. She defends this view against objections that modal knowledge requires knowledge of abstract objects, and argues that we should understand possibility and necessity in a primative way.

Abstract: This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.