Diversity Reading List

Publisher's Note: Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems from logic to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated.

From the Introduction: "Lynn Hankinson Nelson and Jack Nelson extend the work begun in the former’s book Who Knows: From Quine to a Feminist Empiricism, by showing that a Quinean understanding of logic as an empirical field implies that logic remains open to revision in light of fundamental shifts in knowledge. Nelson and Nelson point to the revisions in scientific understandings made possible by the incorporation of women and women’s lives as emblematic of the possible ways that feminist thought can provide a deep reworking of the structures of knowledge and thus potentially of logic. Although they are cautious of any conclusions that logic must change, their work offers a theoretical ground from which the effects of feminist theorizing on logic can be usefully explored."

Publisher’s Note:

Is logic masculine? Is women's lack of interest in the "hard core" philosophical disciplines of formal logic and semantics symptomatic of an inadequacy linked to sex? Is the failure of women to excel in pure mathematics and mathematical science a function of their inability to think rationally? Andrea Nye undermines the assumptions that inform these questions, assumptions such as: logic is unitary, logic is independenet of concrete human relations, and logic transcends historical circumstances as well as gender. In a series of studies of the logics of historical figures--Parmenides, Plato, Aristotle, Zeno, Abelard, Ockham, and Frege--she traces the changing interrelationships between logical innovation and oppressive speech strategies, showing that logic is not transcendent truth but abstract forms of language spoken by men, whether Greek ruling citizens, or scientists.

From the Introduction: "Andrea Nye is also concerned with the role of logic in science, linking the adequacy of logic with its applicability in a domain of scientific knowledge. Nye argues that the dominant predicate logic cannot adequately represent the issues surrounding attempts to divide organisms into species. Feminist critiques of the extensional theory of meaning lay the ground for alternative theories of categorization. Without renewed models of categorization, Nye submits, science is in danger of becoming a self-enclosed “logical” system, rather than an instrumental model of reality."

From the Introduction: "Dorothea Olkowski’s chapter offers an analysis of the need to develop a logic of sense. Drawing on the work of Gilles Deleuze, Olkowski defends formal logic against feminist theorists who have urged that we organize thinking around the principles of embodiment. She warns us against the complete merging of bodily functions and sense-making activities. In Olkowski’s view, feminists need to acknowledge the usefulness of logical analyses at the same time that they must insist on formal systems that reflect and are tempered by human and humane values."

Abstract:

¿Seguimos reglas de inferencia al razonar? Por más intuitiva que resulte la respuesta positiva a esta pregunta, hay una serie de dificultades para vincular reglas lógicas y prácticas inferenciales. El Problema de la Adopción de Reglas de Inferencia constituye un desafío para todo aquel que proponga que podemos seguir nuevos patrones inferenciales a partir del reconocimiento de reglas. En esta sección temática se exploran diversos asuntos conectados a si podemos seguir un nuevo patrón inferencial en virtud de una regla.

Abstract:

Much has been said about intuitionistic and classical logical systems since Gentzen’s seminal work. Recently, Prawitz and others have been discussing how to put together Gentzen’s systems for classical and intuitionistic logic in a single unified system. We call Prawitz’ proposal the Ecumenical System, following the terminology introduced by Pereira and Rodriguez. In this work we present an Ecumenical sequent calculus, as opposed to the original natural deduction version, and state some proof theoretical properties of the system. We reason that sequent calculi are more amenable to extensive investigation using the tools of proof theory, such as cut-elimination and rule invertibility, hence allowing a full analysis of the notion of Ecumenical entailment. We then present some extensions of the Ecumenical sequent system and show that interesting systems arise when restricting such calculi to specific fragments. This approach of a unified system enabling both classical and intuitionistic features sheds some light not only on the logics themselves, but also on their semantical interpretations as well as on the proof theoretical properties that can arise from combining logical systems.

Abstract:

The author argues that there is a strong connection between the dualisms that have strengthened and naturalized systematic oppression across history (man/woman, reason/emotion, etc.), and "classical" logic. It is suggested that feminism's response should not be to abandon logic altogether, but rather to focus on the development of alternative, less oppressive forms of rationality, of which relevant logics provide an example.

Introduction: Plumwood’s second essay uses logical distinctions to map the difficult terrain of feminist theories of difference. By carefully distinguishing among forms of difference, Plumwood refutes attempts by some feminist theorists to identify dichotomous thinking with oppressive thinking.

Abstract:

Being a pragmatic and not a referential approach to semantics, the dialogical formulation of paraconsistency allows the following semantic idea to be expressed within a semi-formal system: In an argumentation it sometimes makes sense to distinguish between the contradiction of one of the argumentation partners with himself (internal contradiction) and the contradiction between the partners (external contradiction). The idea is that external contradiction may involve different semantic contexts in which, say A and not A have been asserted. The dialogical approach suggests a way of studying the dynamic process of contradictions through which the two contexts evolve for the sake of argumentation into one system containing both contexts. More technically, we show a new, dialogical, way to build paraconsistent systems for propositional and first-order logic with classical and intuitionistic features (i.e. paraconsistency both with and without tertium non-datur) and present their corresponding tableaux.