Stewart Shapiro

Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle.

Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends that English draws a distinction between uncountable “stuff” and countable “things”. There are two mereological relations, related in specific ways. Our primary question here is whether an empirically adequate account of the mass/count distinction really does require distinguishing “things” from “stuff”, and thus postulating two corresponding mereological relations, or if instead positing only one sort of entity and corresponding mereological relation is sufficient, as other semantic theories would have it. This question is meant to be one of what we call natural language mereology. We are asking about the mereological commitments of English, or perhaps competent speakers of English, and not about ultimate reality as such. There is no pretense that we will definitively solve the metaphysical puzzle of the statue and clay.

Graham Priest introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might for a dialetheist, then the consequent may also fail to hold. One somewhat friendly purpose of my ‘Incompleteness and inconsistency’ was to improve the technical situation for the dialetheist, eschewing reliance on an informal provability predicate. Another, less friendly purpose was to bring out what I took to be some untoward consequences of the situation. It seems that Priest accepted at least some of the improvements that I attempted. In the second edition of In contradiction, he responded to the alleged untoward consequences. One purpose of this note is to revisit the technical and philosophical situation. There were some errors in my original presentation, brought out by discussion with Priest and by Hartry Field’s analysis of the second incompleteness theorem in such contexts. A second task here is to present a sort of Curry version of the Gödel incompleteness situation. I tentatively conclude that even for a dialetheist, an interesting and complete theory is not as easy to come by as it may look—at least not for theories of arithmetic that are plausible for a dialetheist.

We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that various logics are constitutive for thought within particular practices, but none are constitutive for thought as such.

Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends that English draws a distinction between uncountable “stuff” and countable “things”. There are two mereological relations, related in specific ways. Our primary question here is whether an empirically adequate account of the mass/count distinction really does require distinguishing “things” from “stuff”, and thus postulating two corresponding mereological relations, or if instead positing only one sort of entity and corresponding mereological relation is sufficient, as other semantic theories would have it. This question is meant to be one of what we call natural language mereology. We are asking about the mereological commitments of English, or perhaps competent speakers of English, and not about ultimate reality as such. There is no pretense that we will definitively solve the metaphysical puzzle of the statue and clay.

This edited collection covers Friedrich Waismann's most influential contributions to twentieth-century philosophy of language: his concepts of open texture and language strata, his early criticism of verificationism and the analytic-synthetic distinction, as well as their significance for experimental and legal philosophy. In addition, Waismann's original papers in ethics, metaphysics, epistemology and the philosophy of mathematics are here evaluated. They introduce Waismann's theory of action along with his groundbreaking work on fiction, proper names and Kafka's Trial. Waismann is known as the voice of Ludwig Wittgenstein in the Vienna Circle. At the same time we find in his works a determined critic of logical positivism and ordinary language philosophy, who anticipated much later developments in the analytic tradition and devised his very own vision for its future.

In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or that neither is. If there is reason to accept the view that the set-theoretic universe is open-ended, that will be because such a view is the most compelling one to adopt on the purely ontological front.

This book is just what its title says: an introduction to nonclassical logic. And it is a very good one. Given the extensive interest in nonclassical logics, in various parts of the philosophical scene, it is a welcome addition to the corpus. Typical courses in logic, at all levels and in both philosophy departments and mathematics departments, focus exclusively on classical logic. Most instructors, and some textbooks, give some mention to some nonclassical systems, but usually few details are provided.

The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that our abstraction based theory can be modified to yield much of ordinary mathematics, indeed everything needed to recapture all branches of mathematics short of set theory itself. However, our conclusions are mostly negative. The theory will fall far short of the power of ordinary Zermelo-Fraenkel set theory. It is consistent that our set theory has models that are relatively small, smaller than the first cardinal with an uncountable index. More important, there is a strong tension between the idea that the iterative hierarchy is somehow ineffable, or indefinitely extensible, and the neo-logicist theme of capturing mathematical theories with abstraction principles.

The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives.

This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book includes a comprehensive bibliography.