The Fruits of Logicism

2021

We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for terms and predicates of a mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given a true reading in the logical framework. ∗This paper originated as a presentation that the third author prepared for the 31st Wittgenstein Symposium, in Kirchberg, Austria, August 2008. Discussions between the co-authors after this presentation led to a collaboration on, and further development of, the thesis and the technical material grounding the thesis. The authors would especially like to thank Daniel Kirchner for suggesting important refinements of the technical development. We’d also like to thank . . ....

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions, and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the “logic” assumed rather than from Hume’s principle. It is shown that Hume’s principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only few rudimentary facts of arithmetic are logically derivable from Hume’s principle. And that hardly counts as a vindication of logicism.

This paper provides a study and assessment of the neo-logicist view that mathematical knowledge is essentially logical knowledge, as advanced by Hale and Wright. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism - a general conception of the relation between language and reality; (2) the method of abstraction - a particular method for introducing concepts into language; (3) the scope of 2nd order logic - second order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explored and assessed. The issues discussed included reductionism, rejectionism, the Julius Caesar objections, and the charge that second order logic is set theory in disguise.

The paper starts with an examination and critique of Tarski's wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L ∞,∞ . Also characterized similarly is a natural generalization of Tarski's thesis, due to Sher, in terms of bijections between domains. My main objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M 0 of individuals and {T, F } at their base) is introduced to accomplish the latter. The main result is that an operation is definable from the first-order predicate calculus without equality just in case it is definable from homomorphism invariant monadic operations, where definability in both cases is taken in the sense of the λ-calculus. The paper concludes with a discussion of the significance of the results described for the views of Tarski and Boolos on logicism.

dialectica, 1974

2006

To all these institutions we express our warm gratitude. We are also grateful to the members of the PILM scientific committee for their invaluable help in preparing the program and reporting on so many lectures, as well as to the staff of the Poincaré Archives for their help in the preparation of the conference and their expert assistance in various ways. We particularly express our gratitude to Dr. Prosper Doh (Poincaré Archives) who has taken on the job of technical editor for this book and who has realized the index and the camera-ready copy. Finally, the editors are indebted to the editorial board of "Logic, Epistemology, and the Unity of Science" for accepting this volume in their series. They would also like to thank Springer Publishers and, in particular, Floor Oosting.

Erkenntnis, 2014

This talk surveys a range of positions on the fundamental metaphysical and epistemological questions about elementary logic, for example, as a starting point: what is the subject matter of logic-what makes its truths true? how do we come to know the truths of logic? A taxonomy is approached by beginning from well-known schools of thought in the philosophy of mathematics-Logicism, Intuitionism, Formalism, Realism-and sketching roughly corresponding views in the philosophy of logic. Kant, Mill, Frege, Wittgenstein, Carnap, Ayer, Quine, and Putnam are among the philosophers considered along the way.

Erkenntnis, 2014

In logic, there are no morals. Everyone is at liberty to build his own logic, i.e. his own form of language as he wishes. (Carnap, Logical Syntax of Language, 1934, §17) 1 What is the Status of Logic?