Robert May

Frege's Other Program

Frege’s logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the “neologicist” approach of Hale & Wright. Less attention has been given to Frege’s extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of a concept C , together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume’s Principle and its connections to the root of the contradiction in Frege’s system.

Logic as Science

Frege’s logicist program is a program of scientific unification of arithmetic and logic via the reduction of arithmetic to logic. Logic on this view is the prior science, indeed, the most fundamental of all sciences. The coherence of this picture has been questioned, based on the claim that the Basic Laws of logic are not justifiable as judgements. That Frege’s conception of logic suffers from this fatal flaw is incorrect, and in this paper I explore why. The discussion has three primary parts. The first explores Frege’s view of logic, distinguishing pure from applied logic. The second delves into Frege’s conception of science as applications of logic, and how arithmetic is understood as such an application. The third concerns Frege’s account of judgement, and how it carefully distinguishes a judgement from judging, the cognitive act of recognizing truths. The final section turns to how on Frege’s account of judgement, the Basic Laws of logic are in fact justified as judgements, and accordingly can serve as the axioms of the science of arithmetic. The paper concludes with reflections on Frege’s understanding of scientific unification and reduction, and its relation to his logical realism.

Discussion Note On: “Semantic Nominalism: How I Learned to Stop Worrying and Love Universals” by G. Aldo Antonelli

Editorial NoteThe following Discussion Note is an edited transcription of the discussion on G. Aldo Antonelli’s paper “Semantic Nominalism: How I Learned to Stop Worrying and Love Universals” (this volume), held among participants at the IHPST-UC Davis Workshop Ontological Commitment in Mathematics which took place, in memoriam of Aldo Antonelli, at IHPST in Paris on December, 14–15, 2015. The note’s and volume’s editors would like to thank all participants in the discussion for their contributions, and Alberto Naibo, Michael Wright and the personnel at IHPST for their technical support.

Response to Edwin William's Review of Indices and Identity

Our response to Edwin Williamss review of our book Indices and Identity (II before there can be meaningful debate it must be clear to both participants what the issues are. In I&I, our concern is with the nature and representation of syntactic identity. The central issue we address is a foundational one: to determine the conditions under which two syntactic inscriptions are occurrences of the same syntactic object. Any answer to this question will have implications for a broad range of topics, including, we argue, for the theory of anaphora. It is a central consequence of our approach to syntactic identity that if two syntactic representations, or inscriptions, have the same index, they are thereby indicated to be occurrences of the same syntactic expression. If two syntactic inscriptions do not have the same index, they are thereby indicated to be occurrences of different syntactic expressions. Since the identity conditions given are SYNTACTIC identity conditions, it is to be expected that two inscriptions might be syntactically identical but lexically distinct. This circumstance holds when, for example, an occurrence of Cicero and an occurrence of he have the same index. We refer to the relationship between syntactically identical but lexically distinct inscriptions as VEHICLE CHANGE. This relation is not arbitrary, and a good part of I&I is devoted to investigating the constraints on Vehicle Change. (See especially the discussion on pp. 218-27 and 275-88.) We will consider this relationship in more detail below. Our conception in I&I raises a question as to the relationship between the indication of syntactic identity and the theory of anaphora. What we hold on this point is that the representation of syntactic identity, expressed as a theory of indexing, is sufficient to express the syntactic relation of anaphora: expressions that have the same index are syntactically indicated to be anaphoric. The intuition behind this is easy to grasp: if an expression is repeated, it follows that the two occurrences have the same semantic value, in the cases at hand, its reference. If two inscriptions are coindexed, and hence repetitions in the appropriate sense, they are coreferential. On the other hand, if two inscriptions are not coindexed, whether they are coreferential or not depends on the structure of context, formally defined. The initial chapters of I&I formally develop the syntax and semantics of our treatment of anaphora, at the heart of which lies the intuition mentioned, and the theory is applied to a variety of empirical puzzles.