Frege, logic, and arithmetic

Abstract

IN Frege's Logical Theory' Robert Sternfeld discusses in detail Frege's variant of the logistic thesis regarding the establishment of the foundations of arithmetic within logic. I believe that his account is inconsistent. Professor Stemfeld ascribes all of the following assertions to Frege: 1. Logic is coextensive with arithmetic (p. 58). 2. The content or subject matter of arithmetic is definable within (the context of) logic (pp. 58, 81 ). 3. The foundations of arithmetic are established firmly in logic (p. 34). Arithmetic is reducible to logic (p. 63). 4. Arithmetic is derived from logic (pp. 30, 39, 40, 61). 5. Logic expands into (or is expansible into) arithmetic (pp. 39, 40, 61 63, 70, 87). 6. Numbers are arithmetical objects and hence the content of arithmetic (pp. 69, 81, 152). (Numbers are individual, specifiable entities (pp. 40, 54).) 7. Numbers serve to extend logic into arithmetic (p. 61). &. (Cardinal) numbers are definable within logic (pp. 36, 40, 54, 69, 81). 9. Extensions and, generally, value ranges are logical objects (p. 32). 10. A number is the extension of a second-level concept (p. 41). 11. Numbers are the content or subject matter of logic (pp. 54, 58). Numbers are the substantive content of logic (p. 87). Numbers are the substantial content of logical concepts (pp. 54, 81). Numbers are logical entities (p. 62). Numbers are logical objects (p. 152). Also, ultimately arithmetic is the subject matter of logic (p. 1 56). 12. Numbers are the proper objects which ground logic and give it its par ticular range or scope (pp. 58, 61 ). 1 3. Logic has its own content or subject matter (pp. 78, 79). 14. Logic is concerned with concepts such as negation, quantification, identity (pp. 78, 80). 15. Logical principles are ultimate simples and hence indefinable. Assertions 2, 3, and 4 appear to be equivalent, or at least can be so interpreted as to be equivalent. Assertion 5 is the converse of 4. Assertions 2-4, or 2-5, are assumed to be reasons for 1. Assertion 6 is undeniably true; however, its interpretation depends upon the nature and success of the logistic reduction. Assertion 7 is the converse of 8. Assertions 9 and 10 are clearly held by Frege; the logistic accomplishment rests first on 10, and then 8 is achieved by way of 9 and 10. Assertion 14 is a more detailed specification