The Basic Laws of Arithmetic 1st edition by Gottlob Frege, Montgomery Furth ISBN 0520312368 9780520312364

The Basic Laws of Arithmetic 1st edition by Gottlob Frege, Montgomery Furth ISBN 0520312368 9780520312364

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ISBN 10: 0520312368
ISBN 13: 9780520312364
Author: Gottlob Frege, Montgomery Furth

Basic Laws of Arithmetic: Exposition of the System by Gottlob Frege is a seminal work that aims to establish arithmetic and mathematical analysis as logical systems derived from pure logic. Published in 1893, it represents a cornerstone in the history of mathematical and philosophical thought. Frege’s primary objective was to substantiate logicism, the view that truths of arithmetic are not irreducibly mathematical, synthetic a priori, or empirical, but are instead expressions of logical truths. The book lays out three core tasks: defining logical propositions and rules of inference, and deriving arithmetic's fundamental truths from these logical principles. While Frege’s meticulous approach to these tasks helped establish mathematical logic as a discipline, his work ultimately failed to achieve its purpose, as the set theory underpinning his system proved inconsistent, a flaw brought to his attention by Bertrand Russell.

Despite its failure as Frege envisioned it, the work remains profoundly influential. Frege’s exploration of logical truth and inference pioneered formal logic, including propositional calculus, quantification theory, and set theory. His philosophy of language, embedded within the system’s semantics, offers a deep and nuanced understanding of meaning that continues to resonate within analytical philosophy. Moreover, Frege’s precise and rigorous standards of reasoning surpass many subsequent works, including the more widely adopted Principia Mathematica. Although his logicism is untenable in its original form, Frege’s ideas remain a vital resource for understanding the intersection of logic, mathematics, and language, making his Grundgesetze a crucial study for philosophers, logicians, and historians. This translation of key sections emphasizes its ongoing relevance to modern philosophical inquiries into meaning and language.

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1964.

The Basic Laws of Arithmetic 1st Table of contents:

§ 0. Our task. Demands on the conduct of proof. Dedekind's system. Schroder's class

Exposition of the Begriffsschrift

1. Primitive Signs

i. Introduction: Function, Concept, Relation

§1. The function is unsaturated

§2. Truth-values. Denotation and sense. Thought. Object

§3. Course-of-values of a function. Concept. Extension of a concept

§4. Functions of two arguments

ii. Signs for Functions

§5. Judgment and thought. Judgment-stroke and horizontal

§6. Negation-stroke. Amalgamation of horizontals

§7. The identity-sign

§8. Generality. Gothic letters. Their scope. Amalgamation of horizontals

§9. Notation for the course-of-values. Small Greek vowels. Their scope

§10. The course-of-values of a function more exactly specified

§11. Substitute for the definite article: the function ξ

§12. Condition-stroke. And. Neither-nor. Or. Subcomponents. Main component

iii. Inferences and Consequences

§14. First method of inference

15. Second method of inference. Contraposition

§16. Third method of inference

§17. Roman letters. Transition from Roman to Gothic letters

§18. Laws in symbols of Begriffsschrift (I. IV. VI.)

iv. Extension of the Notation for Generality

§19. Generality with respect to functions. Function-letters. Object-letters

§20. Laws in symbols of Begriffsschrift (lla. III. V.)

§21. Functions and concepts of first and second level

§22. Examples of second-level functions. Unequal-leveled functions and relations

§23. Types of arguments and argument-places. Second-level functions of arguments of type 2 and type 3

§24. General explanation of the use of function-letters

§25. Generality with respect to second-level functions. Basic Law lib

2. Definitions

i. General Remarks

§26. Classification of signs. Names. Marks. Proposition of Begriffsschrift. Transition-sign

§27. The double-stroke of definition

§28. Correct formation of names

§29. When does a name denote something?

§30. Two ways to form a name

§31. Our simple names denote something

§32. Every proposition of Begriffsschrift expresses a thought

§33. Principles of definition

ii. Particular Definitions

§34. Definition of the function ξ-ζ

§35. Representation of second-level functions by first-level 54 functions

§36. The double course-of-values. The extension of a relation

§37. Definition of the function 1 ξ

§38. Definition of the function ) ξ

§39. Definition of the function ξ

§40. Definition of the function ξ

§41. Definition of θ

§42. Definition of 1. Concept of Number

§43. Definition of f

§44. Some propositions of Begriffsschrift as examples

§45. Definition of the function - £ Following and preceding in a series

§46. Definition of the function u ξ

3. Derived Laws

§47. Summary of the Basic Laws

§48. Summary of the Rules

§49. Derivation of some propositions from (I)

§50. Derivation of the main principles of the function ξ = ζ

§51. Derivation of some propositions from (IV)

§52. Derivation of some propositions from (V) and (VI)

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