University Park, PA 16802
Netherlands Institute for Advanced Study (NIAS)
2242 PR Wassenaar
I A Focused Exposition
This is an engaging, controversial, and refreshingly well-written book about Frege's logic from the Begriffsschrift to the Grundgesetze. Danielle Macbeth explains Frege's complicated two-dimensional logical notation more patiently, accurately, and with a greater variety of examples than I have previously seen. She does so, moreover, not merely by relating Frege's innovations to the linear Principia Mathematica style logical symbolism that most logicians these days take for granted, but by delving into the deeper reasons why Frege choose to represent logical relations in this special pictorial way.
II Organization of Themes and Topics
The book is divided into five main chapters of four-five sections each, preceded by a Preface and Introduction, and followed by an Epilogue, notes, list of abbreviations and index. The chapters describe a historical progression in Frege's thinking about the expressive capabilities of the original Begriffsschrift logic, as they introduce 'The Starting Point,' and cover a succession of topics on 'Logical Generality,' 'A More Sophisticated Instrument,' 'The Work Brought to Maturity,' 'Course of Values and Basic Law V.'
The chapters track Macbeth's inquiry into specific problems in Frege's Begriffsschrift logic. She begins with a detailed examination of the specific concept of generality Frege sought to formalize. She proceeds to Frege's notion of function and higher-level generalizations (functions of functions) [End Page 609] first recognized in advanced mathematics. This background leads her to an account of Frege's logicism and the use of the begriffsschriftlicher symbolism with three orthographic styles of variables for three levels of generalizations required for the logical foundations of mathematics. Macbeth's discussion culminates finally in a discussion of Frege's later distinction between Sinn (sense) and Bedeutung (reference, meaning), in relation to the so-called Julius Caesar problem, and the disastrous inclusion of Basic Law V in the Grundgesetze der Arithmetik. This is the principle concerning the extensions of predicates and the comprehension of objects by begriffsschriftliche descriptions of property combinations, against which Russell's famous paradox of 1901 was directed.
After preliminary discussion of what makes Frege's logic distinct from contemporary formalisms, I consider four interrelated subject areas that Macbeth emphasizes as central to her new interpretation of Frege's logic: (1) Frege's evolving conception of logical laws, which Macbeth distinguishes from generalized conditionals and characterizes as inference licenses. (2) The proper interpretation of Frege's Begriffsschrift and begriffsschriftliche notation in the Grundgesetze, including Frege's use of three orthographic styles of variables in generalizations. (3) The question of whether Frege's Begriffsschrift constitutes a quantificational or proto-quantificational logic sufficiently similar to linear Principia-style predicate-quantificational systems. (4) Frege's post-1891 distinction between Sinn and Bedeutung, presented in relation to his later understanding of the nature of a concept and the relation among concepts expressed in generalized conditionals.
III Uniqueness of Frege's Logic
Macbeth encourages the reader to appreciate the uniqueness of Frege's system as a different kind of logic than the Principia logics it later inspired. Whereas it is too easy otherwise to regard Frege's concept-script as a crude idiosyncratic prototype of linear formulations in mathematical logic, Macbeth challenges the conventional wisdom that Frege's logic is a strange-appearing precursor to ordinary quantificational logic. She emphasizes two features that have not otherwise received adequate attention in critical commentary on Frege.
First, Macbeth explains Frege's use of the two-dimensional display of propositional logical connectives. She argues that Frege's notation is not merely an eccentric way of formulating primarily the material conditional and negation in forms that are logically equivalent to the more familiar propositional connectives including conjunction, disjunction and biconditional, but more significantly as a way of pictorially representing the...