Handbook of the History of Logic, volume 3: The Rise of Modern Logic from Leibniz to Frege (review)

The Handbook of the History of Logic, under the general editorship of Dov Gabbay and John Woods, is intended as an extensive and encyclopedic survey of the entire history of logic from earliest times to the present. Every contribution to the Handbook is written by a specialist in a particular era or topic, and most of the articles run well over fifty pages, in many cases twice that much. Because of the range of inclusion and the diversity of contributors, the styles vary greatly from essay to essay. So also does the perspective from which the contributors work. Thus, some essays are largely philosophical, others are mathematically technical, ranging from exposition to critical analysis to rational reconstruction, and even to unadulterated speculation. (For example, in an earlier volume, covering ancient Greek, Indian, and Buddhist logics, there is an effort to rewrite Buddhist logic in Aristotelian terms.) This kind of diversity does not necessarily depend upon the specialized topic or upon the logician whose work is being considered, but rather upon the interests and focus of the authors of the essays.

Additionally, the editors have allowed "logic" to be defined quite broadly, to include what many would doubtlessly consider more properly to be philosophical logic, or even philosophy of logic; and while there is little, in the volumes published thus far, to satisfy readers who would be interested in non-deductive logics—Hilpinen alone gives any attention to abduction or induction, and only in a most cursory manner—nonclassical deductive logics are represented, both within the context of specific logicians and chronological frameworks, or in distinct volumes of the series devoted specifically to the history of non-classical logics.

This expansive definition of logic coincides in many respects with Peirce's own, which takes within its compass the methodology of the sciences and critical thinking, and includes abduction and induction along with deduction.

There are three essays in the volume under review that impinge to a significant extent upon the work in logic of Charles Peirce, each by well-known contributors. They are:

• • Theodore Hailperin, "Algebraical Logic 1685–1900," pp. 323–88;

• • Victor Sánchez Valencia, "The Algebra of Logic," pp. 389–544;

• • Risto Hilpinen, "Peirce's Logic," 611–58.

The first two essays include significant discussions of Peirce's work in logic against the broader historical context in which that work was carried out. The third essay is not a complete account of Peirce's work in logic, but carefully avoids duplicating those aspects of the Peircean corpus that have been discussed by the other two authors. Thus it is absolutely essential to work through all three essays in order to gain a comprehensive overview of the full extent of Peirce's work in deductive logic.

There is a wealth of technical detail in the surveys of Hailperin, Sánchez Valencia, and Hilpinen, and much of their treatment may be daunting for readers without a solid background in algebraic logic. The incorporation of criticisms along with exposition makes the task the more difficult for those not already acquainted with the technical aspects of the work being discussed. Because many of the readers of this review will not necessarily be either historians of logic or trained as logicians, as well as constraints imposed by limitation of space in this journal as compared with the length of the contributions being reviewed, I will provide merely a sketch of a survey of the scope of the contents of the contributions and a general overview of the character of the essays.

Hailperin (p. 323) defines "algebraic logic" as "a style of doing logic, a style in which concepts and relations are expressed by mathematical symbols." He provides a general survey of the evolution of algebraic logic, beginning with Leibniz and ending with Whitehead's Treatise. The most prominent figures in his account are Leibniz, De Morgan, Boole, and...