The Rise of Modern Logic: from Leibniz to Frege

Dov M. Gabbay, John Woods

• Publisher: North Holland
• Year: 2004
• eBook ISBN: 9780080532875
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

With the publication of the present volume, the Handbook of the History of Logic turns its attention to the rise of modern logic. The period covered is 1685-1900, with this volume carving out the territory from Leibniz to Frege. What is striking about this period is the earliness and persistence of what could be called 'the mathematical turn in logic'. Virtually every working logician is aware that, after a centuries-long run, the logic that originated in antiquity came to be displaced by a new approach with a dominantly mathematical character. It is, however, a substantial error to suppose that the mathematization of logic was, in all essentials, Frege's accomplishment or, if not his alone, a development ensuing from the second half of the nineteenth century. The mathematical turn in logic, although given considerable torque by events of the nineteenth century, can with assurance be dated from the final quarter of the seventeenth century in the impressively prescient work of Leibniz. It is true that, in the three hundred year run-up to the Begriffsschrift, one does not see a smoothly continuous evolution of the mathematical turn, but the idea that logic is mathematics, albeit perhaps only the most general part of mathematics, is one that attracted some degree of support throughout the entire period in question. Still, as Alfred North Whitehead once noted, the relationship between mathematics and symbolic logic has been an "uneasy" one, as is the present-day association of mathematics with computing. Some of this unease has a philosophical texture. For example, those who equate mathematics and logic sometimes disagree about the directionality of the purported identity. Frege and Russell made themselves famous by insisting (though for different reasons) that logic was the senior partner. Indeed logicism is the view that mathematics can be re-expressed without relevant loss in a suitably framed symbolic logic. But for a number of thinkers who took an algebraic approach to logic, the dependency relation was reversed, with mathematics in some form emerging as the senior partner. This was the precursor of the modern view that, in its four main precincts (set theory, proof theory, model theory and recursion theory), logic is indeed a branch of pure mathematics. It would be a mistake to leave the impression that the mathematization of logic (or the logicization of mathematics) was the sole concern of the history of logic between 1665 and 1900. There are, in this long interval, aspects of the modern unfolding of logic that bear no stamp of the imperial designs of mathematicians, as the chapters on Kant and Hegcl make clear. Of the two, Hcgel's influence on logic is arguably the greater, serving as a spur to the unfolding of an idealist tradition in logic - a development that will be covered in a further volume, British Logic in the Nineteenth Century.

The Algebra of Logic

Victor Sánchez Valencia

Introduction

In this essay I am principally concerned with the nineteenth-century mathematization of logic initiated in Britain by A. De Morgan (1806–1871) and G. Boole (1815–1864). The logical framework in which this novel process resulted came to be known, presumably since [MacFarlane, 1879], as the algebra of logic. But the phrases logical algebra and algebraic logic have also been attached to it. For the sake of convenience I shall myself indulge in the occasional use of these alternative names.

A rough imperative description of Boole’s methodology in dealing with a logical problem will help to explain these labels:

1. Translate the logical data into suitable equations.

2. Apply algebraic techniques to solve these equations.

3. Translate this solution, if possible, back into the original language.

In other words, symbolic formulation of logical problems and solution of logical equations constitute defining features of Boole’s method.

De Morgan initiated the systematic logical study of binary relations. A subject Boole did not touch on. His was chiefly a logic of monadic terms. The algebra of logic was later cultivated in America by C. S. S. Peirce (1839–1914) and in Germany by E. Schröder (1841–1902). Peirce unified in his work features of the unary and binary systems of Boole and De Morgan. By common consent the work of Schröder may be regarded as the final instalment of the Boolean-De Morgan development [Couturat, 1905, 3]. These four writers are the main subjects in our narrative. A minor one is W. S. Jevons (1835–1882). He took the first steps towards the mechanization of reasoning. The focus of my concern, however, is the work of Boole and De Morgan. I consider the logical contributions of Jevons, Peirce and Schröder only inasmuch as they set forth the tradition inaugurated by those two Victorian friends.

At the beginning of the twenty-century the algebra of logic was superseded by the mathematical logic of G. Frege (1848–1925) and G. Peano (1858–1932). In defense of his logic, Schröder exchanged views with both of them. As the preface to Principia Mathematica shows, it was to no avail. A. N. Whitehead (1861–1947) and B. Russell (1872–1970) singled Peano out as the one who made of received logic a useful mathematical instrument. And as far as logical analysis was concerned, they recognized that their main debt was to Frege. In their hands, the logical tradition that started with Boole was going to be discontinued. But not before it had achieved its most immediate goal, the mathematization of logic.

The fact that the algebra of logic became to be regarded as not suitable for the new research goal, the logical foundations of the whole of mathematics, does not mean that it disappeared from mathematics. To begin with, [Huntington, 1904] regarded one of the products of the algebra of logic, the so-called Boole algebras, as structures satisfying a set of equations. This is the first application outside geometry of Hilbert’s notion of a set of statements defining a structure class. A product of the algebra of logic had been transformed into a mathematical subject. In the second place, according to [Goldfarb, 1979], the modern view of logic was codified in the nineteen twenties by combining the algebraic and mathematical logical traditions. Finally, as the following description of the modern algebraic logic by [Andréka et al., 2001] shows, there is a conceptual continuity between Boole’s initial steps and contemporary logic algebraic efforts.

Algebraic logic can be divided into two main parts. Part I studies algebras which are relevant to logic(s), e.g. algebras which were obtained from logics (one way or another). Since Part I studies algebras, its methods are, basically, algebraic. One could say, that Part I belongs to “Algebra Country”. Continuing this metaphor, Part II deals with studying and building the bridge between Algebra Country and Logic Country. Part II deals with the methodology of solving logic problems by (i) translating them to algebra (the process of algebraization), (ii) solving the alge braic problem (this really belongs to Part I), and (iii) translating the result back to logic.

The algebra of logic was, then, driven out the central stage by the logic of Frege and Peano. Obliterated it was not.

What the algebra of logic had accomplished before the arrival of mathematical logic was a cultural change of profound significance. It is important to pause briefly at this stage in order to stress this fact. Around 1850 logic was still regarded as a part of philosophy and not of mathematics. At the end of the century logic had become the shared concern of these disciplines. The pre-eminent role played by Boole in this cultural shift can never be in doubt. It is true that the range of his logical systems strikes us as rather restricted vis-à-vis the Fregean system [Dummett, 1959]. It is also true that the rigor of his presentation did not match the rigor di...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Front Matter
• Copyright page
• Preface
• Contributors
• Leibniz’s Logic
• Kant: From General to Transcendental Logic
• Hegel’s Logic
• Bolzano as Logician*
• Husserl’s Logic
• Algebraical Logic 1685–1900
• The Algebra of Logic
• The Mathematical Turns in Logic
• Schröder’s Logic
• Peirce’s Logic
• Frege’s Logic
• Index