csp-frege-revolu

HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? 6
the connectives, including the conditional. Of cardinal importance was the realization that, if circularity is to be
avoided, logical derivations are to be formal, that is, have to proceed according to rules that are devoid of any
intuitive logical force but simply refer to the typographical form of the expressions; thus the notion of formal
system made its appearance. The rules of quantification theory, as we know them today, were then introduced.
The last part of the book belongs to the foundations of mathematics, rather than to logic, and presents a logical
definition of the notion of mathematical sequence. Frege’s contribution marks one of the sharpest breaks that
ever occurred in the development of a science.
We cannot help but notice the significant gap in the choices of material included in FFTG—all of the algebraic logicians
are absent, not only the work by De Morgan and Boole, some of which admittedly appeared in the late 1840s and
early 1850s, for example Boole’s [1847] The Mathematical Analysis of Logic and [1854] An Investigation of the Laws
of Thought, De Morgan’s [1847] Formal Logic, originating algebraic logic and the logic of relations, and Jevons’s
“de-mathematicizing” modifications of Boole’s logical system in his [1864] Pure Logic or the Logic of Quality apart
from Quantity and [1869] The Substitution of Similars, not only the first and second editions of John Venn’s (1834–
1923) Symbolic Logic [Venn 1881; 1894] which, along with Jevons’s logic textbooks, chiefly his [1874] The Principles
of Science, a Treatise on Logic and Scientific Method which went into its fifth edition in 1887, were particularly
influential in the period from 1880 through 1920 in disseminating algebraic logic and the logic of relations, but even
for work by Peirce and Schröder that also appeared in the years which this anthology, an anthology purporting to completeness,
includes, and even despite the fact that Frege and his work is virtually unmentioned in any of the other selections,
whereas many of the work included do refer back, often explicitly, to contributions in logic by Peirce and
Schröder. Boole’s and De Morgan’s work in particular served as the starting point for the work of Peirce and Schröder.
The exclusion of Peirce and Schröder in particular from FFTG is difficult to understand if for no other reason than that
their work is cited by many of the other authors whose work is included, and in particular is utilized by Leopold
Löwenheim (1878–1957) and Thoralf Skolem (1887–1963), whereas Frege’s work is hardly cited at all in any of the
other works included in FFTG; the most notable exceptions being the exchange between Russell and Frege concerning
Russell’s discovery of his paradox (see [van Heijenoort 1967a, 124-128] for Russell’s [1908] on the theory of types),
and Russell’s references to Frege in his paper of 1908 on theory of types ([Russell 1903]; see [van Heijenoort 1967a,
150-182]). The work of the algebraic logicians is excluded because, in van Heijenoort’s estimation, and in that of the
majority of historians and philosophers—almost all of whom have since at least the 1920s, accepted this judgment,
that the work of the algebraic logicians falls outside of the Fregean tradition, and therefore does not belong to modern
mathematical logic. Van Heijenoort the distinction as one primarily between algebraic logicians, most notably Boole,
De Morgan, Peirce, and Schröder, and logicians who worked in quantification theory, first of all Frege, and with Russell
as his most notable follower. For that, the logic that Frege created, as distinct from algebraic logic, was mathematical
logic.
The work of these algebraic logicians is excluded because, in van Heijenoort’s estimation, and in that of the
majority historians and philosophers—almost all of whom have since at least the 1920s, accepted this judgment, the
work of the algebraic logicians falls outside of the Fregean tradition. It was, however, far from universally acknowledged
during the crucial period between 1880 through the early 1920s, that either Whitehead and Russell’s Principia
Mathematica nor any of the major efforts by Frege, was the unchallenged standard for what mathematical logic was or
ought to look like. 6
Van Heijenoort made the distinction one primarily between algebraic logicians, most notably Boole, De Morgan,
Peirce, and Schröder, and logicians who worked in quantification theory, first of all Frege, and with Russell as his
most notable follower. For that, the logic that Frege created, as distinct from algebraic logic, was regarded as mathematical
logic. ([Anellis & Houser 1991] explore the historical background for the neglect which algebraic logic endured
with the rise of the “modern mathematical” logic of Frege and Russell.)
Hans-Dieter Sluga, following van Heijenoort’s distinction between followers of Boole and followers of Frege,
labels the algebraic logicians “Booleans” after George Boole, thus distinguishes between the “Fregeans”, the most
important member of this group being Bertrand Russell, and the “Booleans”, which includes not only, of course, Boole
and his contemporary Augustus De Morgan, but logicians such as Peirce and Schröder who combined, refined, and
further developed the algebraic logic and logic of relations established by Boole and De Morgan (see [Sluga 1987]).
In the last two decades of the nineteenth century and first two decades of the twentieth century, it was, however,
still problematic whether the new Frege-Russell conception of mathematical logic or the classical Boole-Schröder calculus
would come to the fore. It was also open to question during that period whether Russell (and by implication
Frege) offered anything new and different from what the algebraic logicians offered, or whether, indeed, Russell’s