Chapter 4 addresses one of the principal ways in which Cassirer's assimilationist ambitions became out of step with the dominant "either/or" tendencies of 20th century philosophy. Skidelsky asserts that in the early 20th century the "symbolic logic" of Bertrand Russell and Gottlob Frege, which focuses on the logic of "relations," abolished from philosophical circles the mathematics-affiliated logic used by Kant:
Russell extended the logicist enterprise from arithmetic to geometry, eliminating altogether Kant's synthetic a priori (p. 53).
Skidelsky also observes that in contrast to Herman Cohen, the doyen of the Marburg school, Cassirer was able to assimilate symbolic logic into his burgeoning system, as represented by his "first work of systematic philosophy," SUBSTANCE AND FUNCTION (1910), a book I have not read.
Skidelsky summarizes Cassirer's ability to see that Kant's dependence on the traditional logic of mathematics had been superseded:
The central, defining feature of traditional logic... is that it takes as its starting point a class of individuals. Concept formation is then envisaged as the abstraction of common properties from the members of this class. Higher-order concepts can be formed through a further process of abstraction, until finally one reaches the ultimately extensive and empty concept of "being." Traditional logic is perhaps adequate to classificatory systems such as botany and zoology; these were, after all, Aristotle's original prototypes. But it cannot do justice to mathematics or mathematical natural science, for the concepts of mathematics, as serial concepts, are not abstracted from independently existing classes of objects but rather constructed through the iteration of some basic relation... (p. 59)
Skidelsky credits Russell with having "shown how the serial concepts of mathematics might be expressed in purely logical terms," in contrast to Kant, who believed that man's inability to reduce mathematics to logic demonstrated that the former's status as "a priori."
However, Russell viewed the logic of relations simply as "the way in which the truth of certain sentences is entailed by the truth of others." Cassirer, Skidelsky claims, never understood that Russell was entirely focused on using the logic of relations to express "truth-value," while Cassirer "was interested in concepts not simply as components of possible propositions but as tools for the organization and mastery of experience." In this Cassirer was very much still in the tradition of Kant, with his belief in the supremacy of man's reasoning powers.
Skidelsky faults Cassirer for not having perceived the philosophical gulfs separating him from Russell and other, more popular proponents of symbolic logic. This counts as a very small fault in my book, however, since I believe that even if Cassirer had been aware of that gulf, it would not have changed his conviction in the power of reason to subsume all differences without eliminating them:
In contrast to Aristotle's generic concept, which obliterates all specific differences, the functional concept retains these differences; indeed, it generates them from out of itself (p. 68).
Anyone familiar with Cassirer will see in this summary the same attitude the philosopher displayed with respect to his "symbolic forms:" a wide-eyed, pluralistic appreciation of both diversity and unity.
I will note in passing that I find it interesting to re-conceive my own current attempts at taxonomy-- that of formulating the generic concepts that unify all forms of the heroic and superheroic idioms-- has probably been informed by the traditional logic of Aristotle. Still, my purpose in so doing is one that applies more toward a Cassirerean appreciation of unity in diversity, and vice versa, than to Bertrand Russell's concentration on "truth-value."