Early Work of Sheffer and Lewis (1908-1918)
Huntington’s Postulate Theory, Royce, and Absolute Pragmatism
Sheffer worked closely with the mathematician E. V. Huntington (1874— 1952) from the time he was a Harvard undergraduate (1902-1905). Huntington had been a doctoral student of Heinrich Weber, a friend of Dedekind’s, and wrote an early work on foundations of set theory (1905). He founded the American school of “postulate theory,” a branch of axiomatics. In Huntington’s view, assumed principles are “postulates” rather than “axioms” because they are taken to be arbitrary conditions for truth or demands, rather than self-evident truths or statements of obtaining fact. According to Huntington, the distinctive terminology of “postulate” was needed because the use of “axiom” “among German writers” assumes that axioms state facts, or self-evident truths, whereas “postulates,” representing the “modern, abstract” view of mathematics, express, “in Russell’s terminology” (as Huntington says) “propositional functions” rather than propositions (Huntington 1911,171).10 This view echoes Frege’s way of responding to Hilbert in his essay on the foundations of geometry (1906): Hilbert might call his axiomatic explorations “reinterpretations,” but in Frege’s mind this really means that Hilbert is working with second-level concepts, at which point, Frege argues, we are not dealing any more with an “axiomatic” system (Sheffer was especially taken with Frege’s essay.).11
Huntington’s view of the method of postulates left its mark on both Lewis and Sheffer: both reverted to the language of “postulates” when discussing the nature of logic (as did Quine and Carnap later on). The Harvard school in logic shared a certain distinctive conception, subscribing to what Huntington held, more broadly, about logic: logic can only prevent errors, playing a negative role, it cannot dictate truth. This view, Kantian about general logic but dubious about Kant’s transcendental logic, implies that the uses of logic require scientific imagination and cannot be prescribed a priori or inferred from other principles (Langer 1938). Logic is embedded, in its nature, in the evolving system of science,
Sheffer, Lewis, and the “Logocentric Predicament” 33 forming a branch therefore of scientific method: this view, derived from Peirce, lies at the heart of pragmatism.
In general, Sheffer was more preoccupied with finding a unitary foundation for general philosophy of science - from cosmology and physics to mathematics and psychology - than was Lewis. In this way they divided problems between them. Most importantly, from the time he was a graduate student Sheffer was Russell’s most enthusiastic representative at Harvard. Whereas Lewis was a pragmatic Kantian, Sheffer was a pragmatic Russellian: realistic, comprehensive in approach, looking to science, and critically naturalistic, though not positivistic. He was not inclined, as Lewis was, to think of logical empiricism as a threat. Like his friend Philipp Frank and other emigrés from the Vienna and Berlin Circles, Sheffer believed in scientific philosophy as a way of life. He continued exchanges with European mathematical logicians when he could, trying to keep up with the developments in Italy and Germany.12
In their salad days, Sheffer’s and Lewis’s responses to Royce and James bore certain affinities to one another. For Royce logic was the study of the various types of order. He regarded truth as absolute, beyond “human opinions, ideas . . . [and] . . . the effort of a live creature to adapt himself to his natural world” (Royce  1969, 684). Logic, on this view, expresses the absolute forms of thinking and truth as such. Influenced by Wundt and Lotze, Royce also taught Dedekind’s work on the foundations of mathematics, Peirce, Russell, and Principia Mathematica to his students. On Royce’s view the universe is infinitely rich and speaks to us in signs. Given the infinite richness of reality and the potential for signs to be misunderstood, logic is essential, for it affords us an absolute, “ahuman” perspective on reality, allowing us to surmount our human conventions of language through a generalized grasp of forms of order. For Royce, Kant had been wrong to suggest that formal logic was a completed science and wrong to think that Euclidean geometry held a privileged role in shaping the forms of human intuition. After the advent of the modern abstract methods, Kantian “intuition” had no further role to play in epistemology, because mathematics had become abstract, a matter of postulational technique. After Principia, Kant’s account of general logic had to be broadened to encompass modern mathematical logic.
Lewis and Sheffer followed Royce in many of these ideas. Their respective extractions of Roycean and Jamesian themes were sophisticated. Both were defending an updated response to Kant according to which the a priori categories of the understanding are dynamic and natural, rather than transcendental, evolving under the pressure of evolution. This however necessitated a rethinking of the nature of general logic, as well as “transcendental” logic. And in their hands, it necessitated a critique of Royce’s “absolute” pragmatism as well.
Royce’s famed “refutation” of James’s pragmatism, his “reaffirmation through denial,” claimed to have established the absoluteness, not only of truth but of logic (Royce  1969). This earmarked Royce’s “Absolute Pragmatism.” Royce argued that an affirmation of James’s pragmatic view of truth presupposes an absolute cleavage between denial and affirmation and thus reaffirms absolute truth through the very attempt to deny it. In response, Sheffer and Lewis relativized the notions of “denial” and “affirmation” to different systems of logic, thereby denying that Royce’s argument forces us to accept any absolute set of principles for logic. By drawing logocentrism into the very activity of doing logic, they thereby made the principles of logic part and parcel of the “subtle” phase in which the conditions of the validity and significance of basic notions, the “meanings,” are made clear. Lewis argued that since all logical systems embed their own version of the distinction between affirmation and denial, Royce’s argument cannot force us to accept any single set of absolute principles.13 Sheffer, as we shall see in Section 1.2, rejected Royce’s argument already in his dissertation (1908), again on purely logocentric grounds. Unlike Lewis, Sheffer retained Royce’s conception of logic as the direct and unfettered study of the most general types of order. He focused on achieving an understanding of the universe through forms, rather than through concepts and procedures, as did Lewis. The contrast between Lewis and Sheffer may be put, at least provisionally, in the following way.
Lewis was always interested in the intuitive-, the “given,” received sensory elements of human knowledge, which he contrasted with the generalizing, conceptual elements and our purposes in desiring to know - which are never pure (Lewis 1910, 26ff., 73). For Lewis, this contrast directly tracked the distinction between passive (received, “intuitive”) and active (made, “conceptual”) aspects of mind in its contribution to our evolving system of knowledge. Logical symbolism belongs with the pragmatically useful, evolutionarily conditioned, framework elements of our thought: it reflects conventional categorial choices. For Lewis, the so-called egocentric predicament is not a predicament, there being no point in claiming to prove that our representations “must” somehow correspond to a fixed “reality” (Lewis 1910, 26ff.). Royce’s “absolute pragmatism” must be abandoned.
Sheffer too regarded the “egocentric predicament” as nonsensical. But above all, at least until 1923, for Sheffer logic is philosophy: it is not one optional or a partial philosophical method, but something neutral among philosophical conceptions. All genuine philosophy and mathematics must rely upon logic or be “ineffable” (1908; 1909). In contrast to Lewis, Sheffer focused on the cosmological and the infinite - areas of physics, pure mathematics, and logic - where Lewis’s conventionalism about categories did not provide a guide. Logic, in Sheffer’s view, can provide a “foundation” for mathematics and physics, although - as he believed, in contrast to Royce - not any particular theory of the universe. Instead, it delineates the study of the different forms of order that actually obtain, given science.
Royce had taken the universe to allow us to read its complex structures with signs that are meaningful from the start. The difficulty, as Sheffer saw it, was that due to upbringing and education, some pieces of notation (e.g., those concerning the transfinite of Cantor) would be experienced by thinkers as meaningless masses of “printer’s ink.” Sheffer’s methodology would always “hitch [its] wagon to a star,” that is, a “meaning,” always distinguishing “real propositions” from “mere postulates,” never confusing them “in the pragmatic sense” (1908,115). The resonance with Frege’s critical remarks on Hilbert’s conception of the foundations of geometry (1906) are striking: for Frege too, any “reinterpretation” of fundamental axioms would take place within a higher-order, meaningful theory. Aiming at a comprehensive treatment of an infinitely rich universe through abstract algebraical form, Sheffer hoped to surmount the limitations of human procedures and concepts with a generalized theory of possible orders, in terms of which one could then mathematically articulate the ultimate logical structure of our best scientific theory of the universe, the universe’s true order.
Given this Roycean conception of logic, notions of form, order, infinity, and notation were embedded in the heart of Sheffer’s work from the start. An essentially a-metaphysical, pragmatic, and scientific mind, he rejected Royce’s idealistic cosmology, just as Lewis did. But he followed Royce’s and Russell’s philosophies of logic and mathematics more closely than did Lewis. Sheffer regarded logic as necessary for handling the infinite; mathematical work was needed to be able to comprehend an “ahu-man” form of truth.
For Sheffer this implied that logic’s forms had to be characterized in a way that would surmount the vagaries of human notations, concepts, conventions, procedures, and symbolisms. How would it do this? Not by identifying and analyzing concepts or procedures that are fundamental, as in Lewis’s Kantian approach. Not by taking the notion of “concept” or “human logical procedure” as basic in epistemology at all. Instead, Sheffer proposed that logic would abstract “forms” or “patterns” of invariance in knowledge. For Sheffer, as for Lewis, Royce’s Absolute Pragmatism needed to be subjected to criticism from the side of what is “given” in knowledge. But the role of the “given” in their respective philosophies, as well as their view of the “givenness” of logic itself, differed.
Sheffer sought a critical naturalism, that is, a non-reductive, pluralistic system of the cosmos articulated in logic by means of a generalized postulate theory (eventually, “superpostulates” for postulate theory itself) reflective of the results of empirical science.14 In this sense, like Quine later on, he unfolded the filaments of Russell’s approach to philosophy throughout his life, developing a kind of cosmic perspective on what there is. In the end, he held - as Lewis never did - that “we may define philosophy as the science which attempts to get postulates which shall embrace the widest possible field, - the Universe” (Sheffer 1918, 64).
He believed in an empirical study of “implies,” and to that extent, he regarded himself as a pragmatist (Sheffer 1922, 40). In particular, he agreed with Lewis’s criticisms of Russell’s claims to have fully analyzed the “meaning” of “implies,” citing Lewis’s 1921 paper approvingly, and holding that it is “plausible,” given Lewis’s arguments, for us to speak of “logics” in the plural (Sheffer 1922, 62). He gave his students an introduction to “strange logics,” analogizing them to “non-Euclidean logics” and telling his students that they could “make logic as strange as you please - but keep it logic” (1922, 61ff.). Yet he also fashioned a kind of Russellian response to Lewis’s criticisms of Russell, denying that Lewis himself had succeeded in analyzing “implication” either (1922). (We shall discuss these remarks later in Section 2.2.)
In the end, Sheffer wanted to try to get past “analysis” of concepts, digging deeper. He took features of logical categorization and the nature of types to reflect “formal properties,” but these he regarded as ultimately “empirical” in nature, even though also “ahuman” in the sense of invariant with respect to alternative representations of the facts.15
On Sheffer’s view, postulates are to be tested empirically, by way of the theories whose forms they articulate. In this he follows Principia, where it is explicitly stated that the ground for holding the particular axioms of logic to be basic is their ability to comprehensively deduce mathematics -that is, the ground is in the end “inductive” (Whitehead and Russell 1925, Preface, v). Moreover, it is an “empirical” matter to determine which orders of relations are actually exhibited in reality (Whitehead and Russell 1925, Introduction, xv). In Sheffer’s generalized postulate theory is holistically beholden, even if indirectly, to physics and biology and the other sciences.16 Sheffer’s theory of postulate theory - “logic” - would be a kind of higher-order synthesis which would not be reductive due to the hierarchies and types that - wary of paradox - he knew lurked if one aimed to define one level wholly in terms of another. Sheffer lectured comprehensively on classics of philosophy of science from early on, always emphasizing the holistic role of hypothesis and revision in science, the intertwining of deduction and induction.17 Unlike Lewis - and more like Quine later on - he was a staunch aposteriorist about knowledge, even about logical and mathematical knowledge.