Sheffer’s Dissertation (1908): “Neutral” Logocentric Philosophy
Sheffer’s dissertation of 1908, “A Program of Philosophy, Based on Modern Logic,” was signed by Royce, Huntington, and psychologist and realist E. B. Holt (a psychologist and protégé of William James).18 The areas on which Sheffer focused in his dissertation were philosophy of logic, mathematics, and philosophy of mind. Holt credited Sheffer with developing a “neutral” theory of mind as a philosophical position; Russell
Sheffer, Lewis, and the “Logocentric Predicament” 37 would later adopt it.19 Although the term “neutral monism” is popular today, we shall see that Sheffer’s view was logocentric and neither monistic nor materialistic.20
Sheffer’s philosophy was an original synthesis of Huntington’s “postulate” theory, Russellian logic-centered philosophy, Royce’s interest in order, infinity, and the algebra of logic, James’s empiricism and introspective study of psychology, and - after he journeyed to the Continent in 1911 - philosophy of science and mathematics (Mach, Duhem, Poincare, Einstein). His program aimed to recast the foundations of logic and mathematics in light of algebra, incorporating the whole into a foundation for all of science, including physics, biology, and psychology, and even sociology and economics. This was of course kindred in spirit to the general aims of the Vienna Circle, avant la lettre.
Very much in the spirit of Russell’s essays from the early 1900s rejecting mysticism on the basis of logic, Sheffer argued trenchantly against philosophies claiming to find foundations for logic lying outside logic itself. In his dissertation he drew a strong contrast between the “Method of Temperament” (James) and “The Method of Reason,” arguing that all philosophy, insofar as it must be written in propositions and argued for, should avail itself of logic.21 Logic, rather than forming an organon, is a canon, just as Kant had urged; but for Sheffer it is constitutive of philosophy:
To the most casual observer of philosophical doctrines it will be evident that we are sadly in need of a general discipline which shall decide what a philosophy can and cannot accomplish, precisely as the canon of a special science outlines the limits of the method and procedure of that science. To illustrate by examples from the common world - one cannot be a carpenter without strictly following the rules of carpentry; one cannot play chess without limiting one’s self to the rules of the game; likewise, without definite rules of procedure, one cannot play - if you please - the game of philosophy.
(1908,3-4)
Sheffer used this “neutral” logocentrism to attack the “neo-Fichteanism” of Rickert and Miinsterberg, which, he argued, took judgment to be capable of transforming reality through will-acts. He also rejected Hinduism and Emersonian transcendentalism, as well as all forms of what he called “Completion Philosophy,” the Idealism of Joachim and Hegel. He rejected Bergson’s philosophy of “immediacy,” in which all logical content is imagined to be stripped away from experience altogether, anything subject to logic becoming regarded as a mere “abstraction.”
Sheffer’s overall strategy was to argue that none of these views has any way of arguing for the fundamentality of their approach, and each generates unanswerable questions about the nature of reality. Feelings and will-acts are themselves abstractions, from the “neutral” point of view. Which are we to say is more fundamental, propositions or will-acts or experiences? Sheffer insists that philosophy must be carried out rationally, and therefore in propositions, if it is not to be a matter of “temperament.” His neutral, logocentric philosophy develops a conception of logic as a matter of method designed to encompass, pluralistically, all views, including those of “temperament” as treated by psychology. The activity of logic is to replace pure metaphysics by logic, and the use of logic is to classify the foundations of all sciences, including psychology.
Sheffer’s dissertation investigates two ways of structuring the “neutral” (logical) realm: mathematics and mentality. In his philosophizing about mathematics he developed critical responses both to Dedekind’s (1888) structuralist approach to systems that regarded the numbers in terms of the idea of a “free creation of the mind” and to Russell’s “if-thenism” about mathematics. For Sheffer, our knowledge of mathematics is a matter of discovery, of “selection according to an order” from within the neutral realm, it is neither a “free” creation nor an axiomatic stipulation nor an appeal to any “self-evident” deliverance of intuitions, concepts, laws, or axioms. This reworked and logicized James’s view of knowledge as “selection” as expressed in his “Pluralistic Universe” (1909). Sheffer’s “selective” structuralism of multi-dimensionally striated form is what he had in mind by a method of logic that would provide a “neutral,” non-reductive philosophy that is at the same time realist. Lewis too would adopt a picture of “selection” from a wider space of what is “given” in his dissertation as opposed to the idea of imposing human forms, but he would do so by denying that any meaningful knowledge - including geometric or arithmetical knowledge - can be “pure,” that is, wholly abstracted from the intuitive materials of experience (Lewis 1910, 26, Appendix).
Throughout his life Sheffer was acutely aware that as the number of logical primitives and axioms are minimized, the surveyability of notational structure for particular sentences, as well as the very idea of the range of application of logic, becomes explosive and potentially unmanageable. (Consider what happens when ordinary paraphrases of statements into truth-functional notation are fully written out using only one of the Sheffer strokes: an explosion in complexity and length in representing information occurs.) His postulational approach, dedicated to extracting form from extraneous detail in notations, was directed at surmounting too-fine-grained structure, getting to truths about order relations. Postulates allowed for the idea of an ultimate ordering principle for a domain, thereby allowing us to extrude arbitrariness from our formalization. They allowed for pluralism in logic, while adhering to logocentri-cism. However, ultimately Sheffer had to struggle with the fundamental logico-philosophical problem of the entanglement of logic in this sense with human-designed symbolic instruments.
As a mathematician and student of Russell’s and Peano’s, Sheffer well appreciated the importance of controlling ideas by means of symbolic cognitive instruments - especially those about the infinite. His approach was, like Royce’s, to develop a structural conception of logic, but not implemented in a top-down way, through a prioristically stipulated structures of “axioms,” or “self-evident truths.” Instead, empirical theory would have logical form extracted from it, bottom up, using a generalized form of “postulate theory.” “Forms” and “laws” would be compared and examined for symmetries and asymmetries, as well as dualities and invariances, of order. The deliverances of science would lead, and logic would abstract and follow.
Lewis and Sheffer shared certain interests in epistemology. Both were greatly preoccupied with the question of how much of what we experience and perceive is “superadded” by us (Sheffer 1922, 65). Huntington’s postulates for sphere-inclusion, for example, are superadded, because, as Sheffer said, “we speak of geometry through channels of a human language - that’s added” (Sheffer 1922, 65; see Huntington 1913, discussed in Mancosu 2006,214). Both took it that what is “given” is experience: perception and results of scientific theorizing - as well as concepts and propositions - are part of this channeling. But everything in philosophy, from their logocentric point of view, belongs already to the context of articulation. The “meanings” are implicit in what is “given” and need only be made explicit. The question is how best to do this.
For Sheffer, unlike Lewis, mathematical logic was the scientific instrument by means of which this explicitness would be achieved in the most comprehensive manner for logic. “We in philosophy,” he said, “are beggars: [we have] few concepts” and need help (1908, 66). He believed that one would need to penetrate beneath the surface of atomic and “molar” [molecular] analysis of propositions, finding further structural forms and relations. By focusing on what is “superadded,” the issue is how to get at the foundations of logic itself. “If we realize,” he told his students, “that in talking of logic we assume all the paraphernalia of logic, etc. then, [it is] as feasible to state the foundations of logic fundamentally in [the] molecular as in the atomic level,” and the question will be whether one way is simpler (1908, 65).
The idea was to use logic to rid ourselves of metaphysics - a Kantian theme. As Sheffer wrote already in his dissertation:
As for our starting point of philosophy, with regard to the “given,” we may as well be frank and confess that all philosophies without exception beg this question of givenness. However we may desire to circumvent the fact it remains one of the most solid truths of philosophy that the primal starting point, the data, we simply accept. Why there should be being rather than non-being we do not try to explain; that there is being we all grant. Hence neither [my] neutral philosophy nor any other can reasonably be reproached for begging the initial states. “Something is.” “There are entities.” Behind some such starting point no sane philosophy attempts to go. And there is nothing more wonderful in the proposition “things are” than in the proposition “God is,” or “The Absolute Mind is.” With regard to this given the only questions are: How are we to conceive this given; what is the method of procedure with regard to it?
(1908,68-69)
These remarks presage Lewis’s discussion of “data” and “the given” throughout his life, culminating in Mind and the World Order (1929). But the method, first, last, and always, was that of logic so far as Sheffer was concerned. We seek order but cannot presuppose or demand or stipulate when we begin precisely what we will find. Nevertheless, we articulate forms and organize the results of our theorizing. In a sense, like Russell’s second-order “propositional functions,” Sheffer’s “forms” are higher-order structures, not merely organized fashionings of first-order concepts and objects (we are reminded here also of what Frege (1906) held the true study of the foundations of geometry would have to be: a second-order study of concepts, not the study of reinterpretations of first-order theories). Sheffer declared “monism” a “purely metaphysical” doctrine and argued that it is not based upon science as it stands. A metaphysical doctrine might be speculative, but in no way foundational: it cannot get back behind logic. If materialism were true, Sheffer said, we could not know it, for if everything is merely atoms in motion then the arguments for and against Materialism are equal; “materialism,” he said, “is as dogmatic as any philosophy” (1918-1919, 12).
As a student of James Sheffer had to grapple with the idea of “temperament” in philosophy. His dissertation shows, in fact, far less sympathy with James’s idea that this matters in philosophy than did Lewis’s.22 The first part of his dissertation concerns types of philosophy and basic approaches. According to his logocentric philosophy, logic comprises all positions in philosophy insofar as they use the method of “Reason” instead of “Temperament.” Logic is not one method among others, it is the only method. Bergsonianism, Russellian views about analysis, idealism, and phenomenology are to be considered and scrutinized; but they all are placed at a different level than logic.23 It alone may inspire a “neutral” philosophy, as Sheffer argues. As he wrote:
We find ourselves in a world of being: in a world of entities of all kinds that simply are. Of this world we may well ask, Is the heterogeneous mass of details anything like a deductive system, is there any unity of orderly arrangement in it? There may be. But the unity and rationality we must not presuppose or predetermine; we must actually find it. We may, to be sure, leave all this mass of hopes, fears,
Sheffer, Lewis, and the “Logocentric Predicament” 41 chairs, triangles, logarithms and trolley cars, just as we experience them and say nothing more about them. Or we may wish to understand them, to reason about them.
As soon as we do regard this set of entities deductively - as soon as we try to philosophize about it and to utter rational statements concerning it - we must proceed according to the canons of deductive reasoning. And to inquire at this point whether we shall adopt one method or another, betrays an ignorance of the fundamental methodology involved. We cannot overemphasize the apparently simple fact that while each of us has the option of philosophizing or not, as we see fit, the moment we have chosen the lot there is no further choice in this matter insofar as the method of procedure is concerned; there is but one method, the logical.
(1908, 67ff.)
This point included philosophy of mind. Sheffer’s conception of mind is neither “monistic” nor “materialist” nor “vitalist” but “neutral”: an aspectual conception of mind and body reminiscent of Spinoza’s but without any underlying substance of God - hence also differing from Royce’s Absolute Form of Idealism. According to him, “mind” and “matter” are undifferentiated until postulates describing their modes of existence are formulated. There is no expectation that the two “aspects” of reality will be one-one correlated with respect to their concepts.24
In his dissertation Sheffer developed this neutral view of “mind” through generalized postulate theory. The idea was to rid ourselves of discussions of the nature of “substance” altogether by regarding mind and matter through the lens of the notion of definability in the sense of order relations revealed through the formulation of a set of postulates with respect to a “base” of elements (facts, objects, experiences, concepts, propositions, or relations):
to state the problem of fundamental entities of philosophy in its strictly logical form, we are considering the problem of the relation between the entities a, b, c, etc. of a given class K (a, b,c,...) and that class K itself. For instance, if the elements a, b, c, etc., are mathematical points, and the class of those elements K (a, b, c,...) is defined as a circle, we should be asking the question of fundamental entities in mathematics if we inquired whether these elements a,b,c,... should be spoken of as circle-entities, or as objects composed of circle-stuff. Similarly, we might ask the question for the class K (a,b,c,...) which defines a triangle. Should we say that its elements are composed of triangle-stuff, or that they are triangle-entities because they happen to be defined by this particular set of propositions?
The problem being thus stated, we see how futile and meaningless it is to ask for the substance of which mathematical entities are composed. And not one iota different is the logic of the situation when we ask for the substance or stuff of which any entities, physical, psychical, or other, is composed. The perception of an object is indeed a mental entity, but so is a point a circle-entity when it happens to belong to a particular class K.
“Every point in this manifold belongs to this circle.” “Every content of my psychical experience belongs to my psychical experience.” Each of these sentences expresses a true proposition when, and only when, we take it in the collective and not in the distributive sense. Collectively, all the points compose the circle; distributively, it is not true that each point is composed of circle-stuff. Again, collectively, all the content of my experience constitutes my experience; distributively, it is false to assume that each smallest particle within my experience is therefore composed of experience-stuff [Note: See Holt’s forthcoming (1914)].
(1908,54-56)
The logical way of thinking is “neutral,” on this view. We may infer from the proposition that “there is an even number greater than 4” that “there are even numbers,” but the conclusion does not yield anything metaphysical insofar as we view it neutrally, in terms of forms, from within logic. Metaphysics there may be, but it is subservient to the interests of science as a whole and not determined through dialectic and speculation a priori.
What we seek is a systematization of concepts to characterize the “submanifold” of the neutral realm that is “mind,” or that is “matter,” distinguishing them from one another. Here we utilize the concepts that characterize their classes K (a, b, c,. . .) and K’ (a, b,c,. . .). Finding the “fundamental formulae or laws which differentiate the class of mental entities” is an “empirical problem,” not a speculative one (1908, 115). For we must know the “facts” in order to define a “sub-manifold,” systematically “correlating” the facts in order to find a “guiding principle which will yield the class K (a, b, c, . . .) we are searching for” (1908, 115-116).
According to Sheffer, mental relations and truths couched in terms of external relations among objects of psychology are selected from a “neutral” realm which is neither mental, mathematical, nor material:
We are such stuff as dreams are made of. Exactly: this point is the one to be emphasized . . . that psychical phenomena, perceptions, sensations, ideas, volitions, and similar complexes are really and truly such stuff as dreams are made of. Dreams, hallucinations, normal and abnormal mental phenomena are alike constituted of the same neutral stuff. ... In a neutral philosophy the important problem is simply to find in the light of all experience and exact knowledge, the conditions, characteristics and properties which distinguish this sub-realm, this selection of neutral entities from all others. As for the
Sheffer, Lewis, and the “Logocentric Predicament” 43 content of consciousness, this content is composed of mathematical, logical and physical objects. The one problem for us is to find the exact relationship or K-property expressed by the word mind or consciousness. We must be willing to wait patiently for all the facts to come in, we must wait for scientific results which will show wherein a stone and the perception of a stone do actually differ. Let it not be said that this will be merely a psychological and not a metaphysical result; this supposed distinction we cannot allow to hold. If science reports the fact, that is the fact in so far forth, and all that philosophy can say about it is only with regard to its proper place in the total sphere of facts or truths. It cannot reject any result which science has actually discovered.
(1908,112)
Sheffer’s idea of the “neutral” perspective is grafted on to ideas that appeared in James’s paper “Does Consciousness Exist?” (1904), a view on which mentality is constituted by objects in relations, rather than qualitatively irreducible qualia or properties (1908, 64-65). Sheffer generalized and logicized James’s position, holding that physical relations, couched in terms of external relations among bodies, are constituted by different arrangements of experience, properties, and laws than those belonging to the mental or mathematical realms. Entities are given through experience and domains characterized through conceptual structures of the propositions that define, characterize, and individuate them. Truths are ultimately discovered through characterizations of experience in language with structurally robust, formally delineable contours. Of course, “complete tabulations” of the concept of these realms may not be forthcoming, but that is an empirical question (1908, 117). Sheffer mentions Perry’s definition of mind as “any neutral entity which is complex enough to exhibit the K-properties which would characterize what [Perry] terms an ‘interested action’” (1908, 118). Sheffer does not take sides here, but focuses on methodology. He points out that “a falling stone” and a “quaternion” are not mind because “their defining properties are simpler than those of mind” (1908, 119).
The “mind” is thus a “submanifold” of the neutral realm, as are the “submanifolds” of “mathematics” and “physics” (1908, 85). We reveal these submanifolds through postulates, which systematize the experiences we characterize and the entities we thereby discover. The point is “selectivity,” rather than “consistency.” First, differing selections of submanifolds might, on the surface, create apparently “inconsistent” statements. So we need to attend to the notational methods by which we grasp and represent the bases of knowledge in their formal relations. “The truth is,” Sheffer writes,
that the postulate-maker is as much and as little free as the artist is. The artist is free to choose and select his material from a given set of already-existing objects; and just so the mathematician is free to make selections from among already-existing entities.
(1908,107)
Sheffer fully realized the difficulties posed for his position by the Canto-rian transfinite. Building on his reading of the early set theoretic work of Hessenberg (1906) and Zermelo (1908, 109-110),25 he writes that “the worry about “Mengelehre,” is that
here, if anywhere in mathematics, we are dealing with what may turn out to be nothing but counters. The simple infinite is bad enough; for the transfinite we have not even a shadow of possibility of every being able to prove the consistency - that is, the existentiality - of its defining postulates.
(1908,109-110)
On Russell’s (1906) view of truth - which Sheffer cites approvingly in his thesis (1908, 36-37) - truth and falsity consist in the holding and not holding of external relations amongst a particular subject, particular relations, and particular objects. But the transfinite raises the prospect that such relations are barred in principle from any possible connection with actual procedures, judgments, or empirically observable events or laws, or even any forms we can take to determine the principles. It would seem as though no physical theory could ever utilize a transfinite number. That would entail, conceptually, the “meaninglessness” of set theory. Without a metaphysical jump - one that Sheffer was disinclined to make - the transfinite would collapse, on his view, into a mere game. Throughout his logocentric career, Sheffer envisioned a possible answer. Only if its “forms” could be incorporated into a general theory of the world and purged of all “human” procedures would the “existentiality” of the set theoretic postulates be revealed to us through truths.
Lewis largely ignored the transfinite, leaving the difficulties to Sheffer. He admitted, at least by 1932, that the infinite poses formidable problems for logic - at least “theoretically.”26 As we shall see, for Lewis, the human perspective of doing in the sense of ordering experience with symbols was central to logic, and in fact to mathematics also. So it was with Sheffer as well. However, Lewis was content to regard such procedures as fully captured in step-by-step recursive procedures or operations unfolding in time unidirectionally - hence in a linear alphabetical script. Sheffer doubted that this perspective would prove adequate.
In his dissertation, Sheffer used his logocentric philosophy and technical prowess to reject Royce’s argument for the necessary consistency of his S-system, the postulate of “the Fecundity of the Will” (Royce 1905). He did not argue that Royce had failed to close the door to alternative
Sheffer, Lewis, and the “Logocentric Predicament” 45 logics, as Lewis would do in his (1921).27 Instead, Sheffer argued directly against the whole idea of will-based definitions.
Royce regarded logic from the following point of view:
There is ... a world of ideal objects which especially interest the logician. It is the world of a totality of possible logical classes, or again, it is the ideal world. Equivalent in formal structure to the foregoing, but composed of a totality of possible statements, or thirdly, it is the world, equivalent once more, in formal structure, to the foregoing, but consisting of a totality of possible acts of will, of possible decisions.
(Royce 1904, 459-460; analyzed in Scanlan 2010, 188ff.)
Royce defines the O-relation as “the relation in which any set of exhaustive, but, in their entirety, inconsistent choices would stand to one another” (1905, 359). The order of terms in any O-collection may be varied at will and one still obtains an O-collection. In this sense, an O-relation is very unlike our present-day idea of a relation, in which every «-element sequence is conceived either to fall or not to fall in the extension of a relation. An O-relation in Royce’s sense is “polyadic,” and Royce frames postulates so as to define the characteristics of any O(pqrs ...), in which p, q, and r are thought of as separate acts. The domain of entities and collections of such entities are completely unspecified and instead “will-acts” form the basis. As Sheffer says (1908, 102), they are “exactly the postulates which define a system of will-acts: not your acts or mine, but the general system which makes any action possible.” The idea is that the O-relation is symmetric: if O(pqr) holds, then O(qrp), O(prq), etc., and all other combinations hold. If O(pqr) fails to hold, so does O(qrp), and so on (Scanlan 2010,179).
Royce’s S-system consists of simple, homogeneous elements that may be collected with order in a variety of ways; collections may contain a finite or infinite number of elements; and any element may be repeated any number of times, so that (to use our later notation) {x} and the “multiset” {x, x} are both collections. “Systems” are collections. All collections are either “O-systems or “E-systems.” An “E-system” is not necessarily closed under permutations of its elements, so there is a kind of orderentanglement among the possible relations on elements of its domain, whereas an “O-system” is closed under the permutations of its elements. In Royce’s system, the logic of classes is defined from the postulation of certain O-relations, and then the principles of classes are derived from the postulates of the L-system. The S-system allows for representation of polyadic relations in terms of symmetrical O-relations that express whole sets of so-called F-relations, created through operations such as separation, division, and obverse-operations on sequences of elements.28
Royce took himself to be elucidating the Peircean relation p -< q, taken in the Algebra of Logic tradition to model a variety of relations, or, in Royce’s terms, “orders.” He holds that wherever we have a linear relation with an “origin” of coordinates - for example, in relations of cause and effect, ground and consequence, orientation in space or direction of tendency in time - we may symbolize with the dyadic asymmetrical relation p -<y q, where “y” is the “origin,” p falling “between” y and q and coming “before” q in the ordering, the relation being asymmetrical (Royce 1905,381).
Royce claimed to be able to reduce, with one technique of defining in terms of his postulates for a general O-relation, any asymmetric relation to a symmetric one in his ^-system. His proof involved an operation of “choice”: reassigning a starting point by way of an initial permutation. The idea is that an unsymmetric ordering relation can then be defined on the single symmetric O-relation.
This, Sheffer showed in the Appendix to his dissertation, begs the question. Assuming that such a starting point exists and that one makes the choice of an origin is already tantamount to the result.29 In the end, the motto must be, Sheffer says, “once asymmetric, always asymmetric” (Sheffer 1908, 171; see Scanlan 2010 for a detailed reconstruction of the argument). Sheffer’s use of the methods of postulates shows the power of a modern, abstract and algebraical conception, that is, of modern axiomatics, to squeeze out any inexplicit principles and exhibit the logical architecture of proof explicitly (Scanlan 2010, 181-182).
Sheffer granted that we “undeniably find” a system of will-acts in experience, so that Royce’s Z-system does represent “neutral propositions,” and not “mere postulates” (1908, 115). But “its existence is empirically found,” rather than established by an “reflective-ontological proof.” Sheffer thus rejects Royce’s “Absolute Pragmatism” as an a priori argument, based, as it is, upon an absolute conception of negation.
The idea of Royce’s argument for “reinstatement” of absolute truth “by denial” is this:
[W]hatever actions are such, whatever types of action are such, whatever results of activity, whatever conceptual constructions are such, . . . the very act of getting rid of them, or of thinking them away, logically implies their presence, are known to us indeed both empirically and pragmatically . . . but they are also absolute. And any account which succeeds in telling what they are has absolute truth. . . . What are the necessary “logical entities,” and what are their necessary laws? What objects must the logician’s world contain? What order-systems must he conceive, not as contingent and arbitrary, but as so implied in the nature of our rational activity that the effort to remove them from our world would inevitably imply their reinstatement?
(Royce 1913, 122, 125; quoted and discussed in Scanlan 2010, 197-198)
In response, Sheffer begins by outlining what Royce means by a “reflective” proposition (1908, 101). Examples he gives are:
There is (at least) one proposition.
There is (at least) one truth.
There exists (at least) one process of thought. (“Cogito ergo sum.”)
By the “reflective” method, Sheffer points out, we can also prove the falsity of:
There are no propositions.
There are no truths.
All universal judgments are true.
No knowledge is possible.
As Royce himself wrote ([1908] 1969, 700),
The contrast is not one between intellectualism and pragmatism. It is the contrast between two well-known attitudes of will, - the will that is loyal to truth as an universal ideal, and the will that is concerned with its own passing caprices. If I talk of truth, I refer to what the truth-loving sort of will seeks. If hereupon I define the true as that which the individual personally views as expedient in opinion or in assertion, I contradict myself, and may be left to my own confutation. For the position in which I put myself, by this individualistic theory of truth, is closely analogous to the position in which Epi-menides the Cretan, the hero of the fallacy of the liar, was placed by his own so famous thesis.
Sheffer is interested in Royce’s method of philosophizing. As he characterizes it - drawing on Royce’s (1905) - it involves the two postulates:
(1) To every act exists its opposite;
and
(2) To any two non-contradictory acts there exists at least a third act.
Sheffer takes Royce to hold that these two postulates are
proved consistent by the evidence of empirical experience - the actually existing relations of our will-act system. By trying to deny this O-system of will-acts we only affirm its existence. If we doubt the consistency of the Z-postulates our doubt proves it. Hence, as soon as we observe that the S-system typifies the system of will-acts of any kind, we have here, according to Professor Royce, a proof of the consistency of the system by the reflective method. For whoever affirms or denies, or thinks in any way, thereby affirms the existence of such “yes-no” entities as the Z-system defines. In short, according to this interpretation the existence of the ^-system gets itself established by a kind of “ontological” proof: the mere definition of S-system proves its consistency by the reflective test that:
[(x -< x) -< x]30
Thus we ought to be able to say of any set of postulates whose consistency we are trying to prove: this illustrates the Z-system of will-acts, and therefore it is consistent to think it and to create it, since if we could not do this we could not do anything.
(1908,108)
Against this argument, Sheffer adduces Frege’s criticisms of the whole idea of “creative definitions” in volume II of the Grundgesetze der Arith-metik (1893-1903, §§143-144). He points out that a consequence of Royce’s position is that all mathematical objects are “will acts,” whereas, he insists, a mathematician does not “create,” but “selects” from experience (1908, 114). This means, of course, that in some way Sheffer needs the notion of what we would later call a model, or at least an idea of will-independent realities of truth, as well as a notion of verification. These notions sit uncomfortably, though, with the Roycean view of logic as a study of all possible forms of order. So, what Sheffer did do was to strip Royce’s idea of “loyalty” to absolute truth of its metaphysical and ethical basis, bringing the Roycean view of logic closer to Frege and Russell by way of the de-psychologizing of logic and the generalizing over arbitrarily complex “forms.” The aim was “propositional reference,” or truth in that sense.
Sheffer writes:
When a postulate-maker like Huntington searches for an “existential” system, usually the system of positive integers, which shall verify the consistency of his propositions, he is in reality asking himself the question: Have I here a set of entities selected from the total neutral realm by a definite generating principle - expressed by my “postulates” - or have I only a mass of symbols, a collection of “printer’s ink”31 which, to be sure, has always being as some neutral entity but may or may not have neutral being as a proposition with propositional reference?
By such considerations we are led to the true meaning of the “Problem of the Consistency of Propositions.” What we are really seeking is not consistence but existence. What we want to know about our “postulates” is whether they are neutral entities actually found in the neutral realm, just as we find gold in a piece of ore, or whether on the
Sheffer, Lewis, and the “Logocentric Predicament” 49 other hand they are only “free creations,” that is, a string of words, a “voc praeterea nihil”.
(1908,108)
Like Frege, Sheffer reduces formalism to meaninglessness by treating it as a thesis about marks on a chalkboard.32 “Consistency” is not a notion Sheffer puts much store in, as it is to be expected that we will see a variety of differing forms, and in a pluralistic and “neutral” conception, we should be careful about taking the surface of words or symbolism in one theory to “contradict” those of another theory. Instead, we should compare theories by way of the more abstract approach of logic. This will require analysis.
In general, he believes,
We are searching for a definite classification and labelling of the parts of reality. And philosophy should at any stage represent as complete an inventory as possible of the facts of our experience. Indeed, when a man says “I do not care for the logical philosophy,” he implies that he might have a non-logical philosophy. But such a non-logical philosophy belongs to the realm of Alice in Wonderland; when taken seriously, it is a misuse of language.
Language, we often hear it said, cannot do full justice to the facts at hand; language is at best only a poor instrument which we must employ, realizing all the time that it falls far short of fully serving our purposes.... But we must either make use of language, or if language does not measure fully up to our requirements, the proper attitude to assume is not to use a particular species of language but to quit talking or writing altogether. For life and philosophy are not identical, and for the purposes of active life we need not at all have any philosophy. Granted, however, that we wish not merely to live in the immediate present but also to analyze our experience; that we desire a rational, logical, that is to say, philosophical account of the elements of our experience, it follows that we must limit ourselves to the restrictions of the philosophical and analytical method.
(1908,7-8)
Sheffer always insisted that we may not assume, a priori, anything about the order of the universe or its constitution vis a vis our human powers: not that it will add up to an intelligible whole, or in particular, not that our postulate systems may be final or complete, not that our a priori demands for cohesion are wholly rational. Here he and Lewis agreed. What we can come to see through exploration and scientific work are meanings that are there to be discovered in the neutral, “given” realm. Thus there is “immediacy,” but it is never given to us through a special power of intuiting that somehow eludes logical articulation. This is so especially because there are diverse ways to systematize even our logical experiences. Postulate systems, consisting of algebraical aspects of true propositions, are utilized to formulate theories of the universe, that is, “truth demands” or general conditions as to forms of order and invariance.
On Sheffer’s view, psychology was not preempted from analyzing logic. Truths, insofar as there are any, are complexes of elements of subjects’ judgment facts, and these are presumably facts figuring in the mental system extracted from the neutral realm. Following James, Sheffer takes it that an “if-feeling” is a way of allowing for the “reality” of logic (1908, 65). As he writes,
What a logical philosophy is trying to do is not to beg the question of mind, and not to revert to the feeling of immediacy - and thus to repudiate itself as a philosophy - but fairly and squarely to present a definition of mind in the only terms in which anything can be defined, namely, in terms of the logical propositions, the fundamental formulae or laws which differentiate the class of entities called “mind” from all other groups of neutral entities. And this assuredly cannot be accomplished by dialectical speculation alone. It is a purely empirical problem.
(1908,116)
Here Sheffer mentions Holt’s Concept of Consciousness (1914), which was just being finished as he wrote his dissertation:
[Holt] defines consciousness or mind as “the sum total of all the entities in our neutral realm which a nervous organism specifically responds to.” This specific response we must in each case empirically discover. For instance, if we wish to discover the total extent of the consciousness of a mouse, we must empirically present to it all kinds of stimuli and note those to which it definitely and specifically exhibits a response. If I present it with a certain food, it will react to it in a characteristic way. If I present to it a mathematical theorem or a religious dogma, it will simply take no cognizance of it, from which we empirically conclude that these latter entities are not a part of its consciousness.
(1908,116)
Russell’s multiple relation theory of judgment requires no “correspondence” to an existing physical, mental, or mathematical fact when a true judgment is made: only the being or reality of the fact.33 This is its great advantage over all forms of representationalism. Nevertheless, to compare realities (say between the mental and the mathematical or the physical realm) requires further principles to be articulated, on Sheffer’s view: we must move beyond Principia and gain a wider perspective using all the empirical work of science alongside generalized postulate theory, which will bore down below the level of Whitehead and Russell’s “elementary propositions,” allowing us to taxonomize the variety of possible orders.
