Types of Order and the System Σ

Types of Order and the System Σ

By: C.I. Lewis

It is a commonplace of current theory that mathematics and exact science in general is capable of being viewed quite apart from any concrete subject matter or any system of physical facts to which it may usefully be applied Geometry need not appeal to any intuition of spatial complexes or to a supposititious space form; it has no need to rely upon diagrams or make use of ‘constructions’. Arithmetic makes no necessary reference to the sensible character of collections of marbles or of areas. Dynamics does not require the dubious assumption that the ‘moving particles’ of which it treats are possible of experience or verifiable physical entities. The ‘points’ of geometry and kinematics, the ‘numbers’ of arithmetic, and so on are simply terms x’s, y’s, z’s, entities, anything,—and the question what concrete things may be successfully regarded as such x’s and y’s is a question of application of the science, not one which need be considered while the system itself is in process of development.

If considerations of usefulness and of application are important in determining what assumptions shall be made or what systems developed, still such pragmatic considerations are principles of selection amongst actual and possible systems, and not internal to the systems themselves.

An arithmetic, a geometry, a kinematics, is thus capable of being viewed simply as a complex of relations and operations (relations of relations) which obtain amongst entities the nature of which, apart from those properties which follow from the relations assumed, is wholly indifferent. Such a system may in fact admit of various interpretations and applications more or less useful, all of which satisfy the requirement that these relations and operations be valid. As Professor Royce is accustomed to put it: a system of science is a type of order, the distinguishing characteristics of which are the kind of relations—symmetrical or unsymmetrical, transitive or intransitive, etc.,—which obtain among its terms, and the relations of these relations, by means of which the terms are ‘ordered’ and the relations ‘transformed.’

The growing recognition of the advantages of so viewing systems of pure science is one of the prime motives for the present interest in symbolic logic, or logistic. For logistic is the science which treats of types of order. One may reach the particular type of order which it is desired to portray—the arithmetic or geometry—by further specification of that minimum order which must obtain among entities if they are to ‘belong together’ in a set or system—the order of logic. This can be done in a variety of ways, which may be roughly divided into two groups. These two methods are distinguished by the fact that in the one case the ‘numbers’ of arithmetic or ‘points’ of geometry are treated as (conceptual) complexes having a definite internal structure, while in the other the ‘numbers’ or ‘points’ are the simple and indifferent terms, the x’s and y’s of the system. The former mode of procedure is best illustrated by the investigations of Russell’s Principles of Mathematics and Principia Mathematica of Russell and Whitehead. The other method is exemplified by Dedekind’s Was sind und was sollen die Zahlen, by the Ausdehnungslehre of Grassmann, and by the paper of Mr. A.B. Kempe, “On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points.” But this second method appears in its best and clearest form in the paper of Professor Royce on The Relation of the Principles of Logic to the Foundations of Geometry. Each of these procedures has its advantages and its difficulties. Of late, the first method has received a disproportionate share of attention. For this reason, if for no other, I deem it important to call attention to the second method in general and to Professor Royce’s paper—its notable exemplification—in particular.

Professor Royce generalizes upon certain relations previously pointed out by Kempe, in the paper mentioned above,—certain relations which are fundamental both for logic and for geometry. If ac · b represent a triadic relation in which a and c are the ‘even’ members and b is the ‘odd’ member, ac · b is capable of various significant interpretations. If a, b, and c represent areas, ac · b may be taken to symbolize the fact that b includes whatever area is common to a and c, and is itself included in that area which comprises what is either a or c (or both). The same relation may be expressed in symbolic logic as:

ac⥽b⥽(a +c); or; a̅bc̅ + ab̅c = 0.

This relation may be so assumed that it has the essential properties of serial order. Taking it in the form just given and presuming the familiar laws of the algebra of logic, if ac · b and ad · c, then also ad · b and bd · c. Hereupon we may translate ac · b by ‘b is between a and c,’ and the relation will then have the properties of the points, a, b, c, d, in that order. Further, if a be regarded as an origin with reference to which precedence is determined, ac · b may represent ‘b precedes c,’ and ad · c that ‘c precedes d.’ Since ac · b and ad · c together give ad · b, if ‘b precedes c’ and ‘c precedes d,’ then ‘b precedes d.’ Hence this relation has the essential transitivity of serial order, with the added precision that it retains reference to the origin from which ‘precedes’ is determined.

Professor Royce points out to his students that the last mentioned property of this relation makes possible an interpretation of it for logical classes in which it becomes more general than the inclusion relation of ordinary syllogistic reasoning. If there should be inhabitants of Mars whose logical sense coincided with our own—so that any conclusion which we regarded as valid would seem valid to them, and vice versa—but whose psychology was somewhat different from ours, these Martians might prefer to remark that “b is ‘between’ a and c,” rather than to note that “all a is b and all b is c.” These Martians might then carry on successfully all their reasoning in terms of this triadic ‘between’ relation. For ac · b meaning a̅bc̅ + ab̅c = 0 is a general relation which, in the special case where a is the “null” class contained in every class, becomes the familiar “b is included in c” or “all b is c.” By virtue of the transitivity pointed out above, 0c · b and 0d · b, which is the syllogism in Barbara, ‘If all b is c and all c is d, then all b is d.’ Hence these Martians would possess a mode of reasoning more comprehensive than our own and including our own as a special case.

The triadic relation of Kempe is, then, a very powerful one, and capable of representing the most fundamental relations not only in logic, but in all those departments of our systematic thinking where unsymmetrical transitive (serial) relations are important. In terms of these triads, Kempe states the properties of his ‘base system,’ from whose order the relations of logic and geometry both are to be derived. The ‘base system’ consists of an infinite number of homogeneous elements, each having an infinite number of equivalents. It is assumed that triads are disposed in this system according to the following laws.

1. If we have ab · p and cb · q, r exists such that we have aq · r and cp · r.

2. If we have ab · p and cp · r, q exists such that we have aq · r and cb · r.

3. If we have ab · c, and a = b, then c = a = b. 

4. If a = b, then we have ac · b and bc · a, whatever entity of the system c may be.

To these, Kempe adds a fifth postulate which he calls the ‘law of continuity’: “No entity is absent from the system which can consistently be present.” From these assumptions and various definitions in terms of the triadic relation, Kempe is able to derive the laws of the symbolic logic of classes and the most fundamental properties of geometrical sets of points.

But there are certain dubious features of Kempe’s procedure. As Professor Royce notes, the ‘law of continuity’ makes postulates 1 and 2 superfluous. And it renders entirely obscure what properties the system may have, beyond those derivable from the other postulates without this. For the negative form of the ‘law of continuity’ makes it impossible to assume the existence of an entity without first investigating all the properties of all the other entities and collections in the system, where some of these other entities and collections exist only at the instance of the ‘law of continuity’ itself. Consequently the existence of any entity or set, not explicitly demanded by the other postulates, can be assumed only at the risk of later inconsistency. Also, in spite of the fact that Kempe has assumed an infinity of elements in the base set, there are certain ambiguities and difficulties about the application of his principles to infinite collections.

In Professor Royce’s paper, we have no such ‘blanket assumptions’ as the ‘law of continuity,’ and the relations defined may be extended without difficulty to any finite or infinite set. We have here, in place of a ‘base system’ and triadic relations, the ‘system Σ’ and “O-collections.”

The system Σ consists of simple and homogenous elements. Collections of these may contain any finite or infinite number of elements; and any element may be repeated any number of times; so that x and x-repeated may be considered a collection, x, x-repeated, and y a collection, and so on. Greek letters will signify determinate collections in Σ. Collections in Σ are either O-collections or E-collections. O(——) signifies that (——) is an O-collection; E(——) that (——) is an E-collection, i.e., that it is not an O-collection. Assuming for the moment the principles of the algebra of logic, O(pqrs …) signifies that pqrs … + p̅q̅r̅s̅ … = 0. [Both the laws of the algebra of logic and the properties of O-collections which render them thus expressible are, of course, derived from the postulates and not assumed in the beginning.] It will be clear that the order of terms in any O-collection may be varied at will. ‘x is equivalent to y’ means that in very collection in which x or y occurs the other may be substituted for it and the collection in question still remain an O-collection.

If two elements in Σ, say p and q, are such that O(pq) is true, then p and q are said to be obverses, each of the other. Since it will follow from the postulates of the system that all the obverses of a given element are mutually equivalent, and that every element has at least one obverse, a ‘unique representative’ of the obverses of x may be chosen and symbolized by x̅. Pairs of obverses will turn out to have the properties of negatives in logic.

Any q such that O(βq) is true, is called a compliment of β.

Any r such that O(βq) and O(qr) are both true is called a resultant of β.

The postulates of the system Σ are as follows:

I. If O(α), then O(αγ), whatever collection γ may be.

II. If, whatever element bn of β be considered, O(δbn), and if O(β) is also true, then O(δ).

III. There exists at least one element in Σ.

IV. If an element x of Σ exists, then y exists such that x ≠ y.

V. Whatever pair (p, q) exists such that p ≠ q, r exists such that while both O(rp) and O(rq) are false, O(pqr) is true. 

VI. If w exists such that O(θw), then v also exists such that O(θv) and such, too, that whatever element tn of θ be considered O(vwtn).

From these assumptions the whole algebra of logic can be derived in such wise that the system Σ has the order of the totality of logical classes. To see this, we must first define the F-relation. If O(pqrs …) to any number of terms, we may represent the same fact by (F(p̅/qsr …), (Fp̅r̅/qs …), (r/Fp̅q̅s̅ …), etc., where the rule for transforming the O-collection into the corresponding F-collections is that we introduce a bar, separating any one or more elements of the O-collection from the remainder, and then replace each of the elements on one (either) side of the bar by its obverse. Since the order of terms in O-collections is indifferent, terms on the same side of the bar in any F-relation are independent of the particular order in which they are written. Also, it follows immediately from the definition of the relation that F(pq/r̅s̅) and F(p̅q̅/rs) are equivalent. Where the F-relation holds for three terms, it turns out to be identical with the triadic relation of Kempe, and the Kempean ac · b is thus a special case of the F-relation, namely F(b/ac), or F(ac/b), or F(a/bc̅), or F(a̅/b̅c), or F(b/ca), etc., all of which are equivalent. We may, then, define the “illative” relation,—“b is included in c” where b and c are classes, “b implies c” where b and c are propositions, “b precedes c,” where b and c are points or terms in one-dimensional array,—as the special case of any of the above F-relation in which a is the “zero element,” or “null class,” or “origin.” But these F-relations are equivalent, by definition, to O(ab̅c) and O(abc̅). Hence b⥽ac may be defined to mean O(ab̅c) and b⥽c to mean O(0b̅c). Thus in terms of the totally symmetrical O-relation, the unsymmetrical, transitive dyadic relation which characterizes both serial order and syllogistic reasoning can be defined.

As is well known, the entire algebra of logic may be derived from a class K, the idea of negation, and the illative relation, hence also in terms of the system Σ and O-collections. The ‘zero element’ or ‘null class’ is any arbitrarily chosen member with reference to which all illative relations are supposed to be specified. Such an element o itself bears the illative relation to any other, x, since F(ox/o), or O(oo̅x) holds for any element x. The element i, the “universe” of the algebra of logic, may then be defined as the negative or obverse of the o chosen. In the system Σ, o and i do not differ from any other pair of obverses, apart from the arbitrary choice of a reference element for illative relations. The logical product of two terms, x and y, is then definable as any P such that F(ox/P), F(oy/P), and F(xy/P). The logical sum of x and y is definable as any S such that F(ix/S), F(iy/S), and F(xy/S). P, so defined, will be such that P⥽x and P⥽y, while any w such that w⥽x and w⥽y will be also such that w⥽P. For S it will be true that x⥽S and y⥽S, and any v such that x⥽v and y⥽v is also such that S⥽v. S and P are, in fact, the “lower limit” and “upper limit,” with reference to the chosen zero element, of all the F-resultants of x and y, an F-resultant being any z such that F(xy/z). These definitions for the product and sum of two elements may be extended immediately to any number of elements, or any collection of β, if we replace x an y by “any element of β, however chosen.” The usual laws of the algebra of logic, connecting sums and products, terms and their negatives, and the elements o and i may then be verified for the system Σ. This order of logical entities is contained in Σ in an infinite variety of ways, since any pair of obverses may be arbitrarily chosen for i and o. F-relations and O-relations, not confined to dyads and triads, are capable of representing this order in a generalized form.

There is, moreover, a wealth of order in the system which the algebra of logic, even in terms of any polyadic relation, does not require. It is this difference which renders the system Σ capable of being viewed as a generalized space form.

It follows from postulate V that if p ≠ q, then there is an element ‘between’ p and q. The postulate states: Whatever pair (p, q) exists such that p ≠ q, r also exists such that while both O(rp) and O(rq) are false, O(pqr) or F(pq/r̅) gives, by definition of the illative relation, r⥽qp and r̅⥽pq) or r is “between” p and q. And r̅ must be distinct from p and q both, for otherwise, it follows from the definition of obverses, one of the two O(r̅p) and O(r̅q) will be true. Hence postulate V may be restated in the form: For every pair of distinct elements, there exists an element, distinct from both, between them. It is at once obvious that if the elements be “points,” and p⥽oq mean that p is between o and q, postulate V requires that the order of points in Σ should be dense in every direction (with reference to every pair of points). It is further clear that if we take any pair of distinct points, o and z, and postulate t between o and t, v between t and z, and so on. Owing to the transitivity of the illative relation, we are thus required to postulate for every pair (o, z) an infinite number of elements in the order o⥽or⥽ot⥽ov⥽oz. Such an ordered collection is continuous. We have already seen that it is dense. It remains to see that it satisfies the requirement that every fundamental segment has a limit. Consider two sections from the collection, κ and λ, such that k is any element of κ, every element j such that j⥽ok belongs to κ, and every element l, such that for every element k of κ l⥽ok is false, belongs to λ. There is, then, an element, called S, such that for every element k in κ, k⥽oS, and if l is any element such that, for every element k of κ, k⥽ol, then S⥽ol. Such an element S is the ‘sum’ or ‘upper limit’ of κ, defined above. Hence every fundamental segment has a limit. Any collection thus characterized by a transitive unsymmetrical relation and continuous order deserves to be called a ‘line.’ Every pair of distinct elements in Σ determines such a line.

For every pair of distinct points, o and q, there exists p such that F(oq/p) and hence O(oqp̅). By the definition of the F-relation, if O(oqp̅), then F(o̅q̅/p). Hence if o and q determine a line, o … p … q, there exists also a line, o̅ … p̅ … q̅ or q̅ … p̅ … o̅, in which appear the obverses of all the elements in o … p … q. But it also follows from O(oqp̅) that F(op̅/q̅), or q⥽op̅. Thus if o … l … z be any line determined with reference to an “origin” o, the line containing the obverses of the elements of o … l … z may be determined by reference to the same origin. And if two elements of o … l … z, say m and n, are such that m⥽on, then n̅⥽om̅. If we further consider the order of elements in both lines, o … l … z, and z̅ … l̅ … o̅, with reference to the origin o and its obverse o̅, the two lines appear as a single line which passes from o to o̅ through l, and from o̅ back to o through l̅. Let m and n be any two elements of 0 … l … z such that F(on/m). We have m⥽on. Hence n̅⥽om̅. But if we have F(on/m), then also O(onm̅) and so F(o̅m/n). Hence n⥽mo̅. Thus any two elements, m and n, such that m is between o and n, are also such that n is between m and o̅. From the transitivity of the illative relation, m⥽oo̅. But if m⥽oo̅, then from the above m⥽oo. Thus we have the continuous line, o … m … n … o̅ … n̅ … m̅ … o, or o̅ … n̅ … m̅ … o … m … n … o̅, which has so far the character of the projective line with o as origin and o̅ the point at infinity. And if m, n, r, occur in that order in one ‘direction’ from the origin, then m̅, n̅, r̅, occur in that order in the ‘opposite direction’ from the origin.

Certain further characteristics of order in the system may be mentioned briefly. In general, lines such as those considered above may “intersect” any number of times. From the definition of obverses, O(aa̅) and O(cc̅) always hold. But by postulate I, if O(aa̅), then O(aa̅p), and hence F(aa̅/p), for any element p. Similarly, if O(cc̅), then F(cc̅/p). Thus collections consisting of the F-resultants of different pairs may have any number of elements in common. But in terms of such operations as were in question in the definitions of ‘sums’ and ‘products,’ sets of resultants may be determined such that they have one and only one element in common. Thus certain selected lines in the system intersect once and once only. There are any number of such sets.

In general, if any pair of elements in a set are obverses of one another, all the other elements of the set will be resultants of this pair, and their entire array will be “one-dimensional” so far as dimensionality may be attributed to such a collection. The problem of selecting sets suitable for any space form—any n-dimensional array—is the problem of selecting so that O-collections will be excluded. Such sets, containing no obverses, are the ‘flat collections’ of Kempe. As he pointed out, the excluded obverses will form an exactly similar set, so that ‘spaces’ come in pairs somewhat suggesting companion hemispheres. In terms of “flat collections,” one-dimensional, two-dimensional, n-dimensional arrays, may be specified in any number of ways.

Once the order of the system Σ is generated in terms of O-relations and F-relations, the determination of such more specialized types of order is a problem of selection only. In the words of Professor Royce, “Wherever a linear series is in question, wherever an origin of coordinates is employed, wherever ‘cause and effect,’ ‘ground and consequence,’ orientation in space or direction of tendency in time are in question, the dyadic asymmetrical relations involved are essentially the same as the relation here symbolized by p⥽yq. This expression, then, is due to certain of our best established practical instincts and to some of our best fixed intellectual habits. Yet it is not the only expression for the relations involved. It is in several respects inferior to the more direct expression in terms of O-relations….When, in fact, we attempt to describe the relations of the system Σ merely in terms of the antecedent-consequent relation, we not only limit ourselves to an arbitrary choice of origin, but miss the power to survey at a glance relations of more than a dyadic, or triadic character.”

With this hasty and fragmentary survey of the system Σ, we may turn to considerations of method. It was suggested in the introduction that the procedure here exemplified differs in notable ways from the method of such studies as those of Principia Mathematica. In that work, we are presented at the outset with a simple, though general, order—the order of elementary propositions so related to one another that one is the negative of another, two may be such that at least one of them is true, and so on. In terms of these fundamental relations, more special types of order—various branches of mathematics—are built up by progressive complication. In some respects this is the necessary character of deductive procedures in general; in other respects it is not. In particular, this method differs from that employed by Mr. Kempe and Professor Royce in that terms, as well as relations, of later sections are themselves complexes of the relations at first assumed. The complication thus made necessary can hardly be appreciated by those who would regard a number, for instance, as a simple entity. To illustrate: In Principia Mathematica, the “cardinal number” of x is the class of referents of the relation ‘similar to’ where x is the relatum. The ‘class of referents’ of any relation R is defined as α such that α is identical with x such that, for some y, x has the relation R to y. ‘Relatum’ is similarly defined. ‘m is identical with n’ means that, for any predicative function φ, φm implies φn. I do not pause upon ‘predicative function.’ α is ‘similar to’ β means that, for some one-to-one relation R, α is identical with the class of referents of R and β is identical with the class of relata of R. A ‘one-to-one’ relation is a relation S such that the class of referents of S is contained in i. ‘i’ is defined as α such that, for some x, α is identical with the x. ‘The x’ is my attempt to translate the untranslatable. The attempt to analyze ‘is contained in’ would require much more space than we can afford. But supposing the analysis complete, we discover that the ‘cardinal number of x’ is ——, where —— is the definition first given, with all the terms in it replaced by their definition, the terms in these replaced by their definition, and so on. All this complexity is internal to the terms of arithmetic. And only when this process is complete can any properties or relations of ‘the cardinal number of x’ be demonstrated. An advantage of this method is that the step from one order to another ‘based upon it’ is always such as to make clear the connection between the two. It preserves automatically the hierarchic arrangement of various departments of exact thinking. The process of developing this hierarchy is tedious and taxes our analytic powers, but there is always the prospect of assured success if we can perform the initial analysis involved in the definitions. But the disadvantages of this complexity can hardly be overemphasized. It is forbidding to those whose interests are simply ‘mathematical’ or ‘scientific’ in the ordinary sense. Such a work as Principia Mathematica runs great risk of being much referred to, little read, and less understood.

In contrast with such complexity, we have, by the method of Mr. Kempe and Professor Royce, an order completely generated at the start, and such that the various special orders contained in it may be arrived at simply by selection. Little or no complication within the terms is required. Involved as the structure of the system Σ may seem, it is, by comparison, a marvel of simplicity and compact neatness. With this method, there seems to be no assurance in advance that any hierarchic relations of different orders will be disclosed, but we shall certainly discover, and without difficulty, whatever analogies exist between various orders. Again, this method relies much more upon devices which may be not at all obvious. It may not tax severely the analytic powers, but it is certain to tax the ingenuity.

In another important respect, advantage seems to lie with this method. One would hardly care to invent a new geometry by the hierarchic procedure, or expect to discover one by its use. We have to know where we are going or we shall not get there by this road. By contrast, Professor Royce’s is the method of the path-finder. The prospect of the novel is here much greater. The system Σ may—probably does—contain new continents of order whose existence we do not even suspect. And some chance transformation may put us, suddenly and unexpectedly, in possession of such previously unexplored fields.

Which of the two methods will prove, in the end, more powerful, no one can say at present. The whole subject is too new and undeveloped. Certainly it is to be desired that the direct and exploratory method be increasingly made use of, and that the advantages of studying very general types of order, such as the system Σ, be better understood.

Reading Notes: June 7th, 2022

“After a study of the possible issues, the Committee presented, as the first of its questions for debate, the following: “In cases where a real (and non-hallucinatory) object is involved, what is the relation between the real and the perceived object with respect (a) to their numerical identity at the moment of perception, (b) with respect to the possibility of the existence of the real object at other moments apart from any perception?” This question was to be understood by all who were to cooperate, as determined by the meanings assigned by the Committee to the terms “object,” “perceived object,” and “real object.” The definitions of these terms, as printed in the Committee’s report, are as follows: “By object in this discussion shall be meant any complex of physical qualities, whether perceived or unperceived and whether real or unreal. By real objects is meant in this discussion such objects as are true parts of the material world. By perceived object is meant in this discussion an object given in some particular actual perception.” It appears, from the context, and from the formulation of the question for debate quoted above, that the Committee very naturally laid some stress upon the fact that what is meant by “some particular actual perception” involved an occurrence at some “moment of time,” called also “the moment of perception”; or, again, involved some determinate set or sequence of such momentary occurrences, “in some particular individuated stream of perceptions,” that is, in the mind or in the experience of some person….The Committee did not define what it meant by the adjective “given,” used in the above-cited definition of “perceived object”….As a fact, however, their definition of the term “perceived object,” taken together with their formulation of their question, and the context in which they used the which they are used the word given, involved a very serious interference with the range of the cooperation which they invited. For what is “given” in a “moment of perception,” and what is not “given at a particular moment,” and the sense in which what is “given” can also be an “object”—all these matters are not topics of a merely pedantic curiosity about words. They are matters which have been lengthily, frequently, and momentously discussed, both in the controversies about perception and in other philosophical inquiries.” (Royce, On Definitions and Debates, 237) 

“In The Religious Aspect of Philosophy (1885), ideas are taken to be representations of real objects. As images of what is perceived or thought, a man’s ideas are all that is present to his mind. These ideas are the only content of his thought, and the objects represented remain outside his thought. To this extent the position of subjective idealism is correct, i.e. “my mind can be concerned only with its own ideas.” But an immediate problem for a subjective idealist is to account for the difference between truth and error. If all I think about will be my ideas, and what they represent are but other ideas of mine, then to assert anything about them must be correct. In that case, sincerity and truth are identical, for when I assert anything there is no reference to anything outside of my own thought. As long as I honestly consult my own ideas, I cannot be in error. The truth of my ideas, however, is commonly taken to be their correspondence to the objects they represent. There is a “commonplace assumption” that error is possible, that an assertion can fail to agree with a real object outside of thought. But how is one to judge if this particular assertion is true or false? To answer that, Royce considers what role the judgment plays in human cognition. It is not an act distinct from that of understanding. That is, the judgment by itself has no intelligible object other than the ideas present to all thought. Royce concludes that the judgment synthesizes my ideas—a position he explicitly avows to be neo-Kantian. But if the judgment reaches no object beyond ideas, the common-sense belief in error must either be abandoned or supplemented. The former course is impossible, for in choosing it one would be admitting the common-sense knowledge had been in error. To state “error is impossible” as a remedy for a mistaken assumption is clearly contradictory. So that latter course alone proves viable. Since no single judgment can be an error (for it reaches no object beyond itself), there must be a higher thought that includes both the judgment and its real object. By comparing the two, this higher thought determines whether the first thought was true or false. Left to itself, the latter remains a fragment “neither true nor false, objectless, no complete act of thought at all.” This is a very brief sketch of Royce’s method of presupposition by denial. He begins with the fact of error in the world, and concludes to an Absolute Thought.” (Zanardi, Idea and Absolute in the Philosophy of Josiah Royce, 10-11) [Underlining is mine]

“What follows is a summarized version of Royce’s more lengthy argument. The fact of error is undeniable; to deny this is to contradict oneself, for how else can this fact be refuted if not by proving it erroneous? Each error implies a judgment whose intended object is other than my ideas and so lies beyond my judgment. Such an object will also be an object of a corresponding true judgment. Since the existence of error implies a higher thought, it will be this thought that contains the object of both the true and false judgments. Since the possibilities of error are infinite, the inclusive thought must be infinite. And since error is possible not only as regards objects but also as regards relations, all possible relations in the world must be present to this infinite thought. Finally, to know all relations at once is to know them in absolute rational unity, i.e. as one single thought….Even if one were to find fault with his argument, the error charged to Royce’s position is alleged to prove the existence of Absolute Thought. It alone knows the real and can compare a judgment with its intended object. Royce offers other arguments for the existence of Absolute Thought. The problem of knowing other minds is an instance of employing the already cited view of human understanding. My idea of another person can only be true or false if there is a third party to compare my idea with the real person. There is also a problem of relating a past idea to a present thought. The past idea was unique in its separate existence and in its view of the future. To determine the identity between its conception of the future and the present thought’s conception of what now has become reality requires an inclusive thought which compare them. How else could my past thought have made any assertion about a future moment? Royce refutes a response that rests on verifying a prediction only upon its fulfillment or failure to occur. My memory of an original thought differs from it and so is still in need of a comparison with that original thought. Again, Royce appeals to a higher thought to make a synthesis of what to the human knower are disparate ideas.” (Zanardi, Idea and Absolute in the Philosophy of Josiah Royce, 11-12) [Underlining is mine] 

Royce’s argument is genius. However, the implicit Neo-Kantianism weakens it greatly. I have attempted to transpose the argument into less “subjectivistic” form that still arrives at Royce’s intended result: 

If erroneous judgments are possible, then all of the necessary conditions underlying this possibility actually exist. Erroneous judgments are possible. Therefore, all of the necessary conditions for the possibility of erroneous judgments actually exist. A necessary condition for the possibility of an erroneous judgment is for the erroneous judgment to fail to agree with that which the thought making the erroneous judgment has intended for its object. But that which the thought making the erroneous judgment has intended for its object can only be that which is known to the thought making the erroneous judgment. Therefore, a necessary condition for the possibility of an erroneous judgment is for the erroneous judgment to fail to agree with that which is known to the thought making the erroneous judgment. But an erroneous judgment cannot fail to agree with that which is known to the thought making the erroneous judgment unless the erroneous judgment is known as being in error with respect to its intended object. Therefore, a necessary condition for the possibility of an erroneous judgment is for the erroneous judgment to be known as being in error with respect to its intended object. But an erroneous judgment is known as being in error with respect to its intended object only insofar as the erroneous judgment and its intended object are known to a thought which compares them, and judges the erroneous judgment to be in error with respect to its intended object. Therefore, a necessary condition for the possibility of an erroneous judgment is for the erroneous judgment and its intended object to be known to a thought which compares them, and judges the erroneous judgment to be in error with respect to its intended object. Therefore, since all of the necessary conditions for the possibility of erroneous judgments actually exist, it follows that for every erroneous judgment that exists, there actually exists a thought which knows said erroneous judgment and its intended object, compares them, and judges the erroneous judgment to be in error with respect to its intended object. There exists at least one erroneous judgment that had the Universe as a whole (and every aspect of it) as its intended object. Therefore, there actually exists a thought which knows said erroneous judgment and the Universe as a whole (and every aspect of it), compares them, and judges the erroneous judgment to be in error with respect to the Universe as a whole (and every aspect of it). A thought which knew the aforementioned erroneous judgment and the Universe as a whole (and every aspect of it), compared them, and judged the erroneous judgment to be in error with respect to the Universe as a whole (and every aspect of it) would be an all-knowing, Absolute Thought. Therefore, since all of the necessary conditions for the possibility of erroneous judgments actually exist, and there exists at least one erroneous judgment that had the Universe as a whole (and every aspect of it) as its intended object, it follows that there actually exists an all-knowing, Absolute Thought. Q.E.D. 

Materialism and Representationalism

The following article presents a dilemma that takes aim at contemporary Materialism’s recourse to “representationalism”—a ghost which, so far as I can see, has long been laid to rest in the history of philosophy.

According to contemporary Materialism, we cannot arrive at an understanding of any part of Nature unless it is a “content” of our nervous systems’ representations—said representational states being parts or regions of our nervous systems. However, the relationship between a representation and its “content” is perplexing. For, we cannot arrive at an understanding of the “content” of a representation, X, without having an understanding of X as being the “representation of something” (and this involves having an understanding of X and characteristics of X); indeed, (a) both what is, and is not, the “content” of a representation (i.e., What a representation is a “representation of,” and what a representation is not a “representation of”) is determined by the characteristics and relations of said representation, (b) a representation’s “content” need not exist, (c) a representation need not “correspond” to its “content,” and (d) the relation obtaining between a representation and its “content” is not, and cannot be, merely one of “similarity,” “resemblance,” “co-existence,” or “effect” to “cause.” However, since we cannot arrive at an understanding of the “content” of a representation, X, without having an understanding of X as being the “representation of something” (and this involves having an understanding of X and characteristics of X), it follows that if we cannot arrive at an understanding of any part of Nature unless it is a “content” of our nervous systems’ representations, then we cannot arrive at an understanding of any part of Nature without having an understanding of our nervous systems’ representations. However, since our nervous systems’ representations are themselves parts or regions of our nervous systems, it follows that if we cannot arrive at an understanding of any part of Nature unless it is a “content” of our nervous systems’ representations, then we cannot arrive at an understanding of any part of Nature without having an understanding of parts of our nervous systems. This is the first horn of the dilemma.

This horn can be defended in several additional ways.  Indeed, in his Gifford Lectures, The World and the Individual, Josiah Royce decisively refutes the view that the relation between a representation and its “content” either is, or could be, merely one of “similarity,” “resemblance,” “co-existence,” or “effect” to “cause.” Rather than including Royce’s entire discussion on the relationship between a representation and its “content,” I’ve include a passage that I found to be particularly germane:

“For consider: An object, as we have seen, has two relations to a [representation]. The one is the relation that constitutes it the object meant by that [representation]. The other is the sort of correspondence that is to obtain between object and [representation]. As to the first of these two: An object is not the object of a given [representation] merely because the object causes the [representation], or impresses itself upon the [representation] as the seal impresses the wax. For there are objects of [representations] that are not causes of the [representations] which refer to these objects, just as there are countless cases where my [representations] are supposed to have causes, say physiological or psychological causes, of which I myself never become conscious at all, as my objects. Nor is the object the object of a given [representation] merely because, from the point of view of an external observer, who looks from without upon [representation] and object, and compares them, the [representation] resembles the object. For the sort of correspondence to be demanded of the [representation] is determined by itself, and this correspondence cannot be judged merely from without. Again, my [representation] of my own past experiences may resemble your past experiences, in case you have felt as I have felt, or have acted in any way as I have acted. Yet when my [representations], in a moment of reminiscence, refer to my own past, and have that for their object, they do not refer to your past, nor to your deeds and sorrows, however like my own these experiences of yours may have been. One who, merely comparing my [representations] and your experiences, said that because of the mere likeness I must be thinking of your past as my object, would, therefore, err, if it was my own past of which I was thinking. Neither such a relation as causal connection nor such a relation as mere similarity is, then, sufficient to identify an object as the object of a given [representation]. Nor yet can any other relation, so far as it is merely supposed to be seen from without, by an external observer, suffice to identify any object as the object of a given [representation].” (Royce, The World and the Individual, Vol. I, 297)

I’ve also included several passages outlining contemporary views as to the relationship between a representation and its “content.”

“[If] awareness of an external object is constituted by having an internal representation of it; [then] the perceiver is not aware of the representation itself, but of what it represents (its content). Thus, one perceives the tomato (the object of awareness) in virtue of possessing an internal representation of it (the vehicle of awareness). This move satisfies some philosophers that perception is direct in the traditional sense, yet on this view, the perceiver experiences the content of a representation rather than the living tomato. The representation must somehow be derived from the visual input by a process that establishes its content….If perceptual awareness consists of having representations, how does the perceptual system determine the environmental entities to which they correspond? Without some independent, extrasensory access to the world, there appears to be no way to establish which internal states indicate which environmental properties, or which representations stand for tomatoes and which for elephants. The perceiver is trapped in a closed universe of sensory phenomena or uninterpretable representations. The indirect solution is inference to the best explanation: The perceptual system infers a representation of the world that best accounts for the order in sensory input....However, as Hermann von Helmholtz understood by the mid-19th century, this inference process presumes that the perceptual system already possesses knowledge about (1) the structure of the world, including the sorts of entities that exist and predicates to describe them, and (2) how the world structures sensory input, such as a theory of image formation and transduction. The trouble is that such prior knowledge must somehow be acquired, again, in an extrasensory manner....The [representationalist] position thus appears to be circular. There is a further problem with treating perception as a process of inference. Inference is a logical relation that holds between conscious mental states (beliefs, thoughts, statements) corresponding to premises and conclusions. But as we have just seen, if we are to avoid the representationalist fallacy, perception cannot be based on conscious awareness of internal states. If the perceptual process is unconscious, then whatever else it may be, it cannot be inferential; the same goes for related terms such as hypothesis, clue, evidence, and assumption. The notion of perception as unconscious inference, originally suggested by Helmholtz, is thus inconsistent. Computational theories seek to avoid this objection by treating perception as a process of computation over representations, but this leaves [the semantic or grounding] problem unresolved.” (Warren, Entry on “Direct Perception” in The Encyclopedia of Perception, Vol. I, 367-368) 

To help illustrate the importance of the above passage. I have included a diagram showing an image projected on the retina of the eye (i.e., the representation) and the infinitely many possible sources (i.e., the “content”) that the projected retinal image can map onto.

 

“The states of a computer can be given many different semantic interpretations; indeed, the same symbolic states are sometimes interpreted as words, sometimes as numbers, chess positions, or weather conditions….What determines what a given (syntactically articulated) state represents? [What] causes certain mental events to have certain contents? [According to some theorists], at least some mental contents represent certain things because they resemble them. An image of X represents X precisely because the conscious mental representations, or images, look like X. Such a view probably is not far from the common notion of visual imagery. If you were to ask a group of people how they know their image of a duck actually represents a duck, rather than, say, a rabbit, they might reply that the image looks like a duck. For several reasons, however, this answer does not explain why the image is a representation of a duck. For example, even in the introspectionist approach, the image need not closely resemble a duck for people to take it as a duck since it is their image, they can take it as virtually anything they wish; after all, the word duck refers to a duck without in any way, resembling a duck. As Wittgenstein points out, the image of a man walking up a hill may look exactly like the image of a man walking backward down a hill; yet, if they were my images, there would be no question of their being indeterminate—I would know what they represented. The relation of resemblance is not well defined. Whether one thing resembles another is not a physically (or geometrically) definable property; resemblance depends on what the viewer knows or believes. To me, most birds closely resemble one another, but to a birdwatcher friend they are as different as ducks and rabbits. Resemblance cannot be specified except in relation to a viewer….Resemblance provides no basis for specifying the semantic content of mental representations.” (Pylyshyn, Computation and Cognition, 40-41) 

“Mediational behaviorists and certain speculative neurophysiologists take the position that a brain event can be said to represent something if that event is sufficiently like (possesses a subset of the properties of) the event that takes place when that something actually is perceived. Another, more radically behaviorist version requires that the mediational event evoke an internal “preparatory response” that is sufficiently like the response that would have been evoked by the corresponding stimulus. Neither position is satisfactory, because of the properties of representations we have already noted (for example, we can think about objects we have neither perceived nor have any disposition to behave toward, such as, perhaps, quarks). In any case, the only mechanism behaviorism provides for explicating the representing relation is that of association. Association, in turn, must be established by such principles as contiguity and evoked by the activation of other associated items (otherwise we would not have provided the naturalistic account of the semantics of the functional states sought by behaviorism). A chain of continuous events mediating between a brain state and an object, however, cannot form the basis of representation, for reasons discussed above, namely, that it is neither necessary nor sufficient that the state of an organism be linked by a series of contiguous events to the object that the state represents. Not only can I think of things to which I obviously am not in this sort of relation (for example, nonexistent things), but when I do think of X, I do not thereby think of associates of X; indeed, I need not think of properties that are necessarily coextensive with X, such as shape, size, weight, color, and so on. The basic problem is that representing is a semantic relation, that semantic relations, like logical relations, appear not to be causally definable…” (Pylyshyn, Computation and Cognition, 41-42)

I have also included a diagram from M.D. Vernon’s Psychology of Perception, that illustrates the problem of under-determination. For, two representations may be nearly indistinguishable with respect to their characteristics as “vehicles” and nevertheless have drastically different “content.” 

 

Let’s return to our argument by outlining the second horn of the dilemma:

Our nervous systems—and their parts—are themselves parts of Nature.

The Materialist cannot, on pain of inconsistency, deny or reject the second horn of the dilemma. Taken together, the first and second horns are the following: 

A) We cannot arrive at an understanding of any part of Nature without having an understanding of parts of our nervous systems.

B) Our nervous systems—and their parts—are themselves parts of Nature.

The Materialist is thus caught in a snare : 

We cannot arrive at an understanding of any part of Nature without having an understanding of something that could never be understood by us.

If the dilemma doesn’t make itself explicit at first glance, it can be brought to light in the following way:

We cannot arrive at an understanding of any part of Nature without having an understanding of parts of our nervous systems, N(1). However, our nervous systems, N(1), are themselves parts of Nature; and so we cannot arrive at an understanding of parts of our nervous systems, N(1)—the having of which is necessary for our arriving at an understanding of any part of Nature—without having an understanding of parts of our nervous systems, N(2). However, our nervous systems, N(2), are themselves parts of Nature; and so we cannot arrive at an understanding of parts of our nervous systems, N(2)—the having of which is necessary for our arriving at an understanding of parts of our nervous systems, N(1)—without having an understanding of parts of our nervous systems, N(3). However, our nervous systems, N(3), are themselves parts of Nature; and so we cannot arrive at an understanding of parts of our nervous systems, N(3)—the having of which is necessary for our arriving at an understanding of parts of our nervous systems, N(2)—without having an understanding of parts of our nervous systems, N(4). However, our nervous systems, N(4), are themselves parts of Nature; and so we cannot arrive at an understanding of parts of our nervous systems, N(4)—the having of which is necessary for our arriving at an understanding of parts of our nervous systems, N(3)—without having an understanding of parts of our nervous systems, N(5)…and so on, and so on, ad infinitum. Our arrival at an understanding of any part of Nature is thus thwarted by a vicious, downward spiral of mutually-presupposing terms.

In other words, a vicious regress would render it impossible for us arrive at an understanding of any part of Nature. However, since Materialism asserts that we do have, and indeed have arrived at, an understanding of some parts of Nature, it follows that Materialism is inconsistent with itself. Indeed, the truth of Materialism is incompatible with our knowledge of its truth.

The above argument is inspired by several passages in Chapter 22, “Nature,” of F.H. Bradley’s 1893 magnum opus, Appearance and Reality. Bradley’s argument receives an analysis in W.J. Mander’s article, F.H. Bradley and the Philosophy of Science. I have included Mander’s overview of Bradley’s argument below:

“Bradley has one further objection to physical nature which is rather unusual and worth quoting in his own words. In order to state the problem, says Bradley: 

“We may here use the form of what has been called an Antinomy. (a) Nature is only for my body; on the other hand, (b) My body is only for Nature…the outer world is known only as a state of my organism….And yet most emphatically…my organism is nothing but appearance to a body. It itself is only the bare state of a natural object….[This] gives us one thing as qualified by the state of another thing, each within that known relation being only for the other, and, apart from it, being unknown and, so far, a nonentity….Nature is the phenomenal relation of the unknown to the unknown; and the terms cannot, because unknown, even be said to be related, since they cannot themselves be said to be anything at all.” 

The puzzle is, that nature can only be understood through our sense organs, but our sense organs can only be understood as part of nature, which as before can only be understood through our sense organs. Thus, the physical world turns out to be an unknown relation between two mutually presupposing elements in a vicious downwards spiral. This is a curious argument, which might at first appear to be making a very obvious mistake. Surely Bradley should have referred not to our sense organs, but our experience. Is it not the case that nature, including our sense organs, simply comes to us in experience, in which case where is the circle? But in fact, this objection concedes precisely Bradley’s point. For what he is attacking is the notion of a purely physical world, and experience in order to do the task that it is being given here, must be something more or other than the physical world. His point is that a purely physical world, whatever else it may be able to explain (e.g. facts about us and our behaviour) can never account for its own cognition—that requires something more of a wholly different order. Within that restriction it seems reasonable to say that a knowledge of nature depends on an understanding of our sense organs. Since it is only filtered through them that cognition can take place, we have to understand the sense organs in order to understand what they give us. This is true in the same sense that we have to understand what a Geiger counter is doing in order to understand what it is telling us. But in that case, since our sense organs are part of nature, they too can only be understood in the same way, launching us on a regress. The only solution is to move to something outside of physical nature, like “experience,” for we do not have to understand experience in order to understand what it tells us. Its data comes already interpreted. Thus understood, I would maintain that this argument of Bradley’s is a valid and sound reductio of the idea of a purely physical nature.” (Mander, F.H. Bradley and the Philosophy of Science, 70)

E.E. Harris outlines Bradley’s argument in a similar fashion in his article, Bradley’s Conception of Nature:

“Although the word “Nature,” Bradley tells us (and rightly), has more meanings than one, when he discusses it in Chapter XXII of Appearance and Reality, he takes it in the sense of “the bare physical world.” “Abstract from everything psychical, and then the remainder of existence will be nature.” This forms the object of purely physical science and, we are told, “appears to fall outside of all mind.” It is obvious that such a conception is, as Bradley maintains, a pure abstraction….[We] construct a notion of [the physical world] as independent of our thought, consisting of things with primary and secondary qualities. It is the same for all observers and we regard our bodies with their sense organs as the media of observation which should convey it to us as it is and as it exists apart from them….But this view is full of confusion….Everything revealed to us of such a physical world can be so only as an affection of our own organisms; yet these again are physical things and so must be reduced to affections of themselves. We cannot infer from our affections to the causes which affect us, for to do that would be to conclude to some “thing-in-itself” which is in the nature of the case unknowable and could not therefore help us, nor could it conceivably be related causally to what is knowable so as to validate the inference. Accordingly, the physical world as the complex of relations between physical things turns out to be “the phenomenal relation of the unknown to the unknown.” Bradley develops this paradox at more length, asserting that inevitably the outer world exists only for my organs. If this means only that my perception of the physical world is so dependent, it can hardly be gainsaid; and, of course, my organs can be perceived as physical objects only on the same condition. In these terms any attempt to explain our experience of the world must lead to vicious regress and circularity. But what if we were to say that the world and our organisms are self-existent apart from any affection we may suffer and apart from our perceiving? That could help us in no way at all to comprehend the physical world, to explain what we know of it, or to say how we come to experience it. So, we are brought, Bradley concludes, to an unavoidable result: “The physical world is an appearance; it is phenomenal throughout. It is the relation between two unknowns, which, because they are unknown, we cannot have any right to regard as really two, or as related at all.” But this circular connection and contradictory interrelation between ourselves and nature is no mere mistake to be discounted. It is a necessary and unavoidable feature of our experience giving nature phenomenal reality as a grouping of facts, coexistence of objects and a sequence of events holding good within a section of what appears to us.” (Harris, Bradley’s Conception of Nature, 187-189)