A New Interpretation of

Russell’s Multiple-Relation Theory of Judgment


Gregory Landini

Department of Philosophy, University of Iowa

Iowa City, IA 52242, USA


History and Philosophy of Logic 12 (1991): 37-69


This paper offers an interpretation of Russell’s multiple-relation theory of judgment which characterizes it as a direct application of the 1905 theory of definite descriptions. The paper maintains that it was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes, that Russell’s generated the philosophical explanation of the hierarchy of orders and the ramified theory of types of Principia Mathematica (1910). The interpretation provides a new understanding of Russell’s abandoned book Theory of Knowledge (1913), the ‘direction problems’ and Wittgenstein’s criticisms.


I. Introduction

Between the years 1907 and 1913, Russell advanced and defended a correspondence theory of truth known as the ‘multiple-relation theory of judgment.’ A version of the theory is endorsed in the Introduction to Principia Mathematica, as is commonly construed as part of an epistemological theory supplementing the ramified theory of types. Though its importance to Principia has been debated, the theory is almost uniformly regarded in the literature as unworkable. Two main objections are said to be telling: the ‘narrow direction problem, which was raised almost from the inception of the theory; and the ‘wide’ direction problem, which was raised by Wittgenstein between 1912 and 1913.

Russell made no small effort to avoid the narrow direction problem, modifying the theory in his 1910 article ‘On the Nature of Truth and Falsehood,’ and again in his book of 1912, The Problems of Philosophy. In fact, he was undaunted even by the failure of the attempt in Problems. In 1913 he was working on a book, now published as Theory of Knowledge, whose main aim was to develop a comprehensive epistemic account of all human knowledge (including that of logic). The foundation was the principle that there is a basic dyadic relation of ‘acquaintance’ which a min can have to universals and particulars. The multiple-relation theory had an instrumental place in that account; and accordingly, the book contained what Russell took to be the solution of the direction problem.

The book was never completed. Russell’s progress cam to a halt in June of 1913 in the face of Wittgenstein’s criticism. Unfortunately, the precise nature of the criticism remains largely uncertain and, as Nicholas Griffin has pointed out, it “has usually been completely misunderstood” (1985, 213). With the publication of Theory of Knowledge, there is a new opportunity to shed light on the many perplexing questions concerning the multiple-relation theory. This paper maintains, however, that recent attempts to understand the 1913 book and Wittgenstein’s criticisms are undermined by their acceptance of a characterization of the multiple-relation theory (hereafter called the “standard account”) which is historically mistaken. The paper sets out an alternative account and defends a new interpretation of the role the theory was to have in Principia. Far from an epistemological addition, it is argued that the multiple-relation theory was intended to provide the philosophical explanation of the ramified hierarchy of Principia. Finally, the consequence for understanding Theory of Knowledge and Wittgenstein’s criticisms are drawn out.

2. The Standard Account of Russell’s Multiple-Relation Theory

Though interpretations of the multiple-relation theory differ on matters pertaining to its purpose in Principia, the force and nature of Wittgenstein’s criticisms, and the like, there are two core features by means of which it is commonly or standardly characterized. I call these features the “standard characterization” or “standard account” of the theory.

The first feature is that the multiple-relation theory is intended as an analysis of propositional attitudes such as belief, judgment, understanding, and the like. For instance, Griffin writes that “…the difficulty for Russell in dealing with belief is that he wanted an account of propositional attitudes which would not invoke propositions. Russell felt that there could be no such objects as propositions (in particular, there could be no such objects as false propositions), and thus the relation of believing could not be a dyadic relation between a subject and a proposition…” (1980, 134). With this problem so defined, Griffin continues (p. 134):

On this theory, Othello’s belief that Desdemona loves Cassio, e.g., is analyzed as a four-place belief-relation holding between Othello (the subject), Desdemona, the universal loves, and Cassio. Thus what appears to be a proposition that Desdemona loves Cassio, is broken down in to its components. The propositional expression becomes an incomplete symbol, analogous to a definite description, having no meaning on its own, but requiring completion by a “propositional” attitude in order to be significant.

The connection parallel with Russell’s theory of descriptions is said to be one of analogy. The propositional attitude completes the proposition symbol which has no meaning in isolation.

This interpretation is prima facia corroborated by the version of the multiple-relation theory given in Principia. Whitehead and Russell write that “…a ‘proposition’, in the sense in which a proposition is supposed to be the object of judgment, is a false abstraction, because judgment has several objects and not one” (1910, 44). That is to say, “the phrase which expresses a proposition is what we call an ‘incomplete symbol’; it does not have meaning in itself, but requires some supplementation in order to acquire a complete meaning” (ibid.) What occurs in an “elementary judgment” (i.e. a judgment in whose verbal expression a propositional object occurs which contains no quantifier phrases or truth-functional connectives), is a relation between a mind and the several constituents of the complex that would exist if the judgment is true. For example, the judgment that this is red is to be a relation of three terms: a mind, this, and redness. When the judgment is true there is a corresponding complex whose constituents are the objects of the judgment, viz., the-redness-of-this; the judgment is false when there is no such corresponding complex (p. 43).

Now the multiple-relation theory adopts the view that there is a primitive dyadic relation of acquaintance (or presentation, which is its converse) between the several objects which are the constituents of complexes, and with complexes themselves. Error cannot arise where acquaintance is concerned because the object of acquaintance, even where it is complex, must be something. Similarly, Whitehead and Russell write that “… when we perceive the-redness-of-this there is a relation of two terms, namely the mind and the complex object the-redness-of-this (ibid). Thus, error, cannot arise in immediate perception; nor can it arise in a “judgment of perception,” since the judgment is derived by mere attention to what is perceived.

Universals are also objects of acquaintance. The multiple-relation theory assumes a Platonic account according to which universals have both an individual as well as a predicable nature. Indeed, it is their individual nature that enables them to be objects of the belief relation. As early as The Principles of Mathematics (1903) Russell held that “… terms which are concepts differ from those which are not, not in virtue of self-subsistence, but in virtue of the fact that … they occur in a manner which is different in an indefinable way from the manner in which subjects or terms of relations occur” (1903, 47).

The extent of Russell’s Platonism is by no means certain, however. Is the multiple-relation theory committed to the assumption that every open well-formed formula (“wff”) stands for a universal with both an individual and a predicable nature? That is, are the “propositional functions” of Principia to be included among the universals which Russell allows as terms of the belief relation? Consider, then, the analysis of the judgment ‘a is human’, where (for convenience) ‘a’ is not taken to require further analysis:


Bel{m, a, human(í )}.

Here A’s judging mind m is related to a propositional function.

One might well object, however, that what Whitehead and Russell call a “propositional function” is a linguistic symbol. It is “… and expression containing a variable, or undetermined constituent, and becoming a proposition as soon as a definite value is assigned to the variable” (Russell 1912a, 133). To be sure, in The Problems of Philosophy, Russell maintained that we are acquainted with may universals, for example, sensible qualities and relations of space and time (1912b, 101). But on this view, Principia does not assume that every open formula stands for a universal. Hence, belief would he analyzed as:


Bel{m, a, Humanity}.

It is to universals, not propositional functions, that a mind may be severally related in belief.


But if this adequate to “elementary beliefs,” trouble arises for a belief whose apparent object is a general proposition such as ‘All men are mortal,’ or for a belief whose apparent object is a molecular proposition. As Anscombe puts it: “… what happens when I judge that A is not to the right of B? Do I stand in the judging relation to A, B, to the right of, and not? Similar questions arise for the other logical constants, ‘if’, ‘and’, ‘or’ “ (1959, 46). Russell might be able to analyze “A believes that Fa & Gb” as


m believes Fa & m believes Gb,

where m is A’s mind. But this technique will certainly not do for “A believes Fa v Gb”. One can believe the disjunction without believing either disjunct. Thus, as Griffin points out, even if Russell had a treatment for conjunction and negation, his analyses would not be truth-functionally complete (1980, 163). Molecular propositions involving implication and disjunction cannot be analyzed in terms of negation and conjunction. General propositions pose similar difficulties. What are the terms of the belief relation in such cases?

For some time the assumption that the propositional functions of Principia are universals has been viewed as the only alternative. Accordingly, I take this assumption as the second feature of the standard account of the multiple-relation theory. For instance, in writing on the theory, Nino Cocchiarella maintains that “… without including propositional functions among the single entities combined in a judgment or belief complex, there would simply be no multiple-relation theory of belief at all” (1987, 189). Cocchiarella argues that (at least) by 1910, Russell had identified propositional functions with “universals,” i.e., properties and relations in intension. A belief, he says “… is a multiple relation between a mind and the objects, including propositional functions (now taken as properties and relations in intension) that would otherwise be the constituents of the proposition being judged” (1980, 102). Hence he gives the following interpretation for the general belief that all men are mortal:


Bel{m, (x)( φ^!x É y^!x), x^ is a man, x^ is mortal},

where ‘(x)( φ^!x É y^!x)’ represents the second-order propositional function of material implication between two predicative propositional functions of individuals (1980, 104).


Of course, Cocchiarella recognizes that Principia introduces a hierarchy of senses of “truth” and “falsehood” which are to be grounded in the multiple-relation theory of judgment. The notion of “truth” in the case of general judgments does not lie in correspondence with a single complex whose constituents are the terms of the judging relation. According to Principia (p. 46):

… in any judgment (x).φx the sense in which this judgment is or may be true is not the same as that in which φx is or may be true. If φx is an elementary judgment, it is true which it points to a corresponding complex. But φx does not point to a single corresponding complex: the corresponding complexes are as numerous as the possible values of x.

On Cocchiarella’s interpretation, a general judgment (x).φx has “second-order truth,” because it is true when it corresponds to several complexes. An elementary judgment such as “a is human,” on the other hand, has “first-order (elementary) truth,” since, if true, it corresponds with a single complex. The theory of Principia, he maintains, is quite unlike the theory given in “Mathematical Logic as Based on the Theory of Types” (1908), where propositions, as objective truths and falsehoods, are assumed. That theory solved the Liar Paradox by splitting propositions, and accordingly truths and falsehoods, into a hierarchy of orders based on the admissible ranges of their bound propositional or individual variables. “Instead of taking propositions of different orders as being single entities that are themselves objectively truth or false,” writes Cocchiarella, “Russell now assumes that judgments as particular occurrence, or statements as potential judgments, are vehicles of a hierarchically ordered system of truth and falsehood” (1980, 104).

Principia introduces a hierarchy of senses of “truth” and “falsehood” which apply to judgments (beliefs), there being no propositions.

Thus, in Cocchiarella’s view, judgments are arranged in a hierarchy of orders. The order of a judgment’s truth or falsehood is the maximum of the orders of the propositional functions occurring as objects of the judgment. The notion of “correspondence” appropriate to the judgment is then determined by the order of the judgment. Since the judgment that all men are mortal involves a second-order propositional function as object of the judging relation, it has “second-order truth.” This means that it is true if it corresponds to the several complexes-viz., the-mortality-of-x1, the-mortality-of-x2, the-mortality-of-x3, etc., where all the x’s are men.

By assuming propositional function symbols always stand for universals, Cocchiarella’s interpretation goes a long way in providing an explanation of how the multiple-relation theory could handle a belief whose apparent object is a molecular or a quantified proposition. But I do not think that this is what Whitehead and Russell actually intended. Cocchiarella grounds the orders of judgment on an independently established hierarchy of orders of propositional functions. Yet, to the contrary, Whitehead and Russell do not ground the senses of “truth” and “falsehood” on the order of propositional functions occurring in a judgment or belief-complex. Indeed, there seem to be no orders of judgments (understood as complexes) at all!

Some care must be taken when interpreting Whitehead and Russell’s use of the word “judgment” in Principia. When a declarative sentence is asserted, there occurs a judgment insofar as there is a mental complex involved with the assertion. But the word “judgment” is not consistently reserved for a judgment (or belief) complex; sometimes it is used for the sentences asserted. For instance, they write that “we use the symbol ‘(x).φx to express the general judgment which asserts all judgments of the form ‘φx.’ Then the judgment ‘all men are mortal’ is equivalent to ‘(x). x is a man implies x is mortal…’ ” (p. 46). Here it is the sentence, not the judgment complex, which is intended. Thus, although Whitehead and Russell sometimes speak as if a general judgment is true or false, this should not be taken as sufficient evidence that there are orders of judgment-complexes. On the contrary, complexes such as a-in-the-relation-R-to-b are treated as individuals in Principia; and we maintain that this applies to all complexes, judgment complexes or otherwise.

The hierarchy of senses of “truth” and “falsehood” in Principia is applied to what Whitehead and Russell call “propositions.” It is a hierarchy of orders of propositions which is thereby generated, not a hierarchy of judgment-complexes. Of course, the “propositions” here are not the objective truths and falsehoods Russell formerly endorsed. They must be declarative sentences or statements capable of truth or falsehood. (Hereafter we shall write “propositionL” when this sense, as opposed to an objective truth or falsehood, is intended.) Whitehead and Russell explain (1910, 42):

That the words “true” and “false” have many different meanings, according to the kind of proposition to which they are applied, is not difficult to see. Let us take any function φx^, and let φa be one of its values. Let us call the sort of truth which is applicable to φa “first-truth.” (This is not to assume that this would be first-truth in another context; it is merely to indicate that it is the first sort of truth in our context.) Consider now the proposition (x).φx. If this has truth of the sort appropriate to it, that will mean that every value of φx has “first truth”… thus if we call the sort of truth that is appropriate to (x).φx “second truth,” we may define “{(x).φx} has second truth” as meaning “every value for φx^ has first truth,” i.e., “(x).(φx has first truth)”… Similar remarks apply to falsehood.


Recall that “(x).φx” is read as asserting (the truth of) all the propositionsL which are possible values of the propositional function φx^. Letting first-truth be truth for an atomic propositionL (i.e., a propositionL in which no quantifiers or logical connectives occur), the propositionL “(x).φx” will have second-truth insofar as it asserts the first-truth of all the propositionsL which are admissible values of the function φx^. For instance, “(x)man(x)” asserts the fist-truth of “a is a man,” “b is a man,” … and so on, where each of these is a value of the function “man(x^)”. Thus the orders of propositionsL - that is, the senses of “truth” and “falsehood” as applies to propositionsL - is built up in such a way that they step downward to the foundational base of first-truth and first-falsehood for atomic propositionsL.


In turn, the foundational notions of “truth” and “falsehood” are defined in terms of correspondence between judgment (belief) and fact in accordance with the multiple-relation theory. That is, when an atomic declarative sentence is asserted there occurs an “elementary judgment” insofar as there is a mental complex involved with assertion. Whitehead and Russell write that “we shall call a judgment elementary when it simply asserts such things as ‘a has the relation R to b’ or ‘a has the quality q’ or ‘a and b and c stand in the relation S’” (1910, 44). An elementary judgment is then said to be “true” when there is a corresponding complex and “false” when there is no corresponding complex. The truth conditions for an atomic propositionL such as “a has the relation R to b” consists in the fact that there is a complex corresponding to the judgment-complex which is involved with the assertion of the atomic propositionL.


Of course, a propositionL containing only bound individual variables, such as

“(x).φx” is called a “first-order proposition” in Principia (p. 54). Thus, it is better to say that it has “first-order truth” rather than “second-truth.” But Whitehead and Russell mean only that it has the second sort of truth possible, where “elementary truth” -i.e., “truth” for an “atomic propositionL” is the first sort of truth possible. Seeing this, we realize that the hierarch is also foundational when it comes to propositionsL involving bound variables for propositional functions. A propositionL such as “(φ)f!(φz^ )”, if true, will have “n+1 order truth” because asserts the “n-order truth” of every propositionL which is a value of the propositional function f!(z^). If, for example, f!(z^) is (x)y^!x then the propositionL “(φ)f!(φz^ )” has “second-order truth” (i.e., the third sort of truth possible) since every admissible value must be a propositionL which can be meaningfully said to have “first-order truth.” Because the senses of “truth” and “falsehood” step down in this way, it is clear that the propositional functions in the range of the quantifier must be such as to involve no bound quantifiers except those for individuals, else the values will not be capable of “first-order truth.” It is in just this stepping downward that bound variables are limited to orders, and the ramified hierarchy is to be explained.


Moreover, it is this stepping downward that solved the Liar paradox. That is, the variables of quantification over propositionsL are limited in their range by the hierarchy of meanings of “truth” (correspondence) as applied to propositionsL. Whitehead and Russell explain (p. 45):

…Applying these considerations to the proposition “(p). p is false,” we see that the kind of falsehood in question must be specified. If, for example, first falsehood is meant, the function “p^ is has first falsehood” is only significant when p is the sort of proposition which as first falsehood or first truth. Hence “(p). p is false” … has second falsehood, and is not a possible argument to the function “p^ has first falsehood.”

According to Principia, the propositionL “(p). p is false” asserts all the propositionsL which are admissible values of the function “p^ is false.” But once the notion of “falsehood” is fixed, say as “first-falsehood,” the admissible values exclude quantified propositionsL. Again, the limitation of the range of bound propositional variables to propositionsL of a given order is generated directly from the senses of “truth” and “falsehood” given by the multiple-relation theory.

If this is right, then Cocchiarella has things wrong side up. He takes a top-down approach which makes the order of a propositional function occurring in a belief-complex as fundamental and holds that this determines the nature of “truth” (correspondence). But Whitehead and Russell would appear to be taking a bottom-up approach. The relation of ‘correspondence’ is fundamental. It is the nature of the correspondence involved that establishes the sense of “truth” and “falsehood” and, accordingly, the orders of propositionsL.

When viewed in this way, the multiple-relation theory actually runs counter to the view that Principia includes propositional functions among universals. If Russell and Whitehead held that propositional functions are to be identified as universals, it is difficult to understand why there would not be a single corresponding complex when the belief,

Bel{m, (x)(φ^!x É y^!x), x^ is a man, x^ is mortal}

is true. Insofar as complex contain universals, nothing would seem to prevent there being a single complex man(x^)-in-the-relation-R-to-mortal(x^), where R is (x)(φ^!x É y^!x). Of course, if there were such a complex, then “truth” for the judgment that all men are mortal would presumably lie in its correspondence with this single complex. If would have “first-truth,” and this directly conflicts with the hierarchy of sense of “truth.” On the other hand, regarding propositional functions as expressions, as we have, we can see that the hierarchy of orders (both the propositionalL and the functional hierarchy) is generated, and philosophically explained, by the multiple-relation theory itself.

Now the matter of whether Principia assumes that every open wff stands for a universal has been earnestly debated for some years, and remains largely unsettled. We cannot take up the issue here. But we must recognize that attempts to understand how the multiple-relation theory analyzed molecular and generalized propositions have almost uniformly assumed that propositional functions are admissible as terms of a belief-complex. We shall show, to the contrary, that the theory can be, and indeed is best, understood without the assumption. In fact, the whole problem of the analysis of the truth-conditions for general and molecular propositionsL arises because the standard characterization of the multiple-relation theory obscures the foundational nature of the hierarchy of senses of “truth” and “falsehood.”


4. A New Interpretation of the Multiple-Relation Theory

In the 1907 article “On the Nature of Truth,” Russell sketched, without endorsing an early version of the multiple-relation theory. The article contains and important clue to a new interpretation which will show the way to a treatment of molecular and general propositionsL. Russell discusses an apparently telling argument for propositions, true or false, “…derived from the case of true propositions which contain false ones as constituent parts” (1907, 48). The difficulty goes back to a point made by in “Meinong’s Theory of Complexes and Assumptions” (1904). The symbols, “p” and “q” in “p É q” are apparently names of propositions. (Russell reads “p É q” as “p implies q”, or alternatively “if p is true then q is true”.) But then there must be false propositions, since the whole can be true even though p is false (1904, 74). Asserted propositionsL not containing any propositionsL as subordinate clauses are not problematic in Russell’s view. Analysis is needed only when a propositionL occurs unasserted as a subordinate clause, such as when it is said to be “true” or “false” or when it occurs in a truth-functional context. Here it is grammatically a name; and this seems to require false propositions. In 1904, Russell felt that this argument for false propositions was telling. Now armed with the theory of incomplete symbols, he writes that a solution can be found by “… an extension of the principle applied in my article ‘On Denoting’ (Mind, October 1905), where it is pointed out that such propositions as ‘the King of France is bald’ contain no constituent corresponding to the phrase ‘the King of France’” (1907, 48). This is corroborated in Principia when Russell writes that “… a proposition, like such phrases as ‘the so and so,’ where grammatically it appears as subject, must be broken up into its constituents if we are to find the truth subject or subjects” (p. 48). Taking this lead, we construe the multiple-relation theory as a direct application of the theory of incomplete symbols to contexts in which propositionsL occur as grammatical subjects.

On the present view, Russell is not saying that because a propositional symbol is incomplete it must always be flanked by a propositional attitude verb. A propositionL may occur as a subordinate clause in a truth-functional context where no propositional attitude verb is present. To be sure, it may occur as well in the context of a propositional attitude verb. Russell owes an account of both occurrences. But I contend that he never set out an account of propositionsL ascribing propositional attitudes in any published work, not even in the 1913 Theory of Knowledge.


Inattentiveness to this consideration leads to a failure to distinguish providing the truth-conditions of propositionsL not containing propositional attitudes from providing an analysis of the truth-conditions for propositionsL ascribing a belief such as “A believes that a has R to b.” The task of the multiple-relation theory of judgment in Principia is to analyze the subordinate occurrences of propositionsL embedded in contexts not containing verbs producing propositional attitudes. It is intended as a correspondence theory of truth for such propositionsL. The theory presupposes that there is a belief-complex Bel{m, a, R, b} available in defining “truth” (correspondence) for the propositionL “a is R to b”. Of course, Bel{m, a, R, b} is the fact which exists when the belief ascription “A believes that a has R to b” is true. But Russell is not giving an analysis of the truth-conditions of the propositionL “A believes that a is R to b.” Indeed, Bel{m, a, R, b} cannot be the definiens of a truth definition for a propositionL. Complexes (or facts) are not like Russell’s former propositions. The symbol “Bel{m, a, R, b} ” could alternatively be written as “m-in-the-relation-Believes-to-a-and-R-and-b”. The belief-complex should not be confused with the propositionL “the mind m is in a belief relation to a and R and b,” any more than the complex a-in-the-relation-R-to-b should be confused with the propositionL “a is R to b.” Complexes unlike Russell’s former propositions, cannot be asserted.

I maintain that, in Russell’s view, an unasserted propositionL is a disguised definite description and, like all definite descriptions, it disappears upon contextual definition analysis. It is in forming the description that belief-complexes come in. For example, using italicized “p”, “q” , “r” as variables for complexes, the description for the complex the-redness-of-a is

“(ip)(Bel{m, a, Redness} corresponds to p).”

The propositionL “a is Red” is true just when there is a unique complex corresponding to the belief-complex Bel{m, a, Redness}.

It may seem, however, that the analysis of molecular and general propositionsL cannot proceed in this way. The difficulty, when viewed from the standard account, is that the multiple-relation theory breaks up every propositionL, atomic, molecular, or general, into its logical constituents and takes the referent of each as a term of a single belief complex. When the original propositionL is molecular or general, it is held that the belief complex must contain as relata some ontic counterpart of the logical connectives or quantifiers; and “truth” is taken as correspondence between the single belief complex and one or more facts.

It is the standard account’s assumption that there is always a single mental complex that corresponds to one or more facts hat has prevented commentators from seeing that the multiple-relation theory is a direct (and not a vaguely) “analogous” application of the theory of descriptions. The pattern:

($1p)( Bel{m, x1,…, xn} corresponds to p)

is applicable only for atomic propositionL. (I use “$1 “ to express uniqueness.) The truth-conditions for a propositionL such as “(x)(Fx É Gx)” will not fit the pattern because, when true, a general propositionL does not correspond to exactly one fact; it corresponds with several, according to Principia. But once the assumption is dropped, we can see how Russell could provide truth-conditions for molecular and general propositionL without becoming mired in the problem concerning what terms of the belief relation are associated with logical connectives and quantifier phrases.

The Principia analysis of “truth” for an atomic propositionL such as “Fa” should be as follows:


“Fa” is true =df E!( i p)( Bel{m,a, F} corresponds to p)

By contextual definition the definiens is:


($p)( (q)( Bel{m, a, F} corresponds to q) º ( q = p) ).

(Hereafter, we use “($1).” ) Similarly, for the relational propositionL “a is R to b” Russell has:

a is R to b” is true =df E!( i p)( Bel{m,a, R, b} corresponds to p).

Since Russell reads “~Fa” as “ ‘Fa’ is false,” negation poses no problem. We do not have to look for an ontic counterpart for the logical connective “not” as a term of the belief relation. The proposition symbol “Fa” occurs as a subordinate clause when flanked by the tilde. The tilde completes the expression. We have:

“Fa” is false =df ~ E!( i p)( Bel{m,a,F} corresponds to p).

By contextual definition this is: ~($1p)( Bel{m, a, F} corresponds to p). Russell calls the sort of “truth” involved with elementary judgments “elementary truth.” As we can see, where atomic propositionL are concerned, it operates in just the way E! operates. It can meaningfully attach only to descriptions. An atomic proposition has “elementary truth” just when there is a unique corresponding complex; it has “elementary falsehood” when there is no such complex.

Now Russell’s reading of the propositionL “Fa É Gb” is If Fa is true then Gb is true.” Hence, what is required is that there be two descriptions, one for the (purported) complex

the-F-ness-of-a and one for the-G-ness-of-b. But here we encounter a problem. The descriptions

“( i p)( Bel{m, a, F} corresponds to p)” and “( i p)( Bel{m, b, G} corresponds to p)” will not do. The analysis must not assume that there is a single mind m (belonging to the person making the assertion) which has the mental occurrences Bel{m, a, F} and Bel{m, b, G}. If this were so, m would believe the antecedent and consequent of the implication. Since two descriptions are involved, the appropriate mental relation is not that of belief.

In “Knowledge by Acquaintance and Knowledge by Description” (1911), however, Russell explains that there is a mental state, distinct from belief and judgment, which accounts for our understanding of a proposition without making a judgment as to its truth or falsehood. He introduces the multiple-relation of supposition for this purpose (1911, 159). Thus we can solve our problem by employing supposition or understanding itself, instead of belief as the multiple-relation. With this clarification, we get the following analysis:

“Fa É Gb” is true =df

E!(i p)( U{m, a, F} corresponds to p) É E!( i p)( U{m, b, G} corresponds to p).

Contextually defined, the definiens is:

($1p)( U{m, a, F} corresponds to p) É ($1 p)( U{m, b, G} corresponds to p).

Of course, we still have unasserted occurrences of propositionsL here. But this is easily remedied by pulling quantifiers out front so that the propositionsL is in prenex normal form-a technique which Principia explicitly endorses in explaining the multiple-relation theory (1910, 47). When it is in prenex normal form, no unasserted propositionL occurs and there is no commitment to propositions. The resulting molecular propositionL, it should be noted, still can be said to have “elementary truth.” Though this is now understood in terms of the elementary truth (or better the “atomic truth”) of its subordinate atomic propositionsL.


Similar analyses are possible for the other truth-functions as well. In fact, the system is truth-functionally complete; the other truth-functions can be analyzed in terms of negation and implication. This, I contend, is Russell’s analysis of “elementary judgments” and “molecular judgments.”

After analysis what is asserted is that there is a ‘compresence’ (in a sense appropriated to the truth-functional connective in question) of complexes. Complexes cannot themselves stand in truth-functional relations, for the truth-functions would then require there to be false complexes (and true complexes) all over again. No complex contains an ontic counterpart of a truth-functional connective. But complexes have a structure and can map onto one another when the structures are ‘ordered’ similarly. This allows descriptions of complexes, and thus what would be truth-functional relationships between them can be emulated by means of the contextual definitions provided by Russell’s theory of descriptions.

It is not difficult to see that the analysis of general propositions would be a natural extension of Russell’s analysis of atomic and molecular judgments. We saw that a “first-order” general judgment asserts the elementary truth of a number of elementary judgments. The general judgment “All men are mortal,” Whitehead and Russell write, “collects together a number of elementary judgments” as in of a “radically new kind” (1910, 45). The judgment (i.e., propositionL to which the judgment is associated) is said to have “second-truth” and this is defined as meaning (p. 46):

(x). “x is mortal” has elementary truth, where x is a man”.

Hence the multiple-relation analysis yields:

(x). ($1p)( U{m, a, Mortality} corresponds to p), where x is a man).

The variable is limited to men here. But Whitehead and Russell go on to say that this is avoidable by interpreting “All men are mortal” as ‘(x). “x is a man” implies “x is mortal” .’ Recalling Russell’s reading of “implies,” and noting that “truth” here is elementary truth, we get the following analysis:

“(x)(man(x) É mortal(x))” has second-truth =df

(x){ ($1p)( U{m, a, Mortality} corresponds to p) É ($1p)( U{m, a, Human} corresponds to p) ).

With prenex normal form no unasserted proposition occurs in the analysans.

It is in this way that the multiple-relation theory generates the hierarchy of orders of “truth” and “falsehood” from the bottom up. By regarding unasserted occurrences of propositionsL as incomplete symbols, higher order “truth” and “falsehood” step downwards to the foundation of “elementary truth” and “elementary falsehood.”


5. The Problem of “Direction” and Russell’s Solution in Theory of Knowledge

Our new interpretation of the multiple-relation theory has important consequences for the understanding of Russell’s unfinished 1913 book Theory of Knowledge. The book attempts to deal with what has become known as the “direction problems.” After Griffin, we separate the problems into two kinds, one “narrow” and one “wide” (Griffin 1985, 219). We discuss the wide form of the problem in a subsequent section. The narrow direction problem, according to the standard account, arises when Russell attempts to explain how, e.g., Othello’s belief that Desdemona loves Cassio differs from his belief that Cassio loves Desdemona. The problem is said to be especially difficult for the multiple-relation theory because the theory breaks up the apparent propositional object of the belief into its constituents-viz., Desdemona, loves, and Cassio-and places each separately before the believing mind. If the two beliefs are to be different, then the theory will have to allow for the separate constituents to be arranged before the mind in the appropriate order. But then this looks suspiciously like allowing propositional objects of belief.

The standard account of the narrow direction problem is misleading. A fundamental assumption Russell made from the start was that a complex such as a-loves-b differs from b-loves-a because of its difference in structure. There is no reason why this should not apply to belief complexes as well. Thus, the belief-complexes are individuated. The problem of direction arises because of the difficulties involved in setting out the nature of “correspondence to fact.” Russell’s theory requires a definite description for the complex a-loves-b, and this description must not equally well describe the complex b-loves-a. The narrow direction problem is how to form a description which will capture the structure which individuates the complexes. In many cases, citing the constituents of a complex is insufficient, for there are other complexes with the same constituents.

G. F. Stout raised the direction problem against the early formulation of the multiple-relation theory. Russell was quick to acknowledge it, and changes his theory in “On the Nature of Truth and Falsehood” (1910). He tried to deal with the problem by making a distinction between the “sense” of the subordinate relation. Taking the propositionL “A loves B,” he wrote that the subordinate relation “… must not be abstractly before the mind, but must be before it as proceeding from A to B rather than from B to A. We may distinguish two ‘sense’ of a relation, according as it goes from A to B or from B to A. The relation as it enters into the judgment must have a ‘sense,’ and in the corresponding complex it must have the same ‘sense’ (p. 158). Stout was not satisfied, and Russell was soon to realize that this solution would not do. As he later put it, relations do not come with a “hook” at one end and an “eye” at the other (1913, 86). In its occurrence as a logical subject the so called “sense” of a relation is lost.

In The Problems of Philosophy (1912), Russell grants the need for a change. Russell now imagines the structural difference of complexes as a difference in the “order” of their constituents. Though a relation does not have a “sense” in its occurrence as a logical subject, it does order the constituents of a complex in which it occurs as a relating relation. The complex a-loves-b is viewed as having an ordering in a way that is analogous to the spatial ordering of the linguistic symbols “a” and “b” in the propositionL

a loves b.” The object a precedes b in the complex a-loves-b but not in b-loves-a.

According to Problems, this holds for belief complexes as well. In the belief-complex

Bel{m, a, loves, b} the relating relation believes imposes an order; the object a precedes the object b. This means that it is ordered in a way which is similar to that of the complex a-loves-b, for here the relating relation loves orders its terms so that the object a precedes the object b. On the other hand, in the complex b-loves-a the relating relation loves orders the complex so that the object b precedes the object a (1912, 128). Thus, the belief complex Bel{m, a, R, b} can (at best) correspond to a complex a-R-b and so the description “(i p)( Bel(m ,a ,R, b} corresponds to p)” is proper.

In Theory of Knowledge, Russell came to reject this solution. The problem is that the notion of the “order” of a complex relies on a spatial analogy. In a chapter entitled “On the Acquaintance Involved in Our Knowledge of Relations,” he realized that this is merely an appearance produced by the spatial order of the linguistic symbols in the propositionL “a loves b.” Complexes do not have a “first-term” or a “second-term.” “In a dual complex,” Russell writes, “there is no essential order as between the terms. The order is introduced by the words of symbols used in naming the complex, and does not exist in the complex itself (1913, 87). Even in its relating capacity in a complex a relation does not “order” the constituents in a linear or spatial way. The complexes a-R-b and b-R-a have a difference in structure, but that difference is not one of the “linear order of terms.”

Russell’s new solution is to introduce relations of position that the constituents a complex, aside from the relating relation, have in the complex. Given a dyadic relation R, there are two “position” relations C1 and C2, which are determined by R, and such that if a and be are entities in a complex whose relating relation is R, as has C1 to the complex, while b has C2 to the complex. “Sense is not in the relation alone,” writes Russell, “nor in the complex alone, but in the relations of the constituents to the complex which constitute the “position” in the complex” (p. 88).

The relations of position are paired with another innovation unique to Theory of Knowledge-namely, the notion of the “logical form.” In a chapter on acquaintance with relations, Russell says that “a complex has a property in which we may call its ‘form’ and the constituents must have what we call a determinate position in this form” (p. 81). Later, in a chapter entitled “Logical Data,” he continues (p. 99):

Suppose that someone tells us that Socrates precedes Plato. How are we to know what this means? It is plain that this statement does not give us acquaintance with the complex ‘Socrates precedes Plato’. What we understand his that Socrates and Plato and ‘precedes’ are united in a complex of the form ‘xRy’, where Socrates had the x-place and Plato has the y-place. It is difficult to see how we could possibly understand how Socrates and Plato and ‘precedes’ are to be combined unless we had acquaintance with the form of the complex.

It is important to read these comments with caution. While logical form plays a central role in Russell’s solution of the direction problem, Russell does not intend that the form is a template into which objects are to be fitted, not indeed is the form to be structurally isomorphic with the (purported) fact with exists if the propositionL is true. Russell’s comments obscure this by making it seem as thought the constituents of a complex (excepting the relating relation) have a position “in the form.” But when he speaks fo the relations of position, he makes it clear that the wording is inexact.

Russell tells us that the complex a-similarity-b is “symmetrical” and there fore has only one position; it is identical with the complex b-similarity-a (p. 122). The complex a-before-b, on the other hand, is distinct from the complex b-before-a. It is “unsymmetrical” since its constituents occupy different positions (p. 123). But a-similarity-b, a-before-b, and b-before-a all have the form xRy, the form of a dual complex. Later, in applying the theory of “position” to solve the narrow direction problem, he reaffirms that “… it is to be observed that the relations C1, C2, …, Cn are not determined by the general form, but only by the relation R. So far as the general form “xRy” is concerned, the position of A is the same in

A-before-B as in B-after-A” (p. 146). Thus, an object has a position in a complex, not in a form. The introduction of logical forms plays a quite different role.

Russell’s solution of the narrow direction problem requires the classification of complexes into those that are “permutative” and those that are “non-permutative” (i.e.. not-permutative). The theory of logical form has two main purposes. The first is to ground this classification by establishing the conditions under which a complex is logically possible. A complex is “symmetrical” with respect to two of its constituents if they occupy the same position in the complex (p. 122). Unsymmetrical complexes divide into those that are “homogeneous” with respect to its unsymmetrical constituents if it is logically possible for there to be a complex whose constituents occupy different positions; otherwise it is heterogeneous (p. 123). Permutative complexes are those that are both unsymmetric and homogeneous. The theory of logical form assures that it sis only with permutative complexes that the direction can arise. In all other cases, it is not logically possible for the constituents to occupy positions in the complex different from those they do occupy.

The second purpose of the theory of logical form is to provide the content of our epistemic understanding of what complexes are logically possible. Belief involves understanding of the propositionL believed. In Russell’s view, it is acquaintance with logical form (or “logical intuition” as he sometimes calls it) that is essential for, and logically prior to, understanding words such as “predicate,” “relation,” and “dual complex” (p. 101). Acquaintance with the form, he says, is necessary in all cases where we understand a statement without having acquaintance with the complex whose existence would insure its truth. It is presupposed in understanding a sentence at all (p. 99). For example, understanding the sentence “a is similar to b” requires more than being acquainted with the entities, a, b, and similarity. It requires an understanding of what a dual relation is-i.e., it requires acquaintance with what it is for two terms to have a relation. Like Belief, Understanding is a multiple-relation, and the complex U{m, a, Similarity, b, xRy} involves the mind’s acquaintance with a logical form.

There are many perplexities and difficulties with Russell’s account of logical form. One major question is the ontological status of logical forms and the nature of acquaintance with them. The issue was not settled in Theory of Knowledge. In the chapter entitled “Logical Data,” he notes that “… logical objects cannot be regarded as ‘entities,’ and therefore, what we shall call ‘acquaintance’ with them cannot really be a dual relation” (p. 97). There must then be kinds of acquaintance. Universals, concrete particulars, and complexes, differ logically, and “… relations to objects differing in logical character must themselves be differ in logical character” (p. 100). But Russell recognizes the difficulties here. “For the present,” he writes, “I am content to point out that there certainly is such a thing as ‘logical experience,’ by which I mean that kind of immediate knowledge, other than judgment, which is what enables us to understand logical terms” (p. 97). Whatever the analysis of logical objects, he says that something like “logical experience” or “logical intuition” is required for the understanding of logical words such as “particular,” “universal,” “relation,” “dual complex,” “predicate.” Logical intuition is a kind of immediate knowledge attained through a process of abstraction and generalization carried to its utmost limit. Thus, every logical notion is a summum genus-the limit of this process of abstraction.

Finally, in a later chapter called “The Understanding of Propositions,” the issue of the status of logical forms is decided in a tentative way. “If possible,” Russell says, “it would be convenient to take the form as something which is not a mere incomplete symbol” (p. 114). Recognizing the problems of construing logical forms as incomplete symbols, Russell suggests that the logical form can be regarded as itself a fact-namely, the fact that there are such entities that make up complexes having the form in question (p. 114):

… the form of all subject-predicate complexes will be the fact ‘something has some predicate’; the form of all dual complexes will be ‘something has some relation to something.’

Thus, what accounts for understanding the word “predicate” (and predication) lies in immediate acquaintance with the abstract general fact something has some predicate. This, of course, only pushes the problem back, raising questions as to the nature of such facts themselves.

It is clear, however, that Russell does not intend that abstract facts be understood to contain ontic counterparts of quantifier phrases or logical connectives. Remarking on the logical form of a dual complex, he writes (Ibid.):

… the logical nature of this fact is very peculiar. For ‘something has the relation R to something’ contains no constituent except R; and ‘something has some relation to something’ contains no constituents at all. In a sense, it is simple, since it cannot be analyzed. At first sight, it seems to have a structure, and there not to be simple; but it is more correct to say that it is a structure.

Though it may be puzzling how the “process of abstraction and generalization” yields a logical form construed as a fact with no constituents, the passage shows that logical forms do not contain quantifiers or variables. Moreover, Russell says that “ ‘logical constants’ which might seem to be entities occurring in logical propositions, are really concerned with form, and are not actually constituents of the propositions in the verbal expression of which their names occur” (p. 98). Later he remarks that “… a molecular form is not even the form of any actual particular; no particular, however complex, has the form ‘this or that,’ or the form ‘not-this’” (p. 132). Hence, Theory of Knowledge concurs with our reading of Principia.

It is important to understand as well that although the abstract facts which are (or represent) logical forms are “logical” and therefore subsist independently of the world as the structures of any “logically possible world,” they are not logically true. Properly speaking, facts are neither true nor false (logically or contingently), though (in a sense) they may exist contingently or (if abstract) subsist necessarily. Now since logical forms account for the logically possibility of complexes, they must subsist necessarily. But this is not to say that the propositionL “something has some relation to something” is logically true. For on the present construal of the multiple-relation theory, the abstract fact is not what makes the propositionL true. Indeed, the abstract fact is without constituents, and thus it is not possible to form a description by breaking up the constituents and having each severally before the mind. Rather, the truth-conditions would be:

($z)($u)($S)($1p)( U{m, z, S, u, xRy} corresponds to p) .

Thus, what makes the propositionL true is the existence of a complex such as a-S-b; and surely this is not necessary.

This is not to deny that there are “abstract logical facts” which ground the necessity of logical truths. In 1911, Russell held that his “ultimate dualism” between universal and particular includes all objects (1911, 155). But by the time of Problems Russell recognized that he would need abstract logical facts to account for logic and mathematics (Cocchiarella 1987, 205). According to Problems, “all a priori knowledge deals exclusively with the relations of universals” (1912b, 103). In particular, a priori knowledge of certain logical and arithmetical principles lies in the “… we have the power of sometimes perceiving such relations between universals” (p. 105). Russell continues the view in Theory of Knowledge, writing that “… perceptions of facts (as opposed to the corresponding judgments) must be included among data… and perceived facts are not always particular; general logical facts, for example, are often such as can be perceived” (p. 47).

The presence of such abstract logical facts, however, may appear to conflict with our interpretation of the hierarchy of orders of propositionsL of Principia. Indeed, if Principia does not assume that every open wff stands for a universal, what are the “connections or relations between universals” perceived in knowing and abstract logical fact?

As we saw, the higher-order “truth” and “falsehood” of a propositionL steps downward and eventually lies in the elementary “truth” or “falsehood” of many atomic propositionsL. Consider, for instance, the higher-order “truth” of the propositionL “L Î 0”. According to the logicism of Principia, this is contextually defined in terms of propositional functions as follows:

($φ){ (x)( φ!x º (F)(Fx & ~Fx)) & ($f)( (y)( f!( y!z^) º f!(φ!z^) ) )}.

Now the higher order “truth” of this propositionL steps downward to the truth of the propositionL

f!(φ!z^),” i.e., “(y) ~($F)(Fy & ~Fy).” We could go on and apply the multiple-relation analysis to get the truth-conditions. But Russell needs more than just the truth-conditions for this propositionL, he must ground its logical truth. Moreover, he wishes to give an epistemic account for our a priori knowledge of this logical truth.

As I interpret Russell, he achieves these ends by grounding our a priori knowledge on the direct perception of abstract logical facts. For instance, it is through acquaintance with the logical fact

(a) ~($φ)( φa & ~ φa) that we perceive a logical relationship between universals which assures the truth of the propositionL ., “(y) ~($F)(Fy & ~Fy).” The propositionL will be self-evident to a person A in virtue of the relation of acquaintance holding between A’s mind and this abstract logical fact. Accordingly, our a priori knowledge of the mathematical propositionL “L Î 0” lies in our immediate perception of (or acquaintance with) this abstract logical fact.


As with the fact that are (or represent) logical forms, the subsistence of abstract logical fracts does not conflict with the hierarchy of orders of propositionsL. Again, no constituents occur in any purely logical truth. In Theory of Knowledge, Russell says that the propositionL “if Socrates is human, and whatever is human is mortal, then Socrates is mortal” is not a propositionL of pure logic, for the truth of a logical propositionL “… is in no way dependent on any peculiarity of Socrates or humanity or mortality, but only on the form of the proposition” (p. 98). But by generalization (ibid. ):

… we arrive at a pure logical proposition: “Whatever x and a and b may be, if x is a and whatever is a is b, then x is b.” The only thing that has been preserved is the pure form… It is such pure “forms” that occur in logic.

Thus, while an abstract fact can be the immediate object of perception (or acquaintance), the truth-conditions for propositionsL containing quantifier phrases (even when they are logically true) is not to be analyzed in terms of correspondence with a single fact. For without constituents, the abstract fact cannot be broken up in accordance with the multiple-relation theory.

Difficulties with the ontic status of logical forms, as with abstract facts generally, reflect the largely unfinished condition of Russell’s theory of logical form. Fortunately, the issue of the ontic status of logical forms is incidental to Russell’s solution of the “narrow direction problem.” What is crucial is their metaphysical role. By establishing conditions under which complexes are logically possible, the theory of logical form was to assure that the narrow direction problem can only arise for “permutative” complexes.

Complexes that are “non-permutative” are complete determined by their constituents and their form. Hence, on our interpretation, the analysis of the truth of a propositionL such as “a is similar to b” is much as it was in Principia:

a is similar to b” is true =df ($1p)( Bel{m, a, Similarity, b, xRy} corresponds to p) .

The narrow direction problem cannot arise here, for the complex a-similarity-b which would exist if the sentence is true, is non-permutative. It is symmetric with only one position. Now it may be objected that the definiens use a name of the belief complex Bel{m, a, Similarity, b, xRy} and so the problem of permutation arises here. But I think that Russell regarded this belief-complex as non-permutative and so nameable. Since a-similarity-b has only one position, switching a and b does not generate a new belief-complex. Moreover, as long as there is acquaintance with the logical form xRy the mind understands what it is for something to be related to something; thus, switching other objects in the complex yield no belief-complex at all.

On the other hand, the analysis of the truth of a propositionL such as “a loves b” involves the troubling notion of “correspondence” with a permutative complex and so requires special treatment. Since the complex a-loves-b is permutative, it will not be possible to use a description such as

(i p)(Bel{m, a,Similarity, b, xRy} corresponds to p) .

Enumeration of the constituents a, loves, and b, together with the logical form, is not sufficient to single out the complex a-loves-b as opposed to b-loves-a. Moreover, to confound matters even more, the belief-complex Bel{m, a, loves, b, xRy} is also permutative and thus cannot be named in the description. Russell avoids these problems by showing how to form a definite description for any n-placed permutative complex whose relating relation is R (p. 147). By using the position relations C1, C2, …, Cn determined by R, he has:

(i g )( x1 C1 g .&. x2 C2 g .&.,…, &., xn Cn g )

Applying this to “a loves b,” we get:

a loves b” is true =df ($1p)( a C1 p .&. b C2 p)

where the position relations are determined by loves. Since no unasserted propositionL occurs, we have a complete analysis of “truth” for propositionsL whose truth-conditions involved permutative complexes.

Some commentators have argued that there is an infinite regress here. Hochberg, for example, holds that attempts to apply the theory of descriptions to the problem of false propositions must fail because it will either face and infinite regress or rely upon the ability of atomic propositionsL to show their truth-conditions. He writes that Russell’s sentences “a C1 p” and “b C2 p” requires analysis and thus “… an infinite regress begins or is only stopped by taking such clauses to have a representational role irrespective of questions of truth or falsehood. If the atomic sentences are taken to represent of “show” their truth-conditions, we do not require other sentences of the schema to state them. In effect, Russell tries to construct a schema in which other sentences of the schema are required to express the truth-conditions for every atomic sentence of the schema. This is why he falls victim to a regress” (Hochberg 1987, 433ff).

This is not so. The definiens contains no unasserted propositionsL, so there is not reason why the analysis of “truth” for the definiens should itself be involved in the analysis of “truth” of “a loves b.” (Indeed, according to Russell’s hierarchy of sense of “truth,” the meaning of “truth” for the quantified propositionL of the definiens is different from that of the definiendum.) Moreover, the atomic propositionsL “a C1 p” and “b C2 p” have truth-conditions that do not fit the same schema as the atomic propositionL

a loves b.” Not all propositionL are treated alike. Some have truth-conditions involving permutative complexes, other do not. Those that do not form the basis for those that do.

To see this clearly, we need to understand the analysis of molecular propositionL. Russell defers the issue of molecular propositionL to a later chapter entitled “Molecular Propositional Thought.” Unfortunately, it was never written. But we can apply our earlier interpretation of Russell’s analysis of molecular propositionL given in Principia:

“($1p)( a C1 p .&. b C2 p)” is true =df

($1p)( ($1q)( U{m, a, C1 , p , xRb} corresponds to q) &

($1q)( U{m, b, C2 , p , xRb} corresponds to q) ).

We use the expression “xRb to indicate the logical form of an unsymmetric and heterogeneous complex. The difference in notation between “x” and “b “ is intended to indicate the logical difference between the terms of any complex having this form. That is, the complex , a-C1-p where C1 is a position relation and p is a complex, is heterogeneous and so non-permutative (p. 147). Accordingly, this assures that the propositionL “p has the position C1 in a” is meaningless. It is logically impossible for a complex to have a position in an object which is not a complex. In Russell’s view, no infinite regress occurs here because the whole difficulty of “sense” cannot arise (p. 112). By the same reasoning, a complex such as

U{m, a, C, p , xRb} , where C is a position relation and xRb is the logical form is non-permutative.

Hochberg fails to appreciate that Russell’s theory is two-tiered. The problematic notion of “correspondence” with a permutative complex is defined by analyzing “truth” for permutative sentences in terms of correspondence with non-permutative complexes. In Russell’s words (p. 148):

… the only propositions that can be directly asserted or believed are non-permutative, and are covered by our original simple definition. Owing to the above construction of associated non-permutative complexes, it is possible to have a belief which is true if there is a certain permutative complex, and is false otherwise; but the permutative complex is not itself the one directly “corresponding” to the belief, but is one whose existences is asserted, by description, in the belief…

The nature of “correspondence” for sentences whose truth-conditions involve permutative complexes is thus unique. Where sentences whose truth-conditions involve permutative complexes are concerned, “correspondence” is indirect. It presupposes the direct correspondence involved with non-permutative complexes. The truth of the propositionL “a loves b” is analyzed in terms of the existence of the non-permutative complexes a-C1-p and b-C2-p. That is, in instantiating the quantifiers, the complex q is a-C1-p, the complex r is b-C2-p. Given that the “position” relations are determined by, and particular to, the relating relation loves, the existence of these complexes assure that the complex p is a-loves-b.

In sum, Russell began by reading the unasserted occurrence of “p” in “p implies q” as a name of a proposition. Until 1907 he thought this to yield a compelling argument for propositions as objective truths and falsehoods. The multiple-relation theory attempts to avoid the argument by regarding “p” as a disguised description for a complex (fact). But by Theory of Knowledge Russell recognized that a sentence (propositionL) whose truth-conditions involve “correspondence” with a permutative complex encounters the direction problem. PropositionsL show direction or order by being asserted. Through inflections, the sequential concatenation of symbols, and the like, position relations are shown (p. 148). When propositionsL occur unasserted as subordinate clauses, however, direction is lost. A name is stagnant, inactive; it cannot show direction. A permutative complex cannot be names (ibid.). Neither, however, can it be described simply by enumerating its constituents. Direction cannot be captured in this way. To solve the problem, Russell shows how to describe permutative complexes in a new way. By using the position relations, he is able to associate every permutative complex with certain non-permutative complexes. The technique requires that there are atomic propositionsL which do not have the direction problem; their “truth” involves correspondence with the associated non-permutative complexes. So long as the two-tiered nature of Russell’s analysis succeeds, descriptions for permutative complexes are possible.





6. The Problem of the Content of Belief

After completing their work for Principia, Russell and Whitehead had discussed a plan for their future work in philosophy. According to Russell’s letter to Ottoline Morrell, Whitehead was working on a technical apparatus (to be Vol. IV of Principia) which would apply logical techniques to the definition of points in space, moments in time and the geometry of physics. Russell was to work on the same problems from the psychological (epistemological) end, showing how all knowledge (logical and empirical) could be obtained by synthesis from the foundation of acquaintance with sense-data, sensibilia, and universals (Morell 1912-1916, 667). With Problems of Philosophy Russell had an initial sketch and Theory of Knowledge was to carry out the program.

The work was to have two sections, one “Analytic” and the other “Synthetic.” The first set out the epistemic foundation for the system and placed the multiple-relation theory as the bridge from the foundation to the synthesis involved in judgment. There were three projected parts: “On the Nature of Acquaintance,” Atomic Propositional Thought,” and “Molecular Propositional Thought (Inference).” Only two parts of the analytic section are extant. The first was revised and published in the Monist; the second remained in manuscript; and the work seems to have collapsed under Wittgenstein’s objections before the third was written. The synthetic section was to explain the process of synthesis involved in knowledge. Excerpts were used in such works as the 1914 “On the Relation of Sense-Data to Physics” and the book Our Knowledge of the External World of the same year.

From Russell’s letters to Ottoline Morrell on 4 and 8 May 1913, it seems that he had the whole project worked out in his head (see Morrell 1912-1916, 764, and 768). Thus, it is likely that when he began writing he had dome idea as to how the multiple-relation theory would form the bridge. Though the part on molecular judgment, which was to analyze the inferential relationships involved in logical and synthetic knowledge, was never carried out, our new interpretation affords us with a glimpse.

There are important departures from Problems in Russell’s synthesis. In Problems our knowledge of physical objects (or better “continuants,” since sense-data are physical for Russell) is descriptive in nature. It is inferred on the basis of our immediate acquaintance with the sense-data they cause (1912, 47). For example, the table before me is known only by means of some description such as “the cause of my sense-data so and so.” Moreover, while he tentative assumed that we have acquaintance with the “self,” our knowledge of other minds is also descriptive. It is inferred from acquaintance with sense-data associated with certain verbal and physical behavior (p. 49). Acquaintance is the foundation of all descriptive knowledge. The fundamental principle is that “every proposition which we can understand must be composed wholly of constituents with which we are acquainted” (p. 58).

How, then, was Problems to analyze the sentence “a is G,” where “a” is an ordinary proper name for a continuant? Since a is a continuant, it is not a proper object of acquaintance. The term “a” is to be replaced with a description “ix Fx” which purports to refer uniquely to a as the cause of certain sense-data (and sensibilia). But the analysis

ix Fx is G” is true =df

($x)( (y)(Fy º x = y) & ($1p)( Bel{m, x, G} corresponds to p) )

does not work. The object x must be capable of being an object of acquaintance before the mind m and this is impossible if x is a continuant. A complex corresponding to a mental complex must always contain constituent objects with which the mind of the person having the mental occurrence is acquainted. But another persons’ mind or a continuant is never an object of acquaintance. What then is the content of belief?

Russell may have approached the problem by taking a sense-datum as the value of the variable occurring as a term in the expression for the belief complex. The continuant is, then, the inferred cause of a certain sense-complex which is the immediate complex that corresponds to the belief (when true)-the mediate complex being the complex containing the continuant. In rough sketch, let F be a property a continuant has just when it causes sense-data s with property h, and let G be a property a continuant has just when it causes sense-data s’ with property g. Then “ix Fx is G” becomes “the x which causes sense-complex h(s) causes sense-complex g(s’ ).” We have:

ix Fx is G” is true =df

($x){ (y)[ ($1p)( Ul{m, s, h} corresponds to p) & y causes p) º x = y)]

& ($1q)( U{m, s’, g} corresponds to q) & x causes q )}.

Variables for continuants do not occur as part of any term for a belief complex here.

By 1913, however, some of the ideas of Problems were modified in a way that helps to avoid the trouble. At the suggestion of Whitehead, Russell began to rethink the notion of “inference” espoused in Problems. In the article “On the Relation of Sense-data to Physics,” he remarks that “inference” is to be replaced by “synthesis”-i.e., logical construction from sense-data (and sensibilia). Acquaintance is still the foundation, but Russell now writes: “Wherever possible logical constructions are to be substituted for inferred entities” (1914a, 115). Continuants are no longer inferred as the “causes” of certain sense-data. A continuant is a “logical construct” from sense-data and sensibilia. The innovation of logical construction would presumably solve the problem of acquaintance when continuants are the apparent values of the variables occurring in terms for belief complexes. For the variables would range only over objects of acquaintance such as sense-data.

But the problem of the content of belief also arises in another way. It is central to the analysis of ascriptions of belief. Applying the present interpretation of the multiple-relation theory yields a straightforward truth defintion for propositionsL ascribing beliefs whose apparent objects are atomic propositions. Consider the propositionL “A believes that a is red,” where a is a sense-datum:

“A believes (de dicto) that a is red” is true =df

($1p)( Bel{m*, Bel, m, a, Redness, fx, f(F, x, g, L)} corresponds to p).

Here m* is the mind of the person making attributing the belief to A, and m is A’s mind. We use fx as the logical form of predication and f(F, x, g, L)} as the logical form which accounts for our understanding of the simplest kind of belief-relation. (The symbol “f” represents a mental state such as belief or understanding, “F” represents a mind, “g” a property, and “L” a logical form.) On this analysis, the propositionL is true just when the complex p, viz., Bel{m, a, Redness, fx}, exists. The person A believes truly when (($1q)(p corresponds to q), that is, when the propositionL “a is red” is true. Of course, the person attributing belief cannot be acquainted with A’s mind on Russell’s view. It is known by description, i.e., through the sense-data associated with A’s verbal and physical behavior. Thus, it will have to be treated as above. Moreover the case of belief ascriptions such as “A believes that a is left of b” will be more complicated insofar as they involve correspondence with permutative beliefs. The analysis would presumably require appeal to Russell’s “position relations.” For example, we get a description such as:

(i p )( m C1 p .&. a C2 p .&. b C3 p .&. left of C4 p .&. (xRy) C5 p ),

where C1, C2, C3, C4, C5 are determined by the relating relation ‘Believes.’

On the other hand, the case for an ascription of a belief whose apparent objects is a molecular or general proposition is quite difficult. Attempts to surmise what might have been Russell’s intended analysis (or at least the direct in which he was moving) often appeal to tentative remarks he made when discussing logical form. Russell notes that our understanding of sentences containing the words “all,” “some,” “or,” and “not” plainly involve logical notions. “Since we can use such words intelligently,” he says, “we must be acquainted with the logical objects involved. But the difficulty of isolation here is very great, and I do not know what the logical objects involved really are” (p. 99). This shows that Russell considered handling the problem by means of “molecular logical forms.” Just as atomic logical forms are essential to our understanding of terms such as “predicate,” “relation,” etc., Russell suggests that acquaintance with molecular logical forms would be involved in understanding phrases such as “all,” “some,” “not,” “or” and so on.

These remarks do not, however, support the interpretation that the problem was to be solved by taking propositional functions to be objects of the belief-complexes involved. Not only does this approach conflict with the hierarchy of “truth” and “falsehood,” it also obviates Russell’s labors to solve the narrow direction problem. For the permutative complex a-loves-b, just put Bel{m, a, φx^, fx}, where φx^ is the propositional function ‘x^ loves b’, and the constituents can go together in only one way. Moreover, if Russell intended to allow propositional functions as objects of the belief relation, surely we would not find him saying that “I do not know what the logical objects involved really are”!

Given that Russell adopted a phenomenalistic logical reconstruction of continuants in Our Knowledge of the External World, it might be tempting to suggest that he may have been moving toward a phenomenalistic and (quasi-behavioristic) line when it comes to the content of a belief whose apparent object is a molecular or generalized proposition. In the case of generalized propositions, for instance, the analysis must provide some mental content for the variable and assure that the right “variables” are bound. A phenomenalist (or radical empiricist) could approach the variable through the notion of “reflective abstraction” on a presented pattern of understanding-complexes, U{m, a, F, fx}, U{m, b, F, fx},

U{m, c, F, fx} …, and so on. Generalization “binds” (as it were) the changing term in the pattern. That is, the mental content of the universal generalization lies in being in a state of expectation developed by attention to patterns of sense-experience.

Such an approach does not, however, lend itself to the analysis of belief attributions of propositions involving logical connectives. For instance, “A believes (de dicto) that (x)(Fx É Gx),” the connective is the material conditional and surely Russell would not have held that the mental content of A’s belief should be constructed in terms of habits of expectation derived from sense-experience. Moreover, the approach seems patently too empiricistic for Russell’s Logicism to sustain. Contrasting a genuine a priori judgment wit the empirical generalization “All men are mortal,” Russell wrote in Problems (1912b, 107):

… we understand what the proposition means as soon as we understand the universals involved, man and mortal. It is obviously unnecessary to have an individual acquaintance with the whole human race in order to understand what our proposition means. Thus the difference between an a priori general proposition and an empirical generalization does not come in the meaning of the proposition; it comes in the evidence for it. In the empirical case, the evidence consists in the particular instances… the ultimate ground remains inductive, i.e., derived from instances, and not an a priori connection of universals such as we have in logic and mathematics.

Thus, whatever the analysis, understanding must be the same whether the propositionL is an empirical generalization or a logical truth.

Taking the abstract fact ($f)($θ)(a)( fa É θa) as the logical form which is the object of acquaintance in understanding universal generalization (of a certain kind) an neglecting the narrow direction problem, we offer the following analysis:


A believes (de dicto) that (x)(man(x) É mortal(x))” is true =df

($1p)( Bel{m*, Bel, m, Humanity, Mortality, ($f)($θ)(a)( fa É θa) , f(F, f, g, L)}

corresponds to p).

where f(F, f, g, L) is the logical form before the mind m* of the person attributing the belief to A. It accounts for his understanding of the predicate “belief” and the nature of belief-complexes of the sort he is attributing to A. ( “f” represents a mental state such as belief, “F” indicates a mind, “f ” and “g” indicate universals, and “L” indicates a logical form.) The belief-complex asserted to exist is:

Bel{m, Humanity, Mortality, ($f)($θ)(a)( fa É θa)}.

The person A believes truly, however, not when this complex corresponds to many facts, but when the propositionL “(x)(man(x) É mortal(x))” is true. In this way, our interpretation accords with the hierarchy of orders in Principia.

When we have a logical truth such as “(F)(x)(Fx É Fx),” understanding no longer involves a multiple-relation. In this case, Russell tells us, there is no difference between “understanding” and acquaintance with the logical form (1910, 132). This follows because there are no non-logical constants in the propositionL. We now have:

A believes (de dicto) that (F)(x)(Fx É Fx)” is true =df

($1p)( Bel{m*, Bel, m, (f)(a)( fa É fa) } corresponds to p).

The belief-complex which exists when the ascription is true is:

Bel{m, (f)(a)( fa É fa)}.

Once understood, the logical truth is self-evident. The connection between universals is directly perceived through acquaintance with the logical form. The person believes truly, however, not because the belief corresponds to this logical form, but because “(F)(x)(Fx É Fx)” is true.

Note, of course, that since we do not assume every wff stands for a universal, we do not analyze the truth-conditions for a propositionL such as “A believes (de dicto) that (x)(Fx É (Gx É Fx)) ” in terms of the existence of a belief complex:

Bel{m, Fz^ É Gz^, ($f)($θ)(a)( fa É θa)}.

Instead, we use the logical form

($f)($θ)(a)( fa É (θa É fa).

and put

Bel{m, F, G, ($f)($θ)(a)( fa É (θa É fa).}.

Thus, we are committed to a proliferation of logical forms. But the present interpretation is fully extendible to the analysis of propositionsL attributing beliefs whose apparent objects are propositions involving heterogeneous quantifiers and the like.

Such an analysis, we maintain, was probably what Russell had in mind when he wrote Theory of Knowledge. It is significant, however, that even if this fails and leaves Russell with no answer to the question of the analysis of belief ascriptions, the multiple-relation theory in Principia is not jeopardized. The two problems are separable. In fact, even the “synthesis” required for analyzing inference, which is involved in acquiring empirical knowledge, need not be mired in the problem of the content of belief. Accordingly, though Theory of Knowledge came to a halt at the part on molecular propositional thought, I do not think that the reason was the problem of the content of belief. As we shall see, what failed to survive Wittgenstein’s criticism was Russell’s multiple-relation theory-even at the point where it deals with non-relational and non-permutative complexes.


7. Wittgenstein’s Criticisms and the Abandonment of Theory of Knowledge


The “wide” form of the direction problem was raised by Wittgenstein through letters and discussions prior to and during the writing of the book. Russell admits that he was not always certain of Wittgenstein’s points. Confessing to Ottoline Morrell on 27 May 1913 (Morrell 1912-1916, 787) he writes:

I showed him a crucial part of what I have been writing. He said it was all wrong. … I couldn’t understand his objection-in fact, he was very inarticulate-but I felt in my bones that he must be right, and that he has seen something I have missed. If I could see to I shouldn’t mind, but as it is, it is worrying, and has rather destroyed the pleasure in my writing…

At this time, Russell was working on logical form and the theory of position. Russell got over this criticism, if only “superficially and by act of will” as he confessed to Ottoline on the first of June (p. 793), and by 6 June he had finished the part on atomic judgment. But Wittgenstein sent another letter to arrange a luncheon for 18 June. Griffin has shown that it is likely that the letter should be correlated with the early letter of 27 May, for in it Wittgenstein proclaims: “I can now express my objection to your theory of judgment more exactly.” The (now infamous) letter reads (1974,, R12):

I believe it is obvious that, from the prop[osition] “A judges that (say) a is in the Rel[ation] R

to b,” if correctly analyzed, the prop[osition] aRb v ~aRb” must follow directly without the use of any other premise. This condition is not fulfilled by your theory.

The consequence was that by the third week of June and after 350 folios written in little more than a month, progress on the book which Russell had “already worked out in his head” ceased.

The wide direction problem questions the ability of the multiple-relation theory to exclude nonsensical beliefs-i.e., beliefs which cannot be either true or false. If the constituents of a complex are each separately before the mind, then presumably nothing in the theory excludes taking a property such as morality in the wrong way. Nothing seems to exclude faulty substitutions which generate the meaningless “A believes that mortality is Socrates.” Similarly, nothing seems to prevent taking a relation such as loves as a subject, Desdemona as a prediate, and generating the nonsensical “Othello believes loves Desdemona Cassio.”

At first blush, however, the importance of the problem is difficult to pinpoint. Why not simply allow meaningless beliefs? If would certainly seem as though confusion can produce meaningless beliefs, and (sadly) that history is replete with them. What is it about this criticism that would have led Russell to abandon Theory of Knowledge?

In an interesting article, Griffin offers and important interpretation based on the work of Sommerville 1981. On this view, the problem of meaningless beliefs pushed Russell into using the theory of logical types of Principia to assure logical distinctions in type between universals, particulars, and logical forms (Griffin 1980, 175). Sommerville draws a connection between the dyadic analog of Principia *13.3 which establishes type-theoretic significance constraints on propositionsL. Thus, in even an elementary non-permutative judgment “A judges that a is R to b,” the person A will be able to judge that

a is R to b only in virtue of further premises to the effect that a and b are individuals, R is a first-order relation and xRy is the form of a dyadic first-order complex (Griffin 1985, 242). But this will not do. These extra premises would themselves require judgments which A has made prior to making his original judgment. Worse, they are second-order and, according to Principia, higher-order judgments are to be defined cumulatively on lower-order judgments. The result is that elementary non-permutative judgments do not provide the foundation for higher order judgments after all. In relying on the theory of types to exclude meaningless beliefs, they presuppose judgments of higher order. On the other hand, Russell cannot simply allow beliefs whose constituents are not appropriately typed, for the theory of belief is to do justice to what would be propositions and no proposition fails to be regimented by type-theory.

One might reply that the multiple-relation theory grounds only ‘orders’ and not types. But this affords no escape in Griffin’s view. Each object of the belief relation is an object of acquaintance. Where objects differ in logical type (order), acquaintance is not sufficient to reveal the difference; “that a is of different logical type than b is something that could only be discovered as a result of judgment (not simple inspection)” (1985, 243). Griffin concludes that the reason Russell abandoned the multiple-relation theory of was “the impossibility of using it to underpin the logic of Principia; and the main stumbling block here, as Wittgenstein pointed out, came from type theory” (1980, 169).

On the present interpretation, however, Griffin’s “further premises” could not have been what Wittgenstein had in mind. First or all, Griffin assumes that the hierarchy of orders applies to belief or judgment complexes. And, insofar as he speaks of loves as a “first-order” relation, he identifies universals with the orders of propositional functions of Principia. We contend, however, that there are no orders of belief-complexes and no orders of universals. There are only orders of propositionsL and propositional functions in Principia. Secondly, Griffin’s interpretation focuses on the problem of analyzing what it is to believe a propositionL which is regimented by the type-theoretical significance constraints of Principia. When A judges that such a propositonL is true he will have to understand the propositionL and know the significance constraints. But on the present view, Russell was concerned with providing truth-conditions, and this should (as we have seen) be separated from the problem of the mental content involved when a propositionL is believed. For the truth-conditions of higher order propositionsL step downward to the foundational base of elementary (atomic) truth and falsehood; and here are no type-theoretical regimentation occurs.

Of course, Griffin holds that this pristine picture ran afoul with the problem of ruling out meaninglessness. For instance, he says the multiple-relation theory fails to show “{(x)Man(x)} is a man” to be meaningless; and (without using type distinctions) provides no analysis of “ ‘ja is true” which could not be applied also to “’ja’ is a man” (1980, 143). But, with our new interpretation, the multiple-relation theory does not need to rely on type distinctions to show that “(x)Man(x)” does not have “{(x)Man(x)} is a man” as a proper instantiation. Every value of “x^ is a man” is an atomic propositionL which must have elementary truth. As Principia points out, in a statement such as “p is a man,” where p is a propositionL, it is not possible to break up the constituents to find the true logical subjects (p. 48). In the analysis,

“{(x)Man(x)} is a man” has elementary truth =df

($1p)( Bel{m,?, ?, ?} corresponds to p),

the definiens cannot be filled in. There is no way to find constituents for an appropriate belief-complex forming a definite description for a purported corresponding complex. Hence the expression is meaningless. In contrast, we have seen how the definiens of a propositionL such as “ ‘ja’ has elementary truth” and “ ‘(x)jx’ has first-truth” are to be given. The theory of types of Principia is nowhere needed here.

Griffin does have another argument to show that Russell came to make a logical distinction between universal and particular. He takes us back to Russell’s early letter to Ottoline Morrell of 21 May 1913. There Russell has remarked that Wittgenstein came to him on the day before “… with a refutation of the theory of judgment I used to hold. He was right, but I think the correction requires is not very serious” (Morrell 1912-1916, 782). Griffin argues convincingly that the theory Russell “used to hold” was that of “On the Nature of Truth and Falsehood,” given in Philosophical Essays. There is corroboration from the 1917 footnote which Russell added to “On Acquaintance and Description” when it was reprinted in Mysticism and Logic (p. 159n), and also from the 1918 article “The Philosophy of Logical Atomism.” Both sources credit Wittgenstein with revealing the early version was “unduly simple,” in the second Russell explaining that “I did then treat the object verb as if one could put it as just an object like the terms, as if one could put ‘loves’ on a level with Desdemona and Cassio as a term for the relation ‘believes’ (p. 226).

Now the encounter with Wittgenstein came when Russell was writing the section of Theory of Knowledge which treated acquaintance with relation, so Griffin holds that the “not very serious correction” Russel envisioned involved changes in the status of the subordinate verb. Russell supposedly relies on the theory of types of Principia so that the subordinate relation is viewed as of a different logical type form that of an individual. This would rule and expression such as “Mortality is Socrates” meaningless. Griffin quotes from a chapter of Theory of Knowledge written after Wittgenstein’s visit: “… the way in which the relating relation occurs in an atomic complex is quite different from the way in which its terms occur…” (Russell 1913, 90), and later “Subject and predicate obviously differ logically, and not merely as two particulars differ… (ibid.).

Note, however, that Russell’s remarks are not about the subordinate verb; they concern the relating relation. Perhaps, then, Wittgenstein objected to Russell’s use of the expression “a-in-the-relation-R-to-b” for a dual complex; the expression makes it appear as if the relating relation R is just another term. In any case, Wittgenstein no doubt demanded that the special status of the relating relation is a complex be represented in the corresponding belief-complex. Theory of Knowledge does reflect this in maintaining that there is a logical difference between universal and particular. Indeed, Russell says of a complex such as

a-R-b that “the position of R, unlike that of the other constituents, can be assigned relatively to the form: this is what enables us to speak of it as the relating relation “ (p. 146). But this logical difference, grounded in the theory of logical form, is not a “distinction in logical type” in the sense of the logical types of Principia. The distinction is just that universals have both a predicable and an individual nature. That a concrete particular is not a predicable might be described as a type distinction, but only in a different sense of “type.” Hence, I do not think the connection to the Principia theory of types is correct. An understanding of Wittgenstein’s criticism must come from another quarter.

If (as seems likely) the version Wittgenstein refuted was that of Philosophical Essays, then Russell’s early relaxed attitude can probably be attributed to the fact that the theory had already been modified. The first time was in Problems; and with Theory of Knowledge, Russell was surely feeling confident that he had a satisfactory solution of this (by now “not very serious”) problem. But Russell had not yet fully grasped Wittgenstein.

I shall argue that the letters that Wittgenstein sent to Russell prior to and during the writing of Theory of Knowledge reveal that Wittgenstein is attacking Russell’s theory of logical form on grounds that it is unable to establish that there are non-permutative complexes. In particular, Wittgenstein is espousing what will become the “Doctrine of Showing” in the Tractatus. According to this doctrine:

Propositions can represent the whole of reality, but they cannot represent what they have in common with reality in order to represent it-logical form. [4.128]

Propositions cannot represent logical form: it is mirrored in them … [4.121a,d]

Propositions show the logical form of reality. [4.1212]

Undoubtedly, the famous doctrine of showing was in its infancy when Russell was writing Theory of Knowledge. Nonetheless, Wittgenstein’s letters show the doctrine unfolding and place it, not the alleged conflict with types at the heart of his attack on the multiple-relation theory.

To see this, let us begin with Wittgenstein’s letter of 16 August 1912. He wrote

(1974, R5):

Now as to “p v q,” etc.: I have thought that possibly-namely, that all our troubles could be overcome by assuming different sorts of Relations to signs of things-over and over again! for the last 8 weeks!!! But I have come to the conclusion that this assumption does not help us a bit. In fact, if you work out ANY such theory-I believe you will see that it does not even touch our problem.

Griffin regards this passage as a criticism of the notion of typical ambiguity espoused in Principia (1980, 234). But the problem shared by Russell and Wittgenstein concerns the significance of the logical connectives and the bound variables. On the present interpretation, the view that there are no logical constants (i.e., that the logical connectives do not occur as constituents of complexes) was not original with Wittgenstein. It was Wittgenstein who inherited it the view from Russell. As we saw, in Principia the truth-conditions for an atomic propositionL differ from that of a molecular and a quantified propositionL insofar as there are different relations of “correspondence” involved. An atomic propositionL points to a single complex, while a molecular or generalized propositionL points to many. Wittgenstein’s letter maintains, to the contrary, that a proper analysis of what called “unasserted occurrences” will not be found in this way. Differences in the relation between the sign “p v q” and the complexes signified does not touch the problem.

In a letter, dated Summer of 1912, Wittgenstein claims to have made new progress on the problem (1974, R6):

I believe that our problems can be traced down to the atomic prop[osition]s. This you will se if you try to explain precisely in what way the Copula in such a prop[osition] has meaning. I cannot explain it and I think that as soon as an exact answer to this question is given the problem of “v” and of the app[aren’t] var[iable] will be brought very ear their solution if not solved. I therefore now think about “Socrates is human.: (Good old Socrates.)

The focus is now on the role of the copula in an atomic propositionL. Reducing the issue in this ay foreshadows the Tractatarian view that all propositionsL, including those containing bound variables, are truth-functions of atomic propositionsL. Atomic proposition intrinsically “picture” or show their truth-conditions without any further premises. Logical “truths” have no truth-conditions. They are the logical structure or “scaffolding” which establishes the possibility of the truth and falsehood of atomic propositionsL, and thereby, of molecular propositionsL.

In a letter of January 1913, Wittgenstein applies this idea to the copula as well

(1974, R9):

I have changed my mind on “atomic” complexes: I now think that Qualities and Relations (like Love), etc., are copulae! … I want a theory of types to tell me that “Morality is Socrates” is non-sensical, because if I treat “Morality” as a proper name (as I did) there is nothing to prevent me to make the substitution the wrong way round…

The point here is that all logical differences are part of logical structure. Since the difference between a universal and a particular is logical, it must be shown through the grammar. “All theory of types,” Wittgenstein wrote in the same letter, “must be done away with by a theory of symbolism showing that what seem to be different kinds of things are symbolized by different kinds of symbols which cannot possibly be substituted in one another’s places” (1974, R9).

In contrast, Russell only conceded to the narrow direction criticism that the “sense” (and by this Russell means the “direction”) of a permutative complex cannot be captured by naming he complex or listing its constituents. His theory of logical form exempts non-permutative complexes by securing that such complexes have no “sense.” It is a matter of logical form, for instance, that “…the relating relation of a complex is always heterogeneous with respect to its other constituents” (1913, 123). This grounds the logical difference between universal and particular. Logical form underlies the impossibility of Socrates being a predicable and assures the meaninglessness of the (purported) propositionL “Mortality is Socrates.” Indeed, we saw that the logical form grounds the non-permutativity of belief-complexes such as

Bel{m, a, Humanity, fx}, Bel{m, a Similarlty, b, xRy} and Bel{m, a, C, p, xRb}.

Wittgenstein disagreed. He held that no complex can be named or described. All meaningful propositionsL show “sense.” This applies to propositionsL which correspond to Russell’s so called “non-permutative” complexes as well. In every case, the propositionL is itself a “picture” of the conditions of its truth and falsehood. The key to unraveling the wide direction problem lies in realizing that Wittgenstein’s notion of “sense” was different from Russell’s notion of “direction.” Logical form establishes the conditions for meaning, i.e., the possibility of having truth-conditions. But to Wittgenstein, logical form is a matter of sense; and “sense” is captured only when it is shown. The logical impossibility of certain combinations can only be shown through logical grammar. There cannot be a theory of logical form.

This is a powerful objection to the multiple-relation theory because, in attempting to give the truth-conditions for a propositionL through an analysis of “correspondence,” the theory requires that we have “logical experience,” intuition, or some kind of acquaintance with logical forms. This makes an understanding of “predicate,” “dual relation,” “complex” and the like possible. For Russell, understanding logical form is presupposed in belief. But such a theory must say what can only be shown. Allowing logical forms as constituents of the belief-complex accomplishes nothing in Wittgenstein’s view. Our understanding (e.g.), of the logical impossibility of a concrete particular occurring as a relating relation, cannot be explained by appeal to our acquaintance with an extra entity, a fact, which is an object before the mind. Wittgenstein writes in the Tractatus, 5.522:

The “experience” that we need in order to understand logic is not that something or other is the state of things, but that something is; that, however, is not an experience.

In failing to immediately show the logical form, Russell’s analysis does not exclude logical impossibility (meaninglessness).

Russell was reportedly “paralyzed” (Wittgenstein 1974, R13). The multiple-relation theory views subordinate propositionsL as occurring “unasserted” and functioning as an apparent name-i.e., a disguised description which, where non-permutative complexes are concerned, directly singles out a complex; and, where permutative complexes are concerned, indirectly singles out a complex (Russell 1913, 148). But Wittgenstein is saying that subordinate clauses preserve what Russell regarded as an “assertoric feature.” It is only through assertion (which uses the grammatical structure of the propositionL) that the “sense” of a propositionL is captured. For example, it is only through assertion that a propositionL such as “aRb” can show that the relation R has a predicable nature. The assertion of a compound propositionL such as

“ ‘aRb’ is true”, or “aRb ÉB” or “A judges that aRb” must preserve assertoric features of the subordinate clause “aRb,” else the “sense” of the subordinate is lost. That is, R must be relating in the assertion of the compound, for only in this way can the subordinate clause show its sense (i.e., its truth-conditions). In Wittgenstein’s words, the subordinate clause must show its truth-conditions immediately in the assertion of the compound, “without the use of any other premise.” This is absent in Russell’s analysis, for the subordinate propositionL is analyzed as a description of a complex. Nothing in the analysis shows that R has a predicable nature and, consequently, proper truth-conditions (the conditions for sense) for “aRb” are not assured. This, I hold was Wittgenstein’s point in remarking that Russell’s theory “…does not makes it impossible to judge non-sense.”

The interpretation concurs with Russell’s later assessment of the problem of belief in “The Philosophy of Logical Atomism.” Russell writes that in belief “… both verbs have got to occur as verbs” (1918, 225). “The subordinate verb (i.e., the verb other than believing),” he continues, “is functioning as a verb, and seems to be relating two terms, but as a matter of fact does not when a judgment happens to be false. That is what constitutes the puzzle about the nature of belief… The discovery of this fact is due to Mr. Wittgenstein” (p. 225). Indeed, belief-complexes pose special problems because, unlike other complexes, they would contain two (apparently relating) relations. As Russell puts it, we have “got on to a new beast for our zoo” (p. 226).

In the Tractatus, Wittgenstein attempted to side-step Russell’s difficulties by challenging his notion of “assertion” and introducing a new explanation of the role of the logical connectives. But in granting that the subordinate clause must somehow retain assertoric features, i.e., that the subordinate verb must function as a verb, Russell was forced to recognize the collapse of the two-tiered structure of the multiple-relation analysis in Theory of Knowledge, and indeed, that the philosophical explanation for the ramified hierarchy of Principia was a failure.


I am indebted to Nicholas Griffin and Nino Cocchiarella for valuable criticisms of an earlier version of this article.


Items marked “” are cited by page number in this article.

Anscombe, G.E. M. 1959 An Introduction to Wittgensteini’s Tractatus, New York (Harper & Row).

Ayer, A. J. 1971 Russell and Moore: The Analytic Heritage, London (Macmillian).

Church, Alonzo 1976 “A Comparison of Russell’s Resolution of the Semantic Antinomies with that of

Tarski,” Journal of Symbolic Logic, 41, 747-760.

Cocchiarella, Nino B. 1980 “The Development of the Theory of Logical Types and the Notion of a Logical

Subject in Russell’s Early Philosophy,” Synthese, 45, 71-115.

________ 1987 “Russell’s Theory of Logical Types and the Atomistic Hierarchy of Sentences,” in his

Logical Studies in Early Analytic Philosophy, Columbus (Ohio State University Press), 193-221.

Geach, P. T. 1957 Mental Acts, London (Routledge & Kegan Paul).

Griffin, Nicholas 1980 “Russell On the Nature of Logic (1903-1913),” Synthese, 45, 117-188.

______ 1985 “Russell’s Multiple-Relation Theory of Judgment,” Philosophical Studies, 47, 213-247.

______ 1986 “Wittgenstein’s Criticism of Russell’s Theory of Judgment,” Russell, n.s. 4, 132-145.

Hochberg, Herbert 1987 “Russell’s Early Analysis of Relational Predication and the Assymetry of the

Predication Relation,” Philosophica, 17, 439-459.

Landini, Gregory 1987 “Russell’s Substitutional Theory of Classes and Relations,” History and Philosophy

of Logic, 8, 171-200.

______ 1989 “Some New Evidence Concerning Russell’s Substitutional Theory,” Russell, n.s. 9, 26-42.

Morrell, Ottoline 1912-1916 “Russell’s letters to Lady Ottoline Morrell,” unpublished mss, in the

Humanitites Research Center, Austin, University of Texas. {Available on microfilm at the Russell

Archives, Hamilton, Ontario (McMaster University).

Pears, David 1977 “The Relation Between Wittgenstein’s Picture Theory of Propositions and Russell’s

Theory of Judgment,” Philosophical Review, 86, 177-196.

Russell, Bertrand 1903 The Principles of Mathematics, 1st ed., Cambridge (University Press).

______ 1904 “Meinong’s Theory of Complexes and Assumptions,” Mind, n.s. 13, 204-219, 336-354,

509-524. Reprinted in Russell 1973, 21-76.

______ 1905 “On Denoting,” Mind, 14, 479-493.

______ 1906 “Les Paradoxes de la Logique,” Revue de mJtaphysique ed de morale, 14, 627-650. [English

original version “On ‘Insolubilia’ and Their Solution by Symbolic Logic,” in Russell 1973, 190-

214. ]

______ 1907 “On the Nature of Truth,” Proceedings of the Aristotelian Society, 7, 28-49. [Sections I and II

reprinted as “On the Monastic theory of Truth,” in his Philosophical Essays, London (Longmans,

Green), 1910, 150-169.

______ 1908 “Mathematical Logic as Based on the Theory of Logical Types,” American Journal of

Mathematics, 30, 222-262. Reprinted in Russell 1977, 59-102.

______ 1910 “On the Nature of Truth and Falsehood,” chapter VII of his Philosophical Essays, London

(Longmans, Green), 170-185.

______ 1911 “Knowledge by Acquaintance and Knowledge by Description,” Proceedings of the

Aristotelian Society, 11, 108-128. Reprinted in Russell 1917, 152-167.

______ 1912a “On the Notion of Cause,” Proceedings of the Aristotelian Society, 12, 1-24. Reprinted in

Russell 1917, 132-151.

______ 1912b The Problems of Philosophy, Oxford (University Press).

______ 1913 Theory of Knowledge: The 1913 Manuscript (ed., E. R. Eames and K. Blackwell), Vol. & of

The Collected Papers of Bertrand Russell, London and Boston (Allen & Unwin), 1984.

______ 1914a Our Knowledge of the External World, New York (Norton).

______ 1914b “The Relation of Sense-Data to Physics,” Scientia, 16, 1-27. Reprinted in Russell 1917,


______ 1917 Mysticism and Logic, London (Allen & Unwin). [Various reprints, including paperback

edition, 1963].

______ 1973 Essays in Analysis (ed., D. Lackey), New York (Brazillier).

______ 1977 Logic and Knowledge: Essays 1901-1950 (ed, Robert C. Marsh), London (Allen & Unwin).

Sommerville, Stephen 1981 “Wittgenstein to Russell (July, 1913): “I am very sorry to hear that my

objection paralyzes you,’” in Language, Logic and Philosophy: Proceedings of the 4th

International Wittgenstein Symposium, Vienna (Holder-Pichler-Tempsky), 182-188.

Stout, G. F. 1911 “The Object of Thought and Real Being,” Proceedings of the Aristotelian Society, 11,


Whitehead A. N. and Russell, Bertrand 1910 Principia Mathematica, vol. 1, Cambridge (University Press).

[Second edition partly reprinted as Principia Mathematica to *56 with Russell’s 1925 introduction

and Appendices A and C, Cambridge (University Press), 1962.]

Wittgenstein, Ludwig 1922 Tractatus Logico-Philosophicus, translated by D. F. Pears and B. F.

McGuinness, London, (Routledge & Kegan Paul) 1961.

______ 1974 Letters to Russell, Keynes and Moore (ed., G. H. von Wright), Ithica (Cornell

University Press).