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What is the problem of the unity of the proposition? Did Russell solve it?
A proposition is made up of a list of words. Yet by itself, a list of words cannot say something the way a
proposition can. The problem of the unity of the proposition (PUP) is essentially a puzzle about what makes
a whole – in this case, a proposition – more than the sum of its parts. What exactly “cements” together the
constituents of a proposition to make it meaningful, and how does it do so? Furthermore, although a
proposition is constituted by a list of words, it is clearly not the case that any list of words, arranged in any
order, will do. Some words, and/or some arrangement of words, form expressions that are (arguably)
meaningless. I refer to this as the Nonsense concern. This concern will not be the focus of this essay; I think
there exists a grey area between a meaningful and meaningless expression – “celery Japan” may be
unanimously agreed upon as nonsense, while “The melody is blue.” is relatively more contentious. This
essay does not make claims about what counts or doesn’t count as nonsense. I will discuss the Nonsense
concern only insofar as a successful account of what makes propositions meaningful should a fortiori
explain what sets a proposition apart from a meaningless string of words.
This essay will explore whether Russell’s Multiple-Relation Theory of Judgement (MRTJ) can account for
the PUP. Following Sainsbury and Lebens, I argue that the MRTJ lays a solid foundation to solving the
PUP, but that Russell ultimately misses the final puzzle piece that would make the MRTJ viable; his
introduction of ‘logical forms’ is clear evidence that he correctly identifies the problem but not the solution
that would address it. The structure is as follows: I first explain how Russell’s MRTJ was motivated by his
rejection of the existence of false propositions. I then outline the MRTJ and show how it led to MRTJspecific concerns that Russell had to address:
a) To account for the dual nature of verbs, such that they can serve as objects in judgements
a) To ensure that the mind unifies the judgement without accidentally reifying the judged complex
b) To demystify the mind’s power of unification
Finally, I critically evaluate Russell’s proposed ‘logical forms’; I explore its merits, its blunders, and argue
that Russell could have saved his MRTJ if he adopted a Davidsonian approach to meaning.
MRTJ
The orthodox view of belief/judgement is as a binary relation between the judger and the judged
proposition. Russell took issue with this view because it led to the reification of false propositions. Consider
the nature of a false proposition:
“Let us take some very simple false proposition, say ‘A precedes B’, when in fact A comes after
B. It seems as though nothing were involved here beyond A and B and “preceding” and the general
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form of dual complexes. But since A does not precede B, these objects are not put together in the
way indicated in the proposition. It seems, therefore, that nothing which is actually composed of
these objects is the proposition; and it is not credible that anything further enters the proposition.”1
Russell is guided by two broader philosophical principles: Logical Atomism, and Realism. The former
dictates that propositions must have as their constituents the entities that they are about.2 This allows Russell
to assert that the false proposition ‘A precedes B’ must be constituted by the entities ‘A’, ‘B’, and
‘preceding’, and nothing else. But no such complex exists in the world because it is not the case. To insist
otherwise would go against Realism – only complexes that feature in a complete description of the world
can be admitted into our ontology.3 These considerations explain Russell’s strong unease regarding the
existence of false propositions, which led him to the MRTJ.
According to the MRTJ, judgement (or belief) is a multiple-relation that holds between the mind (subject)
and the constituents of what the judgement is about. For example, the judgement “Othello believes that
Desdemona loves Cassio,” can be depicted as a quaternary relation: J{Othello, Desdemona, loves, Cassio}.
Russell stipulates one condition: one of the object-constituents of the judgement must be a relation, in this
case ‘loves’ — call this the object-relation R. We judge truly iff the complex entity “Desdemona loves
Cassio.”, i.e. the fact, exists, where ‘loves’ now unites Desdemona and Cassio. In the next section, I’ll
discuss the complications that arise from this treatment of relations.
Observe that there is no longer a singular entity called the ‘proposition’ participating in the relation to the
mind. Russell’s strategy of eliminating the pesky false propositions is effectively to do away with all
propositions.
“I think a false proposition must, wherever it occurs, be subject to analyses, be taken to pieces,
pulled to bits, and shown to be simply separate pieces of one fact in which the false proposition has
been analysed away.”4
Let us return to the PUP: rather than the proposition, Russell now needs to account for the unity of the
judgement. To be precise, Russell needs to explain the nature of both the judging relation and the objectrelation; how do they, combined, cement together the constituents in the judgement? Without this
explanation, we are still left in the dark as to the difference between a list of disparate, unordered
constituents, and a meaningful judgement.
1
Russell, Theory of Knowledge, 110.
Russell, The Philosophy of Logical Atomism, 52.
3
ibid, 56.
4
ibid, 56.
2
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The PUP can be broken down into two related, yet distinct requirements:
Unity:
There must be some relation1 cementing together the list of objects in the judgement.
Directionality: There must be some relation2 holding the constituents in the correct order, such that we can
distinguish between the judgements ‘aRb’ and ‘bRa’.
Which relation, the judging-relation or the object-relation, is responsible for relation1 and relation2? I
discuss Russell’s conception of the dual nature of verbs in order to answer this question.
According to Russell, verbs have a “twofold” nature: verbs can be used as ‘logical verbs’, which express
“a relation actually relating”, or as ‘verbal nouns’, which express “a relation actually relating”.5 To
illustrate: the verb ‘kill’ is used as a logical verb in “Felton killed Buckingham.” versus a verbal noun in
“Killing is wrong.” Russell emphasizes two claims: a) only relating-relations have the power to unite the
constituents flanking them; b) the object-relation, nested inside the judging-relation, occurs as a relationin-itself:
“But this relation, as it occurs the act of believing, is not the relation which creates the unity of the
complex whole structure consisting of the subject and the objects. The relation ‘loving’, as it occurs
in the act of believing, is one of the objects — it is a brick in the structure, not the cement. The
cement is the relation ‘believing’.”6
This tells us it is the judging-relation which plays the role of relation1; the mind unifies the constituents of
the judgement. Russell has important reasons for asserting this. Consider what happens if we allow verbs
to occur as logical verbs, with all their unifying power, in some judgement J{mind, a, R, b}. Then R would
automatically unify a and b to form the complex entity aRb, thus rendering J a binary relation between the
mind and aRb, just as under the orthodox view. If J is a false judgement, we land in the same predicament
as discussed earlier – we are forced to admit the entity aRb even though no such complex exists in the actual
world. This defeats the point of the MRTJ – to reserve the power of unification to the mind, so as to avoid
positing non-existent entities.
Yet, this raises another question – how can the mind unify the judgement whilst keeping the constituents
separate?
“Suppose we wish to understand “A and B are similar”. It is essential that our thought should, as is
said, “unite” or “synthesize” the two terms and the relation; but we cannot actually “unite” them,
5
6
Russell, Principles of Mathematics, §54.
Russell, The Problems of Philosophy, 95.
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since either A and B are similar, in which case they are already united, or they are dissimilar, in
which case no amount of thinking can force them to be united.”7
Somehow, the mind must be able to unify the judgement such that it can say something meaningful, but not
“too much” that the judged complex is “forcibly” reified as a complex entity. Bestowing this reification
power on the mind would effectively land us in the original predicament – the judged complex is reified,
making it akin to a proposition, and we are back to square one. It seems like what is needed is a distinction
between (what I henceforth call) semantic unification – an enabling condition for meaning, and actual
unification, which leads to undesirable metaphysical consequences. Russell never successfully addresses
this, leaving the role of the mind unsatisfyingly ambiguous. I return to this issue later to show how he could
have done so. Before that, I discuss Russell’s approach to tackling the Directionality requirement.
It is less clear whether Directionality is provided by the mind, or whether it is an intrinsic property in the
object-relation. In 1910, Russell claims that directionality is an intrinsic property of the object-relation R:
“We may distinguish two ‘senses’ of a relation according as it goes from A to B or from B to A.
Then the relation as it enters into the judgment must have a ‘sense’, and in the corresponding
complex it must have the same ‘sense’.”8
We can visualise this with the following diagram:
a
a
R1
R2
b
b
Depending on whether we want to judge that ‘aRb’ or ‘bRa’, we would pick either R1 or R2 to feature in
the judgement. Candlish argues that this idea fallaciously “explains direction in a way which involves
unity.”9 To conceive of the object-relation R as possessing an intrinsic property which orders the
constituents a and b is to presuppose the unity of ‘aRb’/’bRa’, which outright contradicts Russell’s rejection
of propositional unity.
By 1912, Russell realises this, and asserts that the mind is responsible for both unity and direction:
7
Russell, Theory of Knowledge, 116.
Russell, ‘On the Nature of Truth and Falsehood’.
9
Candlish, ‘The Unity of the Proposition’, 111.
8
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“It will be observed that the relation of judging has what is called a ‘sense’ or ‘direction’. We may
say, metaphorically, that it puts its objects in a certain order, which we may indicate by means of
the order of the words in the sentence.”10
Thus, a judgment is an ordered list of objects. This is what enables the judger to distinguish between the
judgements J{mind, a, R, b} and J{mind, b, R, a}. It is also what enables us to rule out nonsense statements
that arise from erroneous orderings, like “I believe Desdemona Cassio loves.”
Let’s take stock. A unified judgement consists of a list of constituents which are held together and ordered
by some relation. This relation must be attributed to the mind; it cannot be the object-relation itself as this
approach is incompatible with the possibility of false judgements. To complete the story, Russell needs to
explain how the mind can impose semantic unity on the constituents of the judgement. This explanation
would take care of the Nonsense concern – understanding the nature of this semantic unity involves
understanding its necessary and sufficient conditions, and hence, why it is not the case that any arbitrary
concatenation of words can constitute a meaningful judgement.
Logical Forms
In 1913, Russell recognised the need to be able to entertain the judged complex in the mind, regardless of
whether the judgement turned out to be true or false.
“… in order to understand ‘A and B are similar’, we must know what is supposed to be done with
A and B and similarity, i.e. what it is for two terms to have a relation; that is, we must understand
the form of the complex which must exist if the proposition is true.”11
He dubbed this “form of the complex” ‘logical form’ – an enabling condition for judgement. Russell thought
this was the key to explaining how the mind can impose the required semantic unity on the constituents of
the judgement:
“The process of ‘uniting’ which we can effect in thought is the process of bringing [two objectconstituents] into relation with the general form of dual complexes.”12
Thus, the MRTJ is revised accordingly: a judgement J containing a binary object-relation R consists of five
constituents: J{mind, a, R, b, R(x,y)} where “R(x, y) stands for the form ‘something and something have
some relation’”. This idea is generalised to apply to object-relations that relate an arbitrary number of
objects: for n objects, the logical form might be symbolised as R(x1, x2, …, xn).
10
Russell, The Problems of Philosophy, 94.
Russell, Theory of Knowledge, 116.
12
ibid.
11
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Besides enabling semantic unity, the addition of logical forms is clearly designed to address the Nonsenseconcern. Stipulating the logical form is just a way of making explicit the following conditions for forming
judgements: a) an n-ary relation can only apply to exactly n objects; b) objects and relations cannot be
arranged in any unrestricted manner, but have to fit into a specific template. This dismisses expressions like
“Othello judges that Desdemona loves” (violates a), and “Othello judges that Desdemona Cassio loves”
(violates b) as meaningless, non-judgements.
I think that Russell is on the right track with his idea of logical forms. Arbitrary words can be concatenated
without restriction, but meaning is not automatically created. Thus, the unity of the judgement must
somehow be connected to the restrictions we impose on concatenating words. However, Russell’s execution
of this idea is problematic. Before diving into the objections, I explore a possible motivation Russell might
have for taking this route. Why include logical forms as a constituent to the judgement relation, rather than
treat it as a prerequisite to judgement? I argue that Russell is guided by his Principle of Acquaintance, which
states that all our knowledge — in this case, knowledge of what, or how many terms, are allowed to be
related to each other — must be derived directly or indirectly from acquaintance.13 Since this knowledge
appears to be a priori, Russell hypothesizes that it is given to us directly by acquaintance, and so there must
be a corresponding entity we are acquainted with, namely ‘logical form’.
“I appeal to the principle that, when we [make a judgement], those objects with which we must be
acquainted when we [judge], and those only, are object-constituents … of the [judgement]complex.”14
Reichard offers a neat analogy to illustrate the issue with treating logical form as an additional constituent
to the judgement. Consider a complex object like an arch, made up of many stones. What is it that unites
the stones such that they jointly make up the arch? It is that they are arranged in a particular spatial structure.
Similarly, what unifies the judgement is that the constituents are arranged in a particular structure, but this
structure is not, itself, a constituent of the judgement. He writes that “To assume [the structure is itself a
further stone] would, it seems, mean making a category mistake and would also lead straightforwardly into
Bradley’s Regress.”15 The problem of Bradley’s Regress is this: R(x, y) guarantees the unity (and order) of
the constituents a, R, and b. But by adding R(x, y) as a constituent, we now need another second-order
logical form to ensure the unity of the constituents a, R, b, and R(x, y). And so on, ad infinitum.16 For these
reasons, the inclusion of logical forms is clearly not a viable solution to the problem.
13
Russell, The Problems of Philosophy, chap. 5.
Russell, Theory of Knowledge, 117.
15
Reichard, ‘Philosophy of Language and Linguistics’.
16
Stevens, ‘Russell and the Unity of the Proposition’.
14
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If not using logical forms, how should Russell have stipulated and justified these restrictions? I defend
Sainsbury’s Davidsonian approach. In writing that “...we must understand the form of the complex which
must exist if the proposition is true,”17 Russell was close to grasping the solution. For a string of words to
be meaningful is just for it to be truth-apt. So, how does the judging relation confer truth conditions upon a
string of words? A Davidsonian would argue that the judging relation accomplishes this by definition –
there is no ‘how’ to be explained. In other words, we can entertain a judged complex in our minds iff, and
because we grasp its truth conditions.18 Furthermore, we can specify what these truth conditions are for a
given object-relation:
“In general, for any n objects, o1, o2, …, on, and any n-ary universal, φn, [the judgement] {φn, o1,
o2, …, on} is true if o1, o2, …, on are φn-related.”19
Notice that this is just what Russell tried to do using logical forms. What he should have done instead is
treat each logical form as an axiom governing how a given object-relation can be concatenated with other
objects to constitute a meaningful judgement. What we end up with is a Tarski-style axiom schema that
systematically specifies the conditions of truth-aptness for every predicate in the language.20 And since we
have identified semantic unity with truth-aptness, this wraps up our account of the unity of the judgement.
This simultaneously deals with the Nonsense concern: meaningless strings, which are not truth-apt, are
automatically ruled out as non-judgements.
One may object that Sainsbury skirts around the required explanation by simply making definitional
assertions. I argue, following Lebens, that there is nothing more that needs explaining. The original question
is answered: what cements together the constituents of a judgement is just truth-aptness, and the mind
confers truth conditions onto a judgement by concatenating the constituents in a particular manner – a
manner which we have specified axiomatically. Lebens also solves Russell’s puzzle regarding the dual
nature of relations: in a judgement, the object-relation occurs neither as a relation-in-itself nor as a relatingrelation – it occurs predicatively. In other words, the role of the mind is to “predicate its object-relation of
its object-terms”.21 To demand an explanation of how the mind does this is to demand a general theory of
predication. Insofar as we agree that properties can, in the most general sense, hold of objects, I think that
this demand is unreasonably high.
17
Russell, Theory of Knowledge, 116 (my emphasis).
Sainsbury, ‘How Can We Say Something?’.
19
ibid.
20
Lebens, Bertrand Russell and the Nature of Propositions, 182.
21
ibid
18
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I hope to have shown that Russell’s MRTJ lays the foundation for a robust theory of judgement and
meaning. It demonstrates how one can avoid an ontological commitment to false propositions, via analysis
of a proposition into its constituents, and attributing the unifying power solely to the mind. Though Russell
ultimately falls short in fleshing out the nature of this mental unifying power, the MRTJ is far from doomed.
Sainsbury and Lebens demonstrate its viability under a Davidsonian interpretation – provided we refrain
from placing absurd explanatory demands on the theory. In their words, “What else could one expect?22
[3046 words]
22
Sainsbury, ‘How Can We Say Something?’.
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