Russell’s “On Denoting"

Meggan Payne

            Bertrand Russell’s work in the analytic philosophy of language has been extremely influential. He was one of the leading philosophers of both the logical realist and the logical positivist movements1 , and actually began the latter movement with his paper “On Denoting2 ,” my topic here. “On Denoting” was a fundamental switch in ideas for Russell; as a Realist he thought more along the lines of Gottlob Frege, saying that the sense of an expression is its referent, which lends what Russell thinks is undue ontological status to things such as Pegasus, Santa Claus, and the present King of France, but as a positivist he rejected that Realist claim and its baggage. In this paper I shall examine the theory Russell presents, taking particular care to give a clear (and correct) interpretation of the obscure passage on ‘C’, “C”, and C.

I. “On Denoting” - Summary and Explanation

            Russell’s project in “On Denoting” is to explain and motivate his theory, which suggests a new and better way to evaluate denoting phrases. Denoting phrases are those that begin with quantifier words (such as ‘some,’ ‘all,’ or ‘no’) or articles (such as ‘a’ or ‘the’). He claims that previous theories had not done an adequate job of explaining this kind of phrase.

            His own theory hinges on the notion of the variable. As in quantifier logic, ‘x’ stands for the individual being spoken about in a phrase or sentence, and an upper case letter (‘C’ in Russell’s paper) represents the properties of said individual, or the states or circumstances that the sentence puts that individual in. So in “I spoke to a student,” ‘C’ represents ‘I spoke to,’ ‘x’ represents the individual, and the whole sentence can be translated “ ‘C(x) and x is a student’ is not always false.”

            Phrases or sentence with ‘the’ need to be analyzed differently, because, according to Russell, they claim that the subject of the description is unique. For example, the sentence “The ninth-grader that I tutor is almost six-and-a-half feet tall” not only claims that a certain indivi-dual, who is a ninth-grader and whom I tutor, is quite tall, but also that there is only one ninth-grader that I tutor. If I tutored a classroom of thirty ninth-graders, the sentence would be misleading - not exactly true - on Russell’s theory. The logical form of my example would be as follows: It is not always false of x that x is a ninth-grader and that I (speaker) tutor x, and that x is almost six-and-a-half feet tall, and that “if y is a ninth-grader and y is tutored by me (speaker), then y is identical with x” is always true of y.

            Having stated his theory, Russell moves to showing why a new theory of definite descrip-tions is required. The mistake earlier theorists make is, according to Russell, that they consider the denoting phrase a real part of the actual proposition. Meinong claims that all denoting phrases refer to objects, so that a “non-denoting” denoting phrase denotes real objects that do not exist. This is a difficult view at best. Frege has a better view, Russell claims: he divides the denoting phrase into the sense (meaning, in Russell’s terminology) and the denotation. This is a good fix, except for, again, denoting phrases that do not denote, like “The round square.” According to Frege’s analysis, a sentence like “the round square is purple” has a sense (Russellian meaning) but not a denotation. As “the round clock is purple” is about the round clock, so “the round square is purple” is or should be about the round square, because of the similarity in form between the two sentences. The latter should be nonsense, Russell says, but instead it is “plainly false (p. 214).” Russell sees only one viable way out of this mess: we must drop the presupposition that denoting phrases are genuine parts of the sentences they appear in.

            Russell claims that there are three logical puzzles that his solution solves and that the others do not. The first is the puzzle of identity and information that Frege takes on in “Uber Sinn und Bedeutung,” which is that if ‘a’ and ‘b’ denote the same individual, how can “a=b” be informative while “a=a” is a logically necessary statement, and definitely not informative? The second is that either “A is B” or “A is not B” must be true; what if there is no A? The third is that sentences of the form “the difference between A and B does not subsist” (which Russell claims is equivalent to sentences of the form “there is no difference between A and B”) seem to be self-contradictory, because the phrase “the difference between A and B” denotes a thing and so on Russell’s account must exist, and then says of this existing thing that it does not even subsist.

            Russell is heading towards solving these puzzles with his own theory, but first he needs to show the difficulty with discovering and using the relationship between meanings and denota-tions. The passage in which he does this is quite difficult, partly because of the several typos in the passage, and partly because, I am going to argue, he makes a mistake in symbolization which is fatal for the interesting, new part of what he has to say.

            Russell wants to say what the difference between C and ‘C’ is, such that a logical formula can be worked out for representing their relationship. He works out the following claim about meaning and denotation: “...in order to get the meaning of [the first line of Gray’s Elegy], we must speak not of ‘the meaning of [the first line of Gray’s Elegy],’ but of “the meaning of ‘[the first line of Gray’s Elegy],’” which is the same as ‘[the first line of Gray’s Elegy]’ by itself.” (I have corrected the typos in punctuation so as to give Russell the best opportunity to be correct, and have also put in [the first line of Gray’s Elegy] wherever Russell had put C, because it is easier to understand what is going on if C actually is something). So the meaning (Fregian sense, not any kind of speaker’s meaning) one gets for the first line of Gray’s Elegy is ‘the first line of Gray’s Elegy,’ which coincides with Fregian intuition nicely. About denotation, Russell says the following: “Similarly ‘the denotation of [the first line of Gray’s Elegy]’ does not mean the denotation of [the first line of Gray’s Elegy], but means something which, if it denotes at all, denotes what is denoted by the denotation we want.” ‘The denotation of [the first line of Gray’s Elegy]’ cannot mean anything, however; ‘the denotation of [the first line of Gray’s Elegy]’ is the same thing as ‘the denotation of “the curfew tolls the knell of parting day,”’ and since ‘the curfew tolls the knell of parting day’ is a proposition and not a denoting complex, it therefore does not have a denotation, and so the sentence “ ‘the denotation of [the first line of Gray’s Elegy]’...means something which...denotes what is denoted by the denotation we want” is, similarly to a sentence about the present King of France, false on Russell’s theory, since it is about something that does not exist. So Russell is correct when he says “when C occurs it is the denotation we are speaking about,” but that does not imply that C denotes; C is the object of the denotation, but not the denoting or the thing that does the denotation. ‘Denotation’ as Russell uses it here is a noun, but later on he mistakenly uses it as a verb.

            Russell goes on to say the following: “... to speak of C itself, i.e., to make a proposition about the meaning, our subject must not be C, but something which denotes C. Thus ‘C,’ which is what we use when we want to speak of the meaning, must be not the meaning, but something which denotes the meaning (p. 216).” Here Russell is arguing that to speak of a denoting complex - ‘the first line of Gray’s Elegy,’ for example - we cannot use it itself in a proposition about its own meaning; one couldn’t say “the first line of Gray’s Elegy has x as its meaning.” From this, Russell is claiming here, it follows that ‘the first line of Gray’s Elegy’  is not the meaning of the first line of Gray’s Elegy, but denotes it. This is all rather clearly not true; ‘the first line of Gray’s Elegy’ denotes “the curfew tolls the knell of parting day,” but “the curfew tolls the knell of parting day” is not the meaning of the first line of Gray’s Elegy.

            Again, Russell’s ‘meaning’ here corresponds closely to Frege’s ‘sense,’ not, for example, Grice’s or Davidson’s ‘meaning.’ Take the denoting complex ‘the intersection of lines A and B.’ Senses are, on Frege’s theory, the way in which an object is given. The intersection of lines A and B is given as ‘the intersection of lines A and B,’ and so that must be its sense, or its Russellian meaning. Since the first line of Gray’s Elegy is given as ‘the first line of Gray’s Elegy,’ that is its meaning after all. This part of Russell’s theory is correct only if the meaning of a denoting complex is not the denoting complex itself.

            Because Russell does not allow this relationship of C and ‘C,’ he is going to have to say that their relationship is “wholly mysterious.” Hence he argues that the distinction between meaning and denotation “has been wrongly conceived.”

            Since Russell wants to argue that there is not easily discernible difference between meaning and denotation, he wants to try to eliminate one of them from being necessary parts of sentences. Meaning is uneliminable, however, because the difference between the two sentences from the first puzzle, ‘a=a’ and ‘a=b’ will have to depend on meaning, since each part of the two sentences has the same meaning. So Russell gets rid of denotation.

            He begins with an example. In the sentence “George IV wished to know whether Scott was the author of Waverly,” ‘Scott’ and ‘the author of Waverly’ denote the same individual, and yet if ‘the author of Waverly’ is replaced with ‘Scott’ in the example sentence, the sentence is no longer true; we could not correctly suspect of George IV that he ever wished to know whether Scott was Scott. Russell says that while the sentence “Scott was a man” has Scott as its subject (Scott is the x in the sentence form “x is a man”), the sentence “the author of Waverly is a man” does not have the form “x is a man;” it has the form “one and only one x wrote Waverly, and that x is a man.” So denoting phrases can always be reduced to the form “it is not always false of x that x [-----] and it is always true of y that if y [-----], then y is identical with x; and x [===].” Hence the denotation of a denoting complex will always be the entity x, of which different propositions are true, depending on what the particular denoting complex says. The denoting complex is not a necessary part of the sentence, since it can be expanded into a more analyzable phrase as shown above, so the denoting complex does not really have a meaning, because the rephrasing of the complex is the real piece of language, not the complex itself.

            This solves the first puzzle Russell presented, because while the denotation of ‘Scott’ and ‘the author of Waverly,’ for example, is the same individual, but ‘the author of Waverly is not genuinely part of the proposition. To explain how this can be while the inference from ‘the author of Waverly’ to ‘Scott’ still works in English, Russell introduces the concept of primary and secondary occurrences.

            The sentence “George IV wished to know whether Scott was the author of Waverly” is a proposition about the proposition “Scott was the author of Waverly,” and embedded in the embedded proposition is a denoting complex. When translating such a sentence into what Russell says is the correct form (with x’s), one can give the denoting complex either primary or secon-dary occurrence, depending on what one thinks is the best interpretation of the sentence. Inter-preting the George IV sentence with the primary occurrence of the denoting complex yields the following: “One and only one man wrote Waverly, and George IV wished to know whether Scott was that man.” The denoting complex is primary in this case because the phrase that replaces the denoting complex is the first of two conjuncts and the second conjunct contains a phrase, “that man,” that is modified by the first conjunct. A secondary occurrence of the same denoting complex could be translated as follows: “George IV wished to know whether one and only one man wrote Waverly and Scott was that man.” The occurrence of the denoting complex is secondary in this case because it is the direct object in the sentence, not the subject and verb; George IV is the subject and the denoting complex is the first part of what he wanted to know.

            This distinction will also solve the second problem Russell presented, he claims. The problem was that either ‘A is B’ or ‘A is not B’ must be true, but if there is no A, it will not be found in either the B camp or the not-B camp. Russell can now say exactly what is going on here. The sentence “The present king of France is bald” is translated “One and only one x is the present king of France, and that x is bald.” Since there is no present king of France, the first conjunct of that sentence is false, which makes the whole thing false. “The present king of France is not bald” can either be taken to mean “one and only one x has the property of being the present ing of France, and that x is not bald,” giving the denoting complex primary occurrence, which also turns out to be false; or it can be taken to mean “It is false that there is one and only one x which has the property of being the present king of France and which also has the property of being bald,” in which the denoting complex has secondary occurrence. This latter interpretation is, Russell says, true.

            The third problem, that the sentence “the difference between A and B does not subsist” seems to be contradictory, is also solved by the primary/secondary occurrence distinction; the sentence does not have a denotation, so if the denoting complex is interpreted as having primary occurrence, the sentence is false, and if it is interpreted as having secondary occurrence, it is true.

            This treatment will work for any sentence containing a denoting complex that includes a non-entity. All positive sentences about these characters are false; all negative sentences will be false if the denoting complex has primary occurrence and true if it is secondary. This will avoid what Russell thinks are lamentable theories about the null-class being full of un-real individuals. Russell also reasserts that his theory will explain the importance of the concept of identity, which can no longer be understood as just claiming that one thing is the same thing as another thing.

            While Russell does not get into discussing all the implications of his theory, he does say one last thing that hints of the “logical positivist” direction analytic philosophy of language took after the publication of “On Denoting.” Russell writes on page 219 the following:

                        “...when there is anything with which we do not have immediate acquaintence, but only definition by denoting phrase, then the propositions in which this thing is introduced by means of a denoting phrase do not really contain this thing as a constituent...in every proposition that we can apprehend...really entities with which we have immediate acquaintence...”

II. Some Objections to Russell’s Theory

            In his 1957 article “On Referring3 ,” Peter Strawson raises objections to Russell’s analysis of denoting phrases. One of these is that Russell misformulates sentences containing denoting phrases (“The present king of France is bald,” for example) into sentences that do three separate things; according to Russell, they make an existence claim - there is a present king of France - and a uniqueness claim - there is only one king of France - and also say something about the thing whose existence was claimed - that he is bald. Strawson’s objection to this is that not all denoting phrases seem to make the first two sorts of claims. He says that to use a denoting complex is to (conversationally and not logically) imply an existential claim, but not to make one (page 235), and so the correct answer to the question “Is the present king of France bald?” would be “There is no present king of France,” not “no,” as follows from Russell’s theory.

            Strawson’s theory corresponds to intuition (at least for Fregians) much more closely than does Russell’s, but it does a disservice to formal logic, at least at the level of quantifier logic. If existential import is only a conversational implication of sentences such as “The present king of France is bald,” then sentences such as “Some trees are green” are also going to only imply existential import. This is messing significantly with logic, and while the systems in general use now are not necessarily the last word, there has to be a much better reason for abandoning them than that some implications of them are more counterintuitive than implications of other possible logical systems. Besides, there is a reasonable solution to this same objection that can be reached without attacking quantifier logic.

            My problem with Russell’s theory is basically the same one Strawson identified in “On Referring:” I do not think it is very clear that sentences with denoting complexes necessarily make claims about their subjects’ uniqueness and existence. Consider the following sentence: The elephant is gray. On Russell’s theory, this sentence would be translated into a sentence without a denoting complex as follows: One and only one x is an elephant, and that x is gray. this is almost silly - no one rationally acquainted with the world thinks that there is only one elephant in it, even though this sentence might occur relatively frequently, especially when both elephants and preschoolers are present. So sentences with denoting complexes do not necessarily claim uniqueness.

            Neither can they be considered to necessarily have existential import. Consider this sentence: The science officer of the starship Enterprise is logical. This sentence might also frequently occur among rational persons, especially if those persons happen to be Trekkies. It would generally be considered a true sentence, by Trekkies in particular, and to say so we need not posit a null class that has Federation science officers in it. We also need not say that we are talking about a different possible world. In this world, the correct answer to the question “Is the science officer of the starship Enterprise logical?” is “yes,” not “there is no science officer of the starship Enterprise,” even though everybody knows we are not going to run into this fellow on the street.

            But we also do not want to deny existential import to denoting complexes; that will ruin logic more than is comfortable. Although this might be a messy problem, there is a way to, so to speak, have our existential import and deny it too, using Russell’s primary/secondary occurrence distinction as a tool.

            Returning to the “the present king of France is bald” example, we will say that existential import is actually part of this sentence, but that existential claims can have either primary or secondary occurrences in sentences. A primary occurrence in the case of this sentence corresponds exactly to Russell’s analysis of denoting complexes; “the present king of France is bald” would be rendered “one and only one x has the property of being the present king of France, and that x is bald.” But this won’t necessarily work, for reasons explained just above. A secondary occurrence of the existential quantifier would read “___ one and only one x has the property of being the present king of France ___ that x has the property of being bald,” where the ___’s are filled with some logical operator. Actually, they will be filled with the horseshoe; the symbol that fills them can’t be the tilde because it must be binary, and dot (or ampersand) will not work because then the sentence can be simplified to a sentence with existential import, which will not solve any problems. Wedge does not at all represent what the sentence means in English, and neither do any quantifiers or modal operators. So the horseshoe is the only option left (except for the triplebar, of which the horseshoe is a part anyway). But this is a multivalued logic horseshoe, not a regular truth-functional one. Plugging the horseshoe into the blanks, the sentence now reads “If one and only one x has the property of being the present king of France, then that x has the property of being bald.” The truth values for that sentence will be as follows:

                        -If it is true that there is one and only one x such that x has the property of being                                    the present king of France and true that that x is bald, then the sentence is true.

                        -If it is true that there is one and only one x such that x has the property of being                                    the present king of France and false that x is bald, then the sentence is false.

                                -If it is false that there is one and only one x such that x has the property of being                                    the present king of France, it does not matter whether the second part is true, false,                  or logically indeterminate (neither true nor false); the sentence is logically                                                indeterminate in that case.

            This solution corresponds to Fregian intuitions, and also answers Strawson’s and my objection without messing up logic. It remains to be seen, however, whether or not this formulation of sentences with denoting complexes, with the existential quantifier embedded in a hypothetical sentence, can solve the three problems that Russell solves with his formulation.

            Problem one: On Russell’s formulation, “Scott is the author of Waverly” becomes “One and only one x wrote Waverly, and Scott is identical with that x.” On mine, it becomes “If one and only one x wrote Waverly, then Scott is identical with x.” In this case, it will not make any relevant difference which formulation is used; either can be applied to Russell’s treatment and yield the same result.

            Problem two: The only difference between the result Russell’s formulation yields and the one mine yields is that in the case in which there is no x, the sentence is logically indeterminate for both “the present king of France is bald,” and “the present king of France is not bald,” no matter whether the denoting complex has primary or secondary occurrence. This, again, corresponds to Fregian intuition.

            Problem three: This is a non-problem. “The difference between A and B does not subsist” is not a correct translation of “There is no difference between A and B.” The correct translation (even according to the rules of quantifier logic translation) is “It is false that there is a difference between A and B.” But even so, my view does tell what is going on in all cases in which a denoting complex does not denote, which is the most important thing Russell does with his solution to problem three; my view says that in all such cases, the sentence will be logically indeterminate.