On Denoting

        Russell’s theory takes the “notion of the variable as fundamental;” he uses ‘C(x)’ to mean a proposition, where x, the variable, is essentially and entirely undetermined (213).  Russell says that if we take ‘C(x) is always true,’ and ‘C(x) is sometimes true,’ and we substitute everything, nothing, or something in for x, then we can interpret these two denoting phrases to be as follows (213):

• C(everything) means ‘C(x) is always true’
• C(nothing) means ‘“C(x) is false” is always true’
• C(something) means ‘It is false that “C(x) is false” is always true’.

Here, ‘C(x) is always true,’ is taken as ultimate and indefinable, and the others are defined by means of it (213).  This helps to show what Russell means when he says, “a phrase is denoting solely in virtue of its form,” as stated above.  If we state that C(x) = C(everything), and we take C(everything) as always true, then C of anything else depends on C(everything).  Hence, denoting phrases don’t have meaning in themselves; however there is meaning in every proposition in which their verbal expressions occur (213).  A good example of this is the proposition is one that Russell gives when he says, “I met a man.”  He says that if this were true, he would have had to have met a definite man, but that’s not what he’s affirming.  What he is affirming is, ‘“I met x, and x is human” is not always false’ (213).  

Russell goes on to state a theory introduced by Frege, which distinguishes, in a denoting phrase, the difference between “meaning,” or complex significance of a phrase, verses “denotation,” which is a certain, and rather simple point in a phrase.  Denotations, for Frege, are mere constituents of the meaning of the phrase (214).  For instance, the phrase, “evidence of life on other planets,” is complex in meaning, but its denotation is simple.  Life, planets, etc., are constituents of the meaning, whereas denotation has no constituents at all.  For Russell, when he wants to speak about the meaning of a denoting phrase, as apposed to its denotation, he uses inverted commas (215).  Therefore, if we take C as a denoting phrase, we should take into consideration the relationship between C and ‘C’, where C represents the denotation, and ‘C’ expresses the meaning.  The meaning denotes the denotation; in other words, meanings can have denotations, but denotations cannot have meanings.  But the difficulty of the situation is how to succeed in both preserving the connection of meaning and denotation and preventing them from being one and the same; furthermore, you can only fully get the meaning by using denoting phrases.  Thus, in order to get the meaning we want, we must speak of ‘the meaning of “C”,’ which is the same meaning as ‘C’ by itself, and compare it to C (216).

For Russell, the expression, “Apollo visited Athens” is false.  His reasoning for this is as follows:  If “Apollo,” then the proposition is that he is the sun-god, and if C has the denoting phrase, “Apollo visited Athens”, then ‘C’ expresses the meaning, “‘Apollo, the sun-god, visited Athens’.”  Here, “Apollo visited Athens” is false if the occurrence of “Apollo” is primary.  Therefore, all propositions in which “Apollo” has a primary occurrence are false.  

The difference between A and B’ has a denotation when A and B are different, but not when they’re the same. If A and B are the same, there is no entity x.  Thus, out of any proposition we can make a denoting phrase, which denotes an entity if the proposition is true, but does not denote an entity if the proposition is false (218).  All propositions in which Apollo occurs, are to be interpreted by the above rules for denoting phrases.  If ‘Apollo’ has a primary occurrence, the proposition containing the occurrence is false; and since we’ve shown that there is no difference between C and ‘C’ in this situation, then there is no entity “Apollo,” making the statement, “Apollo visited Athens” false.  With Russell’s theory of denoting, we are able to presume that there are no unreal individuals, because if there is no entity, there is no truth, so the null-class is the class containing, as members, all unreal individuals; and, since Apollo is an unreal individual, he is contained in this null-class, and the statement “Apollo visited Athens” is false because it does not denote an entity (218-219).  Furthermore, if we go back to one of the original statements: C(nothing) means ‘“C(x) is false” is always true’ and we plug in “Apollo visited Athens” for x, then “C(Apollo visited Athens) is false” is always true’, then Russell’s argument that “Apollo visited Athens” is false, is true!