In "On Denoting," Bertrand Russell (1905) asserts that an adequate theory of denoting should be able to uphold the law of excluded middle. In other words, a theory of denoting should be able to determine whether 'p is q' is true or 'p is not-q' is true. When a proposition contains a denoting phrase that fails to denote (hereafter "non-denoting phrase"), such as 'the present King of France', it seems difficult to determine whether the proposition is true or false. Nevertheless, an adequate theory of denoting should uphold the law of excluded middle, even in cases where a proposition contains a non-denoting phrase.
In this paper, I will show that Russell's solution fails to preserve the law of excluded middle because his solution to the puzzle of non-denoting phrases reveals that there was no puzzle in the first place. I will begin by outlining Russell's solution to the puzzle. Then, I will argue that Russell's acceptance of the idea that the law of excluded middle and the principle of bivalence are equivalent leads to the conclusion that Russell's theory fails to preserve the law of excluded middle. Finally, I will consider a possible rejoinder Russell may offer as a defense, and I will show that such a defense will lead to unacceptable consequences.
Russell's Solution to the Puzzle of Non-Denoting Phrases
When we confront a proposition and its negation that both contain non-denoting phrases, there should be a way to determine which proposition is true and which proposition is false. According to Russell, the law of excluded middle requires that every proposition is either true or false, but propositions containing non-denoting phrases are a problem.
By the law of excluded middle, either 'A is B' or 'A is not B' must be true. Hence either `the present King of France is bald' or `the present King of France is not bald' must be true. Yet if we enumerated the things that are bald, and then the things that are not bald, we should not find the present King of France in either list. (Russell 1905, p. 110)
If p and ~p both contain non-denoting phrases, then it is at least plausible that both p is false and ~p is false. Such a conclusion violates the law of excluded middle. Russell believes that his theory of denotation solves this puzzle because it can determine the truth-value of both 'the present King of France is bald' and 'the present King of France is not bald'.
Russell argues that even if the denoting phrase fails to denote, there is a way to uphold the law of excluded middle. Suppose we have two propositions: (x): the present King of France is bald'' and (y) the present King of France is not bald. According to Russell, x is false because there is no unique object which satisfies it. Russell must try to explain how y comes out true.
Russell shows that y is scope amibiguous, and y can be read in two different ways:
One and only one thing is the King of France and that one is not bald; or
It is not the case that one and only one thing is King of France and that one is bald.
In (i), the denoting phrase has a primary occurrence involving a negative predicate, and this sentence is false. In (ii), the denoting phrase has a secondary occurrence, and this sentence is true. Secondary occurrence, says Russell, "may be defined as one in which the phrase occurs in a proposition p which is a mere constituent of the proposition we are considering, and the substitution for the denoting phrase is to be effected in p, not in the whole proposition concerned" (Russell 1905, p. 115). By virtue of the secondary occurrence in (ii), the whole proposition is a complex that denies x, which is false. Therefore, y under reading (ii) is true and the law of excluded middle is preserved.
Criticism of Russell's Solution
Russell uses the scope distinction of propositional negation to figure out which proposition about 'the present King of France' is true. Russell believes that 'it is not the case that one and only one thing is the present King of France and that one is bald' is true since the denoting phrase in this proposition has secondary occurrence.
Critics sometimes point out that the denoting phrase fails to denote because the referent is an incomplete object. 'The present King of France' is an incomplete object, and it seems impossible to conclude that the King is bald or he is not bald (cf. Meinong 1913/1983). Similarly, Ernst Mally said that an object can still be 'so' (Sosein) even though it has no being (Sein) of any sort. These types of objects lack definite characteristics, but they can still be referred to in propositions. What follows is perplexing: it is true that the present King of France is bald and it is true that the present King of France is not bald (Mally 1971).
These criticisms press on an ontological assumption that not many analytic philosophers may be willing to forfeit, the assumption that no statement about what does not exist is true. Russell assigns all statements where non-entities have a primary occurrence the value false. So, we should abandon a critique of Russell's ontological assumption in favour of a criticism that addresses Russell on his own terms.
Russell's solution treats 'the present King of France is not bald' as scope-ambiguous. One way to challenge the adequacy of Russell's solution is through the notion of the law of excluded middle. One interpretation of the law of excluded middle says that a proposition is either true or false. If a proposition is neither true nor false, then it falls outside the scope of the law of excluded middle. The law of excluded middle is preserved because some propositions turn out to be truth-valueless. In other words, one could hold that both 'the present King of France is bald' and 'the present King of France is not bald' are truth-valueless.
This is not to say that these propositions have a 'third' truth-value. This would clearly violate the law of excluded middle as defined here. In fact, the acceptance of a three-valued logic leads to an obvious question: what could the interpretation of a third truth-value possibly be? ( For an answer to this question, see Hilary Putnam, "Three-Valued Logic,'' (1957) reprinted in Mathematics, Matter, and Method: Philosophical Papers, Volume I (Cambridge, UK: Cambridge University Press, 1975), pp. 166-173.) Truth-valueless propositions preserve the law of excluded middle because they fall outside the scope of the law.
One might notice that Russell's notion of the law of excluded middle looks very much like the principle of bivalence. In fact, the law of excluded middle, as Russell presents it, seems indistinguishable from the principle of bivalence. Russell believes that the law of excluded middle is that every proposition is either true or false, which seems equivalent to the principle of bivalence: anything to which truth or falsehood is applicable is either true or false. In logical notation, however, the distinction between the principle of bivalence and the law of excluded middle is evident. Whereas the principle of bivalence says 'p is true v ~p is true', the law of excluded middle says 'it is true that (p v ~p). Russell seems to accept the principle of bivalence as equivalent to the law of excluded middle.
The principle of bivalence seems inapplicable to sentences of the form 'the p is q' when there is no existent p. For instance, the principle of bivalence is inapplicable to the proposition 'the golden mountain is 5000 meters high' when there is no golden mountain. If the principle of bivalence is inapplicable to a proposition containing a non-denoting phrase, then there is no true reading of the proposition `the p is q'. If there is no true reading of the proposition, then there does not seem to be a genuine puzzle for Russell to solve. For example, 'the present King of France' could be either 'one and only one thing is the present King of France and he is not bald' or 'it is not the case that one and only one thing is the present King of France and he is bald'. This works equally well with the proposition, 'the present King of France is bald'. Russell fails to give enough evidence to believe that one of his readings of 'the present King of France is not bald' is true. Thus, there does not seem to be a legitimate puzzle for Russell to solve.
Russell's Rejoinder
The principle of bivalence seems inapplicable to sentences of the form `the p is q' when there is no existent p. For instance, the principle of bivalence is inapplicable to the proposition 'the golden mountain is 5000 meters high' when there is no golden mountain. If the principle of bivalence is inapplicable to a proposition containing a non-denoting phrase, then there is no true reading of the proposition 'the p is q'. If there is no true reading of the proposition, then there does not seem to be a genuine puzzle for Russell to solve. For example, 'the present King of France' could be either 'one and only one thing is the present King of France and he is not bald' or 'it is not the case that one and only one thing is the present King of France and he is bald'. This works equally well with the proposition, 'the present King of France is bald'. Russell fails to give enough evidence to believe that one of his readings of 'the present King of France is not bald' is true. Thus, there does not seem to be a legitimate puzzle for Russell to solve.
Conclusion
This post has tried to show why we should call Russell's solution of the puzzle into question. The scope ambiguous reading of 'the present King of France is not bald' does not solve the puzzle. In fact, the solution seems to point out that there was no puzzle to begin with. By calling Russell's solution into question, we have good reason to believe that his theory fails to preserve the law of excluded middle as he had intended to do.
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