Russell and the Puzzle of Excluded Middle

Frege was able to resolve his linguistic puzzles through his famous sense and reference distinction, yet Russell wanted to develop a theory that could present a solution that does not need to rely on what he considered making arbitrary assumptions (i.e. positing sense when it is not needed). Essentially, Russell's theory of descriptions is predicated upon a purely referential theory of meaning and takes at its heart the understanding that denoting phrases (ordinary names and descriptions) are not singular terms, but are quantifier phrases. On the surface, the puzzle involving the law of excluded middle presents a challenge for Russell's theory because it seems that he would need to reject the important logical law of excluded middle in order to preserve the cogency of his overall theory. However, further analysis shows that this puzzle can be resolved when combining three key issues of Russell's theory: names and descriptions are not logically proper names, but are incomplete symbols that disappear upon analysis, the reduction of these sentences to quantified sentences, and a primary/secondary scope distinction applied to a negation operator.

Russell makes it clear that he wants to uphold the logical principle of excluded middle, which states that every proposition is either true or not true. Applying this principle to disjunctions, it can be concluded that any sentences taking the form (S v ~S) must be true, since one of the disjuncts is necessarily true. Holding this principle as truen, a problem arises when one considers a situation in which the referent of a sentence involves a vacuous singular. For example, one considers two sentences:

(1) The present King of France is bald

(2) The present King of France is not bald

Russell notices that if one were to look at the list of all the people in the world who are bald and the list of all those who are non-bald, the present King of France does not appear on either list. In fact, there does not exist a present King of France, which makes the vacuous denoting phrase of "present King of Frace" neither bald nor non-bald. Using this logic, the sentences are both false (Russell posits Hegel's solution to this dilemma that the king is probably wearing a wig). However, the principle of excluded middle makes it impossible for two contradictory sentences to have the same truth value. After all, how can we say that these two sentences can both be false?

The first step in understanding Russell's solution to this paradox is to realize what Russell believes is the proper way to translate sentences into logical forms. He asserts that denoting phrases are not genuine singular terms occurring in singular subject-predicate propositions, but rather function like general terms occurring in general quantified propositions where the ordinary name or description disappear. The genuine logical form of (1) translated into first-order predicate logic is:

(1') (Ex) (Kx & (Ay)(Ky Ð" (y = x)) & Bx)

In his theory, this sentence should not be translated into a subject-predicate form, but as an existential generalization with conjunctions (general proposition), in which the original description disappears on analysis. In this sense, the proposition is not about what the denoting phrase may or may not denote, since the semantic value of a description is not an object, but a second order function that takes a first order function as argument. With this translation, Russell would resolve that that the claim is false since it is claiming that there exists exactly one thing that has the property of being the present King of France and the property of baldness.

Russell's approach has shown that the occurrence of a description is replaced with a logically equivalent, general quantified proposition. However, there are situations in which the context has an ambiguity in how much of the original sentence the description is removed. For example, when a negation operator is