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