LANGUAGE In the unanalysed sentence ‘the author of Hamlet’ looks like a complex name, and would have been treated as one in Frege’s system As analysed by Russell no such nominal expression appears, and instead we have a combination of predicates and quantifiers The analysis is meant to apply not only when—as in this case—there actually is an object that answers to the definite description, but also when the description is a vacuous one, such as ‘the present King of France’ A sentence such as ‘The King of France is bald’, when analysed along Russellian lines, turns out to be false Consider the following two sentences: (1) The sovereign of the United Kingdom is male (2) The sovereign of the United States is male Neither of these sentences is true, but the reason differs in the two cases The first sentence is plain false, because though there is a sovereign of the United Kingdom she is female; the second fails to be true because the United States has no sovereign ruler On Russell’s analysis this sentence is not just untrue but positively false, and accordingly its negation, ‘It is not the case that the sovereign of the United States is male’, is true (On the other hand ‘The sovereign of the United States is not male’ comes out, like the second sentence above, positively false.) What is the point of this complicated analysis? It is natural to think that since there is no sovereign of the United States, sentence (2) is not so much false as misleading; the question of its truth-value does not arise This is no doubt true of our use of such definite descriptions in ordinary language, but Frege and Russell aimed to construct a language that would be a more precise instrument than ordinary language for the purposes of logic and mathematics They both regarded it as essential that such a language should contain only expressions with a definite sense, by which they meant that all sentences containing the expressions should have a truthvalue If we allow into our system sentences lacking a truth-value, then inference and deduction become impossible Frege proposed to avoid truth-value gaps by various arbitrary stipulations Russell’s analysis, whereby ‘the sovereign of X’ is in no case a referring expression at all, achieves the definiteness that he and Frege both sought, and does so by far less artificial means It is easy enough to recognize that ‘the round square’ denotes nothing, because it is an obviously self-contradictory expression But prior to investigation it may 131