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Extra info for A modern formal logic primer: predicate theory
An apparent counter-example is to be found in the very sort of question with which I have argued that Russell is principally concerned, the question whether entities of different kinds exist. He clearly attaches meaning to the proposition that there are concrete particulars, which he thinks true, and to the proposition that there are classes, which he thinks false, so that it would seem that the expression 'there are' can after all be significantly conjoined with designations of objects of different types.
For instance, if we speak of Mr Smith's only son, and the fact is either that Mr Smith has more than one son, or that he has no son at all, what our words denote, according to Frege, is the class of Mr Smith's sons, which, in the case where he has no son, will be the null-class. Russell observes, I think reasonably, that 'This procedure, though it may not lead to actual logical errors, is plainly artificial, and does not give an exact analysis of the matter. ' 1 A fact which is not often remarked is that if we do take expressions like 'the present King of France' as denoting the null-class, the proposition 'The present King of France is bald' and 'The present King of France is not bald' both tum out to be true, since the nullclass is included in all classes.
Then 'Some man has walked on the moon' becomes 'There is an x such that x is human and has walked on the moon', and 'The author of Waverley was Scott' becomes 'There is an x such that x wrote Waverley, such that for all y, if y wrote Waverley, y is identical with x, and such that xis identical with Scott'. The use of quantifiers not only simplifies the translation, but also avoids the undesirable implication of Russell's earlier formulae that whenever we use a descriptive phrase we are speaking in metalinguistic fashion about propositional functions and, the extent to which they are satisfied.