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Author: Admin | 2025-04-28
(relevant)individual\(y\) is such that if he\(y\) isa boy, then he\(y\) is identical withhim\(x\), and he\(x\) sang. The awkwardmiddle conjunct was Russell’s way of expressing uniqueness withFregean tools; cf. section seven. But rewriting the middle conjunct would not affect Russell’stechnical point, which is that ‘the boy’ does notcorrespond to any constituent of the formalism. This in turn reflectsRussell’s central claim—viz., that while a speaker mayrefer to a certain boy in saying ‘The boy sang’, the boyin question is not a constituent of the proposition indicated.According to Russell, the proposition has the form of an existentialquantification with a bound variable. It does not have theform of a function saturated by (an argument that is) the boy referredto. The proposition is general rather than singular. In this respect,‘the boy’ is like ‘some boy’ and ‘everyboy’; though on Russell’s view, not even ‘the’indicates a constituent of the proposition expressed.This extended Frege’s idea that natural language misleads usabout the structure of the propositions we assert. Russell went on toapply this hypothesis to what became a famous puzzle. Even thoughFrance is currently kingless, ‘The present king of France isbald’ can be used to express a proposition. The sentence is notmeaningless; it has implications. So if the proposition consists of afunction indicated with ‘\(\textrm{Bald}(\ )\)’ and an argumentindicated with ‘The present king of France’, there mustbe an argument that is indicated. But appeal to nonexistentkings is, to say the least, dubious. Russell concluded that ‘Thepresent king of France is bald’ expresses a quantificationalproposition:\[\exists x \{K(x) \land \forall y [K(y) \rightarrow y = x] \land B(x)\};\]where \(K(x) = \textbf{T}\) iff \(x\) is a present king ofFrance, and \(B(x) = \textbf{T}\) iff \(x\) is bald. (For presentpurposes, set aside worries about the vagueness of‘bald’.) And as Russell noted, the following contraryreasoning is spurious: every proposition is true or false; so thepresent king of France is bald or not; so there is a king of France,and he is either bald or not. For let P be theproposition that the king of France is bald. Russell held thatP is indeed true or false. On his view, it is false.Given that \(\neg \exists x [K(x)]\), it follows that\[\neg \exists x \{K(x) \land \forall y [K(y) \rightarrow y = x] \land B(x)\}.\]But it does not followthat there is a present king of France who is either bald or not.Given that \(\neg \exists x [K(x)]\), it hardly follows that\[\exists x \{K(x) \land [B(x) \lor \neg B(x)]\}.\]So we must not confuse the negation ofP with the following false proposition:\[\exists x \{K(x) \land \forall y [K(y) \rightarrow y = x] \land \neg B(x)\}.\]The ambiguity of natural language may foster such confusion, givenexamples like ‘The present king of France is bald ornot’. But according to Russell, puzzles about“nonexistence” can be resolved without specialmetaphysical theses, given the right views about logical form andnatural language.This invited the thought that other philosophical puzzles mightdissolve if we properly understood the logical forms of ourclaims. Ludwig Wittgenstein argued, in his influential Tractatus Logico-Philosophicus,that: (i) the very possibility of meaningful sentences, which can
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