In Russell's 1905 paper "on denoting" in which he introduces his theory of descriptions, he uses his method of analyzing propositions that include denoting phrases (descriptive ones), by rewriting them into their LOGICAL forms. By this, he embeds such propositions to "symbolic logic" to handle.

[ He does this by using the "undefinable" notion of universally quantifier and the "fundamental" notion of variable (and then as in first-order logic defines propositions including other descriptions such as 'some man', 'no man', 'a man', etc. in the terms of universally quantified propositions with logical connectives such as negation) ]

For example, taking M(x) to be the "propositional function" (i.e. property; in nowadays terminology) of being human and D(x) to be the property of being mortal, "All men are mortal" becomes in its logical form as "for all x (M(x) => D(x))". Now it seems Russell is ranging over all objects as the quantifier's domain of discourse : The class the we know (Thanks to Russell himself!) is NOT A SET if we treat sets as objects (And I guess we do so). But the problem is that in using quantifiers, our domain of discourse must be a SET according to definition.

Now don't you think it is a misuse of (first-order) logic if we use its tools to work on such an illegal proposition?(i.e. it is NOT a first-order formula, though may seem so!)

Is there any logic in which the quantifiers range over arbitrary classes (and Russell was probably considering that logic)?

Of course I think we can fix this by determining our universes in different contexts: In the above example the universe will become 'the set of all men' and then the logical form will become: "for all x D(x)" and then we will feel free to use other tools of (first-order) logic to handle this proposition. I'm mostly curious on whether Russell did that mistake!

What do you think? What's wrong here? Please ask me if there is any ambiguity in the question.


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    Russell (and Whitehead) mature work: Principia Mathematica has been devoted to develop logic and math according to Type Theory, precisely in order to avoid the paradoxes. The "original" russellian approach to logic aimed at developing logic as the "science of everything",i.e. the domain of discourse must include all entities : individuals, classes, proposition, and perheaps more. After the paradoxes, this approach (which allowed "unrestricted" quantification) was no more teneable. – Mauro ALLEGRANZA Aug 7 '14 at 13:37

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