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In his book Beyond the Limits of Thought while talking about Russell's solution of paradoxes Graham Priest writes (text made bold, footnotes and references omitted),

Russell solved these problems by means of the Axiom of Reducibility. According to this, if f is any function whose arguments are of order i, then there is an extensionally equivalent function whose order is i + 1 . Russell calls such functions predicative, and states the axiom as follows:

The axiom of reducibility is the assumption that, given any function f, there is a formally equivalent predicative function.

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Russell's solution founders on a much less technical but more fundamental problem. According to the theory, every variable must range over one order of propositional functions. No variable can therefore range over all propositional functions. For the same reason, no variable can range over all propositions. This pushes many claims beyond the limit of the expressible. Take, for example, the law of excluded middle: every proposition is either true or false . Since this has a quantifier over all propositions, it cannot be expressed. Or, closer to home, consider the Axiom of Reducibility itself. This is supposed to hold for all functions, f. Russell's very statement of it (above) therefore violates the theory of orders. Even decent statements of the VCP cannot be made without violating the VCP since they must say that for any function, f, any propositional function which 'involves' f cannot be an argument for f. Such statements are impossible by Russell's own admission.

To add insult to injury, the very theory of orders cannot be explained without quantifying over all functions, and hence violating it. For to explain it, one has to express the fact that every propositional function has a determinate order. Hence, the theory is self-refuting.

Russell was well aware of at least some of these difficulties. To solve them he formulated his theory of systematic ambiguity. This is most easily understood with respect to an example. Take the statement of the Axiom of Reducibility. According to Russell, this has to be understood (for fixed i) as indicating not a single proposition, but an infinitude of propositions: one for each order that f might be. Russell, himself, explains the point with respect to a similar example as follows):

In some cases, we can see that some statement will hold of 'all nth-order properties of a', whatever value n may have. In such cases, no practical harm results from regarding the statement as being about "all properties of a" provided we remember that it is really a number of statements, and not a single statement which could be regarded as assigning another property to a, over and above such properties. Such cases will always involve some systematic ambiguity.

This solution is, frankly, disingenuous. For, however we express it, what we are supposed to understand by a systematically ambiguous formula, such as the formal statement of the Axiom of Reducibility above, is exactly what would be obtained by prefixing the formula with a universal quantifier '∀f', ranging over all functions. There must therefore be such a thought, though it cannot be expressed in the theory of orders.

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Finally, again to rub it in, note that one can express the quantifier 'for every propositional function' as 'for every n and every propositional function of order n'. (And thus systematic ambiguity is often taken to be expressed by schematic uses of order-subscripts when these are made explicit.) Hence the very first sentence of Russell's own explanation of the principle of systematic ambiguity, quoted above, violates the VCP, since it talks of all propositional functions (of a). Moreover, this statement cannot be understood in terms of systematic ambiguity since this universal quantifier does not have the whole of the rest of the statement in its scope.

My questions are as follows,

  1. "Take, for example, the law of excluded middle: every proposition is either true or false . Since this has a quantifier over all propositions, it cannot be expressed." - I don't understand why "this" has a quantifier over all propositions. So far in any modern representations of (Classical) First Order Logic, I haven't seen (nor can I remember) the LEM being represented as such. Indeed, after what Conifold says here, I don't understand what Priest means when he claims that "it cannot be expressed". How is he able to claim so?

  2. "Or, closer to home, consider the Axiom of Reducibility itself. This is supposed to hold for all functions, f. Russell's very statement of it (above) therefore violates the theory of orders." - How exactly does Russell's statement of Axioms of Reducibility violates the theory of orders?

  3. "Even decent statements of the VCP cannot be made without violating the VCP since they must say that for any function, f, any propositional function which 'involves' f cannot be an argument for f. Such statements are impossible by Russell's own admission." - As I understand, here also Priest's objection relies on the use of "for any function". If this is so, then as I have asked in my first question, it is not clear to me as why does he claim that "[e]ven decent statements of the VCP cannot be made without violating the VCP". If this is not the case then what did Priest intend to mean here?

  4. "For, however we express it, what we are supposed to understand by a systematically ambiguous formula, such as the formal statement of the Axiom of Reducibility above, is exactly what would be obtained by prefixing the formula with a universal quantifier '∀f', ranging over all functions." - I am not so sure that "what we are supposed to understand by a systematically ambiguous formula, such as the formal statement of the Axiom of Reducibility above, is exactly what would be obtained by prefixing the formula with a universal quantifier '∀f" (in fact this doubt was partly responsible for my asking the first question). Indeed, to me Russell's suggestion sounds similar (see the quote) to that of Conifold's remark that I mentioned earlier. Am I missing something important and/or subtle here?

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    From a philosophical point of view, R's theory must be understood in the context of the Logicist project of founding arithmetic (and math) on logic alone, project initiated by Frege and stopped by R's discovery of the Paradox. – Mauro ALLEGRANZA Oct 6 '17 at 6:02
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    From this point of view, some ingredient of R's theory, like Ax.of Infinity and Reducibility Axiom are highly suspect, becuase there is no way to support their logical nature (like every existential assertion). At most, they are mathematical axioms. – Mauro ALLEGRANZA Oct 6 '17 at 6:04
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    Russell's theory is discussed in the well know article of Gödel: Russell's mathematical Logic. – Mauro ALLEGRANZA Oct 6 '17 at 6:10
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    Very useful: B.Linsky, The Evolution of Principia Mathematica (2011) as well as B.Linsky & N.Griffin (editors), The Palgrave Centenary Companion to Principia Mathematica (2013). – Mauro ALLEGRANZA Oct 6 '17 at 6:12
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    You using that quote to support what you're arguing for shows that you're forgetting that PM ultimately fails, which is exactly what the links Maruo suggested, especially Gödel's, talk about. Yes, Russell had a serious reason to refrain, a similar reason that he had for assuming the axiom of infinity was a logical truth, because if he didn't then the Logicist program would have fallen apart, which it did because people continued to study PM and found all of the logical issues with it as a formal system, some of which are what Priest is pointing out. – Not_Here Oct 6 '17 at 6:43

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