Idle remarks on Russell's paradox and higher-order entities
Okay, as promised here comes the third and last part of my little series on Rieger's paradox. I will first describe a general version of Russell's paradox, of which Rieger's is a special case. Then I'll discuss whether Frege is already prey to the paradox by his admission of too many concepts. Whether he is will depend on whether it makes sense to say that there are entities which are not first-order entities. I'm sorry that there is probably nothing new in all this.
First, the general version of Russell's paradox.
Let R be any relation. Suppose there is some thing
t such that all and only the (possibly zero) things
which are not R-related to themselves are R-related to
t. Then 
x(
R(x,x)
R(x,t)).
But then R(t,t)R(t,t) -- Contradiction. Hence there is no such thing.
Examples. 1. Where R is the relation of class-membership, Russell's paradox proves that there is no class t of which all and only the classes that are not members of themselves are members. 2. Where R is the relation of satisfaction between things and predicates, Russell's paradox proves that there is no predicate t satisfied by all and only the predicates that do not satisfy themselves. 3. Where R is the relation of instantiation between things and properties, Russell's paradox proves that there is no property t instantiated by all and only the properties that do not instantiate themselves.
Here is an equivalent formulation of what Russell's paradox proves: There is no relation R such that for some thing t, all and only the things which are not R-related to themselves are R-related to t. A fortiori, there is no relation R such that for any things whatsoever, there is some thing t to which all and only those things are R-related. (I'll call such an alleged relation 'Russellian'.)
Examples. 1. There is no membership-relation such that for any things there is some class having all and only those things as members. 2. There is no satisfaction-relation such that for any things there is some predicate satisfied by all and only those things. 3. There is no instantiation-relation such that for any things there is some property instantiated by all and only those things.
Another example: Rieger's paradox. Assume that there is a relation that holds between any concept F and the thought [Fb] that Ben Lomond falls under F. Assume further that 1) for any things there is some concept under which all and only those things fall, and 2) when F and G are concepts under which different things fall, then [Fb] and [Gb] are different thoughts. Then for any things whatsoever, there is a thought t that Ben Lomond is one of those things. This however is impossible, by the weak version of Russell's paradox.
In my defence of Frege, I suggested that he needn't accept assumption (2). One could also deny the very first assumption and claim that for some concepts F, there is no thought [Fb]. This seems rather hopeless though (in particular since for the paradox to be blocked, it would have to be true of the property 'ordinary'). Finally, one could reject assumption (1). There is a good reason to do so, but I'm not quite sure if it is conclusive. The reason is that assumption (1) is inconsistent: By the weak version of Russell's paradox, there can be no relation of 'falling under' such that for any things there is some concept under which all and only those things fall.
The problem with this argument is that it presupposes something that Frege vigorously denies: That concepts are things. Now Russell's paradox obviously doesn't presuppose any particular notion of 'thing'. In fact, I've been using 'thing' as a blanket term: Anything there exists is a thing. So to escape the refutation of assumption (1), Frege has to deny that concepts exist. But if there are no concepts, then certainly there are no concepts corresponding to every plurality of things. So (1) is false either way.
Rejoinder: Existence comes in different flavours. It's true that concepts don't have 'first-order' existence, but they have various 'higher-order' ones. There is no single concept of existence for first-order and higher-order entities. What's more, the same holds for all concepts. E.g. there is no meaningful notion of 'thing' applying both to first-order and higher-order entities. This is all very plain in higher-oder logic: There are first-order quantifiers and second-order quantifiers, but there is no single quantifier that somehow comprises both of them. Likewise all predicates are either first-order or higher-order. Since concepts are precisely the values of higher-order variables, any plausible translation of assumption (1) into higher-order logic will not only be true, but logically true.
I find this view rather disconcerting. As soon as you try to express it in English, you'll end up contradicting yourself: If no concept applies both to first-order and second-order entities, then what about 'entity' and 'first-order entity or second-order entity'? If no single quantifier goes over absolutely all there is, then what about the quantifier used in this very statement? If no concept is both second- and third-order, then what about 'concept'? The source of these difficulties is that ordinary language lacks the means of higher-order quantification. This is why Frege permanently runs into the problem of the concept 'horse'. This is also part of why the interpretation of higher-order logic is such a messy business. There is no problem as long as your domain does not include all the things. But if it does, then you can't take the second-order quantifiers to range over sets or classes or properties or incomprehensible pluralities or what have you. Because if there are any such things, then they are already in the domain, and you can't map the values of second-order variables into the first-order entities. If, on the other hand, there are no such things, then it is hard to see how they can be usefully employed for the semantics of higher-order logic.
But as I said, this all just mirrors the fact that ordinary language lacks higher-order quantification. Of course it is messy to find a first-order interpretation of higher-order logic. Just as it is messy to find a propositional interpretation of first-order logic. Yet couldn't we just acquiesce in a higher-order language, as we do in a first-order language? The fact that English is not a higher-order language certainly can't settle ontological disputes. I'm not sure what to say against this.
Actually, I find George Boolos' argument quite convincing that ordinary language does provide means for (monadic) second-order quantification, namely plurals. Maybe we should really use plurals to state the semantic value of predicates in Tarski-style semantics. This is certainly not what Frege had in mind -- what is unsaturated about pluralities? -- but it might make sense of much of what he says even without forcing upon us a strange and new conceptual scheme.
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