Hintikka's criticism of the Fregean view of quantifiers

On the Fregean view of quantifiers, quantifiers express properties of properties. Thus, ∀ expresses a property U of Universality and ∃ expresses a property I of instantiatedness. So, ∀xFx says that Fness has universality, while ∃xFx says that Fness has instantiatedness.

One of Hintikka's criticisms is that it is hard to make sense of nested quantifiers. Consider for instance

1. ∀x∃yF(x,y).

Properties correspond to formulae open in one variable. But in the inner expression ∃yFxy the quantifier is applied to F(x,y) which is open in two variables.

But the Fregean can say this about ∀x∃yFxy. For any fixed value of x, there is a unary predicate λyFxy such that (λyFxy)(y) just in case Fxy. The λ functor takes a variable and any expression possibly containing that variable and returns a predicate. Thus, λy(y=2y) is the predicate that says of something that it is equal to twice itself.

Now, for any predicate Q, there is a property of Qness. So, for any x, there is a property of (λyFxy)ness. In other words, there is a function f from objects to properties, such that f(x) is a property that is had by y just in case F(x,y). We can write f(x)=(λyFxy)ness.

Now, we can replace the inner quantification by its Fregean rendering:

2. (λyFxy)ness has I.

But (2) defines a predicate that is being applied to x, a predicate we can refer to as λx[(λyFxy)ness has I]. This predicate in turn expresses a property: (λx[(λyFxy)ness has I])ness. And then the outer ∀x quantifier in (1) says that this property has universality. Thus our final Fregean rendering of (1) is:

3. [λx[(λyFxy)ness has I]]ness has U.

We can now ask which proposition formation rules were used in the above construction. These seem to be it:

4. If R is an n-ary relation and 1≤k≤n, then for any x there is an (n−1)-ary relation R_k,x which we might call the <k,x>-contraction of R such that x₁,...,x_k−1,x,x_k+1,...,x_n stand in R if and only if X₁,...,x_k−1,x_k+1,...,x_n stand in R_k,x.
5. If p is a function from objects to propositions, then there is a property p* which we might call the propertification of p such that x has p* iff p(x) is true.
6. There are the properties I and U of instantiation and universality, respectively.

We can think of propertification and contraction as related in an inverse fashion. Given an n-ary relation, contraction can be used to define a function from objects to (n−1)-ary relations, and propertification takes a function from objects to 0-ary relations and defines a 1-ary relation from it (this could be generalized to an operation that takes a function from objects to (n−1)-ary relations and defines an n-ary relation from it).

Observe that if P is a property, i.e., a unary relation, then the contraction P_1,x is a proposition (propositions are 0-ary relations), equivalent to the proposition that says of x that it has P.

With these two rules and the relation R that is expressed by the predicate F, start by defining the function f(x) that maps an object x to the property R_1,x, and then define the function g(x) that maps an object x to the proposition I_1,f(x). Thus, g(x) says that x stands in R to something. Now, we can form the propertification g* of the function g, and to get (1) we just say that g* has U. Thus the proposition that is expressed by (1) will be U_1,g*.

One worry about proposition formation rules is that we might fear that if we allow too many, we will be able to form a liar-type sentence. A somewhat arbitrary restriction in the above is that we only get to form a propertification for functions of first-order objects.

Another worry I have is that I made use of the concept of a function, and I'd like to do without that.