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, ∀*x**F**x* says that *F*ness has universality, while ∃*x**F**x* says that *F*ness has instantiatedness.

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

- ∀
*x*∃*y**F*(*x*,*y*).

Properties correspond to formulae open in one variable. But in the inner expression ∃

*y**F**x**y* the quantifier is applied to

*F*(

*x*,

*y*) which is open in two variables.

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

Now, for any predicate *Q*, there is a property of *Q*ness. So, for any *x*, there is a property of (Î»*y**F**x**y*)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*)=(Î»*y**F**x**y*)ness.

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

- (Î»
*y**F**x**y*)ness has *I*.

But (2) defines a predicate that is being applied to

*x*, a predicate we can refer to as Î»

*x*[(Î»

*y**F**x**y*)ness has

*I*]. This predicate in turn expresses a property: (Î»

*x*[(Î»

*y**F**x**y*)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:

- [Î»
*x*[(Î»*y**F**x**y*)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:

- 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 *x*_{1},...,*x*_{k−1},*x*,*x*_{k+1},...,*x*_{n} stand in *R* if and only if *X*_{1},...,*x*_{k−1},*x*_{k+1},...,*x*_{n} stand in *R*_{k,x}.
- 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.
- 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.