Thursday, September 6, 2012

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. xyF(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 ∀xyFxy. 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:

  1. yFxy)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:
  1. x[(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:

  1. If R is an n-ary relation and 1≤kn, then for any x there is an (n−1)-ary relation Rk,x which we might call the <k,x>-contraction of R such that x1,...,xk−1,x,xk+1,...,xn stand in R if and only if X1,...,xk−1,xk+1,...,xn stand in Rk,x.
  2. 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.
  3. 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 P1,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 R1,x, and then define the function g(x) that maps an object x to the proposition I1,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 U1,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.

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