FOM: Branching quantifiers

[Raatikainen Panu A K]

On 15 Mar 01, at 10:18, JoeShipman at aol.com wrote:

> Can someone please give an ordinary-language semantic 
> interpretation of these quantifiers?  I don't understand  what 
> they're supposed to mean.

In a nutshell: 
This can be easily done with Skolem functions. If you have an 
ordinary first order sentence
 (x)(Ey)(z)(Eu) S(x, y, z, u) 
the corresponding Skolem function form is 
(Ef)(Eg)(x)(z) S(x, f(x), z, g(x, z)), 
i.e. the last existential quantifier depends on the both earlier 
unversal quantifiers. What if one would like to have it depend only 
on the first. This cannot be expressed in the standard FO logic. 
But it can be expressed by Henkin (branching, or partially ordered) 
quantifier:

(x)(Ey)
              S(x, y, z, u)
(z)(Eu)

The Skolem function form of this is:

(Ef)(Eg)(x)(z) S(x, f(x), z, g(z)).

I hope this helped.

Panu Raatikainen 
Dept. of Philosophy
University of Helsinki