[FOM] quantifiers as sentential operators

[G. Aldo Antonelli]

Alasdair wrote:

> It definitely makes sense, because that is exactly the approach
> taken in the theory of cylindric algebras.  It can be
> considered as a form of multimodal logic, where you have
> countably many S5 modal operators, one for each variable.

With the added complication that you have to get the operators to 
commute: Box_x Box_y A must turn out to be equivalent to Box_y Box_x A.

In terms of Kripke frames this is a form of confluence: if you can get 
from world w to world w' by following an x-transition composed with a 
y-transition, then it must be possible to get from w to w' also 
following a y-transition composed with an x-transition.

And since we are pushing the analogy -- what would the worlds be? It's 
clear that they must be assignments to the variables, where an 
x-transition takes you from an assignment w to an x-variant w'.

An interesting line of thought, it seems to me.

-- Aldo

*****************************************
G. Aldo Antonelli
Professor of Philosophy
University of California, Davis
Coordinating Editor, Review of Symbolic Logic
http://philosophy.ucdavis.edu/antonelli
antonelli at ucdavis.edu