Talk by Natasha Alechina
Quantification over Interdependent Variables
Consider the following truth definition for $\exists x \varphi$: `there exists an
object $d$ such that
$\varphi(d)$ holds and $d$ is a {\em possible value} for $x$ given that $y_1,\ldots,y_n$ have taken
values $d_1,\ldots,d_n$, respectively'. Several already known logics arise from this truth definition
by specifying what are the relevant variables $y_1,\ldots,y_n$ and what does `possible value' mean.
One example is a logic where not all assignments of values to the variables are allowed. Then
$d$ is a possible value for $x$ if ${}^{x,y_1,\ldots,y_n}_{d,d_1,\ldots,d_n}$ is an allowed
assignment. An interesting logic arises if $y_1,\ldots,y_n$ above are the free variables of $\exists
x \varphi$, and `possible values' are given by some abstract dependence relation. It can be used to give
an alternative semantics to some generalized quantifiers. This unifying approach to different logics,
based on the idea that the range of a variable depends on the values of other variables, helps to
clarify connections between them. We prove several theorems to that extent.
Paul Dekker, November 2, 1995