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# Evening Lecture

## Jaakko Hintikka

## On the Logic of Interdependent Quantifiers

A fully satisfactory logical language should be capable of
representing any pattern of dependence and independence between
variables. In the usual logical languages, dependencies between
variables are expressed by dependencies between the quantifiers to
which they are bound, while dependencies between quantifiers are
expressed by the nesting of their scopes. But in the received
first-order logic such nesting does not allow us to express arbitrary
patterns of dependence and independence. Hence the received
first-order logic has to be extended to IF first-order logic.

One particularly interesting kind of pattern involves
interdependence. Sequences of interdependent variables can be thought
of as vectors in a logical space. Then the set of all Skolem
functions of a given sentence defines an operator (Skolem operator),
i.e. a mapping of that space into itself. The sentence S states
that those and only those vectors exist that are eigenvectors of one
of the Skolem operators of S. In the logic of
interdependent quantifiers, instantiation becomes tricky and in some
cases impossible. All this is relevant to the applications of IF
logic.