"The Logic of Partitions--with an application to Information Theory"
Abstract:
Ordinary logic is shown by modern category
theory to be essentially the logic of subsets. The notion of a (set) partition
is categorically dual to the notion of a subset, so one might expect there to
be a logic of partitions dual, in some sense, to ordinary logic. This talk
gives an introductory treatment of the logic of partitions: its connectives,
tautologies, and "truth tables" along with a transform (analogous to
the Godel transform in intutionistic logic) to generate a partition tautology
from every classical tautology. There are also partition logic versions of
"Kripke structures" and semantic tableaus but they will not be
covered.
Then using the close analogy between subset logic and partition logic, one can
mimic the development of finite probability theory from subset logic to develop
a logical information theory out of partition logic. Logical entropy is
precisely related to Shannon's entropy so this provides a conceptual foundation
for Shannon's information theory in partition
logic.