  Shannon information theory provides various measures of so-called "syntactic
information", which reflect the amount of statistical correlation between
systems. In contrast, the concept of "semantic information" refers to those
correlations which carry significance or "meaning" for a given system. Semantic
information plays an important role in many fields, including biology,
cognitive science, and philosophy, and there has been a long-standing interest
in formulating a broadly applicable and formal theory of semantic information.
In this paper we introduce such a theory. We define semantic information as the
syntactic information that a physical system has about its environment which is
causally necessary for the system to maintain its own existence. "Causal
necessity" is defined in terms of counter-factual interventions which scramble
correlations between the system and its environment, while "maintaining
existence" is defined in terms of the system's ability to keep itself in a low
entropy state. We also use recent results in nonequilibrium statistical physics
to analyze semantic information from a thermodynamic point of view. Our
framework is grounded in the intrinsic dynamics of a system coupled to an
environment, and is applicable to any physical system, living or otherwise. It
leads to formal definitions of several concepts that have been intuitively
understood to be related to semantic information, including "value of
information", "semantic content", and "agency".
