Symbolic representation of probabilistic worlds

Symbolic representation of environmental variables is a ubiquitous and often debated component of cognitive science. Yet notwithstanding centuries of philosophical discussion, the efficacy, scope, and validity of such representation has rarely been given direct consideration from a mathematical point of view. This paper introduces a quantitative measure of the effectiveness of symbolic representation, and develops formal constraints under which such representation is in fact warranted. The effectiveness of symbolic representation hinges on the probabilistic structure of the environment that is to be represented. For arbitrary probability distributions (i.e., environments), symbolic representation is generally not warranted. But in modal environments, defined here as those that consist of mixtures of component distributions that are narrow (“spiky”) relative to their spreads, symbolic representation can be shown to represent the environment with a relatively negligible loss of information. Modal environments support propositional forms, logical relations, and other familiar features of symbolic representation. Hence the assumption that our environment is, in fact, modal is a key tacit assumption underlying the use of symbols in cognitive science.

► Symbolic representation of the world is a core issue in cognitive science.
► Symbol systems represent the world via combinations of a small number of representational elements.
► Probabilistic models of cognition are increasingly prominent, but lack compositional relations.
► This paper establish conditions under which symbols can represent probabilistic worlds with negligible information loss.
► This establishes mathematical conditions for the validity of symbolic representations.