Some Remarks on Probabilities Over Formalized Languages

Abstract

Some applications of probability theory, its methods and results in the field of automated problem solving and artificial intelligence in general need the possibility to ascribe probability values to well-formed formulas of a formalized language. First, a possibility is studied how to convert, using the model theory, such a formula into a random event in its classical set-theoretic sense. Second an, immediate definition of probability measure as a real-valued function over a formalized language is suggested. The obtained notions and assertions enable to formalize and develop some mathematical tools important when introducing the stochastic methods into the domain of automated problem solving (e.g., the notion of the so called randomized incidental phenomena).