Exogenous Semantics Approach to Enriching Logics

The exogenous semantics approach to enriching a logic consists in defining each model in the enrichment as a set of models in the original logic plus some relevant structure. We illustrate the approach by probabilizing classical propositional logic, including a novel global propositional logic. A model of the probability logic is a probability space where the outcomes are classical valuations. A model of the global propositional logic is simply a set of classical valuations. Syntactically, probabilities appear as constructors of new terms from classical formulas. Soundness and completeness results are proved for the calculi of both logics. A simple zero-knowledge protocol is used to illustrate the capabilities of the proposed probability logic.