Soundness and Completeness Results

Now we can relate the semantical systems of sections 9.1, 9.2 and 9.3 to the syntactical systems of chapter 10 by means of soundness and completeness theorems. We are going to present three kinds of such theorems: (“strong”) soundness and completeness concerning derivability on the syntactical side and entailment on the semantical side, (“weak”) soundness and completeness concerning provability on the syntactical side and validity on the semantical side, and (“strong”) soundness and completeness concerning conditional theories on the syntactical side and conditional theories that are associated with models on the semantical side. The completeness parts of the theorems of the latter kind correspond to the type of completeness theorem by which a consistent classical theory is shown to have a non-empty set of classical models; in the case of probability semantics, such a kind of theorem is of course only available if the probability semantics considered includes the notion of a model.