In this paper we define the polyadic Pavelka algebras as algebraic structures for Rational Pavelka predicate calculus . We prove two representation theorems which are the algebraic counterpart of the completness theorem for RPL∀.

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.

In this paper we study some model theory for Gaifman probability structures. A classical result of Horn-Tarski concerning the extension of probabilities on Boolean algebras will allow us to prove some preservation theorems for probability structures, the model-companion of logical probability, etc. extending some classical results in eastern model theory.\\[0.2cm] Mathematics Subject Classification: 03C20, 03C90, 28A60.\\[0.2cm] Keywords: Gaifman probability structure, existentially closed probability structure, inductive probability, model-companion of a probability.

MV-algebras stand for the many-valued Łukasiewicz logic the same as Boolean algebras for the classical logic. States on MV-algebras were first mentioned [20] in probability theory and later also introduced in effort to capture a notion of `an average truth-value of proposition' [15] in Łukasiewicz many-valued logic. In the presented paper, an integral representation theorem for finitely-additive states on semisimple MV-algebra will be proven. Further, we shall prove extension theorems concerning states defined on sub-MV-algebras and normal partitions of unity generalizing (...) in this way the well-known Horn-Tarski theorem for Boolean algebras. (shrink)

Measures on epimorphic images of the coproduct of a non-empty family of measurable spaces are shown to be equivalent to measures on Boolean algebras. We obtain two sets of sufficient conditions for the existence of probability measures on factor spaces, from which a given measure $\mu$ on the coproduct can be retrieved. The results are applied to probability logic - to prove, reformulate and generalize Los representation theorem.