propositional language; nonstandard semantics
The classical propositional language is evaluated in such a way that truthvalues are subsets of the set of all positive integers. Such an evaluation is projected in two different ways into the unit interval of real numbers so that two real-valued evaluations are obtained. The set of tautologies is proved to be identical, in all the three cases, with the set of classical propositional tautologies, but the induced evaluations meet some natural properties of probability measures with respect to nonstandard supremum and infimum operations induced in the unit interval of real numbers.
 Church A.: Introduction to Mathematical Logic
. Part I. Second edition. Princeton Univ. Press, Princeton, NJ 1956 MR 1435972
| Zbl 1016.03500
 Kramosil I.: Probabilistic first–order predicate calculus with doubled nonstandard semantics. In: IPMU 98 (Information Processing and Management of Uncertainty) – Proceedings of the Conference, vol. 2, Paris, July 6 – 10, 1998. Paris, La Sorbonne, pp. 1390–1397