The Polish logician Jan Lukasiewicz began to create systems of many-valued logic, using a third value "possible" to deal with Aristotle?s paradox of the sea battle. Now many-valued logic has applications in diverse fields. In the earlier years of development of multiple-validity idea, the most promising field of its application is artificial intelligence. This application concerns vague notions and commonsense reasoning, e.g. in expert systems. In this context fuzzy logic is also interesting, because multiple-validity is modeled in artificial intelligence via fuzzy sets and fuzzy logic.

There exist various many-valued logical systems. However non-Archimedean valued logic isn?t yet constructed. The idea of non-Archimedean multiple-validity is that (1) the set of truth values is uncountable infinite and (2) this set isn?t well-ordered. This idea can have a lot of applications in probabilistic reasoning (see [4, 1]).