Formal axiomatic theories on the base of three-valued logic

Formal axiomatic theories on the base of J. Lukasiewicz's three-valued logic are considered. Main notions connected with these theories are introduced, for example, the notion of a Luk-model (i.e., model of a theory in terms of J. Lukasiewicz's logic), of a Luk-consistent theory, Luk-complete theory. Logical calculi describing such theories are defined; analogues of the classical theorems on compactness and completeness are proved. Arithmetical theories based on J. Lukasewicz's logic and on its constructive (intuitionistic) variant are investigated; the theorem on effective Luk-incompleteness for a large class of arithmetical systems is proved which is a three-valued analogue of K. Goedel's famous theorem on the incompleteness of formal theories. Three-valued analogues of M. Presburger's arithmetical system are defined; it is proved that they are Luk-complete but not complete in the classical sense.