  Three problems arose after discovering many-valued logic. The first concerned its axiomatization and metalogical properties, the second its philosophical foundations and intuitive interpretation, and the third its applications. Due to the work of Łukasiewicz himself, Wajsberg and Słupecki, the first group of questions was largely solved. Wajsberg showed that the formulas: CpCqp, CCpqCCqrCpr, CCNpNqCqp, CCCpNppp axiomatize Ł3 (three-valued propositional calculus). The same author proved that a finite Łn is axiomatizable if it includes the theorems: CCpqCCqrCpr, CCCqrCCpqCpr, CCqqCpp, CCpqCNqNp, CNqCCpqNq. If n =  ℵ0, Łn can be axiomatized by (Łukasiewicz’s conjecture, proved by Wajsberg): CpCqp, CCpqCCqrCpr, CCCpqqqCCqpp, CCCpqCqpCqp, CCNpNqCqp. However, all of the above axiom-sets are functionally incomplete. The problem was solved by Słupecki for Ł3. He introduced the new functor T defined by T1 = T½ = T0 = ½ and added the formulas CTpNTp, CNTpTp to Wajsberg’s axioms. All Łukasiewicz’s many-valued logics are consistent. Słupecki proved that Ł3 is Post-complete. Every Łn (n&gt;2) is contained in two-valued logic, although the converse does not hold; for example, the formulas CCNpNp, CCNppp, CCpqCCpNqNp, CCpKNqNp, CcpEqNqNq are theorems in the two-valued system only. If n =  ℵ0, Łn is contained in every finite Łn.