Contents Introduction.................................................................................................................................................. 3 § 1. System $\mathscr{S}$ of a propositional calculus...................................................................... 4 § 2. System $\mathscr{S}*$..................................................................................................................... 5 § 3. $\mathscr{S}*$-algebras.................................................................................................................. 9 § 4. The algebra of set designations of $\mathscr{S}*$.................................................................... 11 § 5. Models of the system $\mathscr{S}*$............................................................................................ 13 § G. Completeness theorem................................................................................................................... 17 § 7. Formalized theory of fields of sets.................................................................................................. 20 § 8. Classical elementary theory of Boolean algebras....................................................................... 23 § 9. Elementary theories of $\mathscr{S}$-algebras based on $\mathscr{S}$-logic.................. 26 References................................................................................................................................................. 29