Some properties of the roots of theories in propositional logic systems

Some properties of the roots of theories, and the relationship between the sets D(Γ)D(Γ) of all conclusions of two different theories in the classical propositional logic system, Łukasiewicz propositional logic system, and the R0R0-propositional logic system are studied. It is proved that D(Γ)D(Γ) is completely determined by its root whenever ΓΓ has a root, and the construction of the roots is given.