Properties of the Logical Consequence Operation and Its Relationship with the Independence of Propositional Logic

We study the properties of the logical consequence operation and the characteristic features of independent sets of formulas. Further, we apply these results to propositional logic. Finally, we show under what conditions the results of addition of a formula to independent sets of formulas and the union of two independent sets of formulas are also independent by the operation of logical consequence, i.e., we establish the relationship between the logical consequence and the preservation of independence in propositional logic.