Defining by Structural Induction in the Positive Propositional Language

The main goal of the paper consists in proving schemes for defining by structural induction in the language defined by Adam Grabowski [11]. The article consists of four parts. Besides the preliminaries where we prove some simple facts still missing in the library, they are: “About the language” in which the consequences of the fact that the algebra of formulae is free are formulated, “Defining by structural induction” in which two schemes are proved, “The tree of the subformulae” in which a scheme proved in the previous section is used to define the tree of subformulae; also some simple facts about the tree are proved.