On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign ο to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicIΔ set up correspondingly by the calculiFor any calculusи we denote byяи the set of all formulae of the calculusи and byℒи the lattice of all logics that are the extensions of the logic of the calculusи, i.e. sets ofяи formulae containing the axioms ofи and closed with respect to its rules of inference. In the logiclɛℒG the sign □ is decoded as follows: □A = (A & ΔA). The result of placing □ in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and λ the decoding of □ is meant), by virtue of which the diagram is constructedIn this diagram the mapsσ, x andλ are isomorphisms, thereforex−1= λ; and the maps △ andμ are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place:σ−1 =μ−1λ△ and σ = △−1xμ. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.