Andrey Kuznetsov

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). 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 the decoding of is meant), by virtue of which the diagram is constructed.

The article deals with the ontological subsystem of the philosophical foundations of the corresponding picture of the world, defining the typical understanding of the world as a unity in a variety of properties and relations typical of a particular historical period on the basis of appropriate philosophical interpretation of such categories as matter, motion, space and time, necessary and accidental, possible and real, and other categories, attracted from the classical philosophical heritage and non-classical philosophical teachings.