After these definitions I will show that most of the known results for formal topologies can easily be adapted to this new framework. Building up the usual completion of the formal topology, weight and distance can be canonically extended to give two maps, still a weight and a distance, defined on the space of formal points. Some elementary analysis is then developed. It allows the coexistence of classical logic and basic orthologic in the same system. The same technique is hopefully useful to face the problem of the control of context in general and hence to obtain translations also for other logics.
Basic logic is the fixed common core containing the operational rules.
The main result is that the cut-elimination procedure for the whole system respects all the underlying logics. The basic idea is to formalize the selection of formulae to be connected by any operational rule. Questo sistema di deduzione naturale riesce a simulare il parallellismo e la sincronizzazione di processi. It also clarifies the relationship between the calculi of extensible and non-extensible objects presented in the recent literature. The basic idea is that fomal opens act as generalized tuth values and formal values of formulae.