Dialectic of the combination of systems

Given two axiomatic-deductive systems S1 and S2, each is an island unto itself, each exists in the world of its own rules, the symbols of which are meaningful only relative to these rules and to the system as a whole. Thus S1 and S2 are neither mutually consistent nor non-consistent - and we can combine them into a new system in a free way.  To speak of consistency we must first identify symbols of both systems, we must somehow postulate a common language.  And we need a notion of contradiction. But all this only makes sense relative to a third system S3 which acts as an arbiter.  If S1 and S2 deduce contradictory things regarding a symbol A, how can we formalize this in S3 ? 

By introducing a symbol for S1 and S2 inside the combined system S1+S2, qualifying symbols representing 'from the perspective of S1' and 'from the perspective of S2'.  The introduction of subscripts or symbols for indexing by S1 and S2 is a projection of the system inside itself, a reflection-into-self.

But then S1 and S2 must be qualified by the arbiter system S3. Why is the perspective of S1 relating to A what it is ? (And the same for S2).  S1 and S2 share context S3 and the quality of S1 depends on that of S2 and vice-versa.  S1 and S2 have become parts of a whole, for-themselves so far as they are for-another. Communication is the outcome of contradiction.