Relating Logic (RL) is a family of logics of relating connectives, just as Modal Logic is a family of logics of modal operators. The basic idea behind a relating connectives is that the logical value of a given complex proposition, with a relating connective as the main connective, is the result of two things: (i) the logical values of the main components of this complex proposition, supplemented with (ii) a valuation of the relation between these components. The latter element is a formal representation of an intensional relation that emerges from the connection of several simpler propositions into one more complex proposition.
Although the simplest model for a relating logic is a pair consisting of a valuation function and a relation, the situation may get more complicated. We can use multi-relating models to represent more types of relations between sentences. In addition, the valuation of relationships between sentences may not be binary but may be many-valued or more subtly graded. Furthermore, we can mix relating semantics with possible world semantics, equipping all worlds with additional valuations of complex sentences. Last, but not least, any semantics may be treated as relating one, when we assume that in case of complex sentences a relationship is represented by a universal relation.
The solution that relating logics offers seems to be quite natural, since when two (or more) propositions in natural language are connected by a connective, some sort of emergence occurs. In fact, the key feature of intensionality is that adding a new connective results in the emergence of a new quality, which itself does not belong to the components of a given complex proposition built by means of the same connective. An additional valuation function determines precisely this quality. Talk of emergence is justified here, because the quality that arises as a result of the connections between the constituent propositions is not reducible to the properties of those propositions. Consequently, if the phenomenon of emergence is to be properly captured, we need additional valuations in a model. The key feature of relating semantics is that it enables us to treat non-logical relations between sentences seriously.