The simplest approach to a logic of context is to treat ist(c,p) as a modal operator with p quantifier free. Sasa Buvac and Ian Mason [BBM95] did this. However, the applications to natural language, to databases and to formalizing common sense knowledge and reasoning require a lot more. Here are some desiderata for a formal theory.
The variety of potential applications of contexts as objects suggests looking at contexts as mathematics looks at group elements. Groups were first identified as sets of transformations closed under certain operations. However, it was noticed that the integers with addition as an operation, the non-zero rationals with multiplication as an operation and many others had the same algebraic property. This motivated the definition of abstract group around the turn of the century. In such a theory, formulas express relations among contexts would be primary rather than the propositions true in the contexts. Thus the theory would emphasize specializes(c1,c2,time) rather than ist(c,p).