Desiderata for a Mathematical Logic of Context

 

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.

• truths(c) is the set of p such that ist(c,p). In some formalizations it will be a first class object. In any case we can think about it in the metatheory.
• The simplest possibility for truths(c) for a particular context c is that it is an arbitrary set of propositions, i.e. not required to be closed under some logical operations.
• The second possibility is that truths(c) is closed under deduction in some logical system--perhaps the theory of contexts.
• truths(c) may be the set of propositions true about some subject matter. We can assert propositions about this set of proposition without knowing what sentences are in it.
• Associated with at least some contexts is a domain domain(c). As with truths(c), domain(c) may be an object, presumably in a higher level context, or it may be only in the metalanguage.

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).