Here are some features of our formalizations.

- We offer no
*definition*of context. There are mathematical context structures of different properties, some of which are useful. Asking what a context is is like asking what a group element is. See section 4 for more on this. - Sentences about propositions and contexts are built up from a
formula
*ist*(*c*,*p*) which is to be understood as asserting that the*proposition**p*is true in the*context**c*. When we have entered the context*c*, we can write -
Once a program has inferred a sentence
*q*from*p*, it can*leave*the context*c*and have*ist*(*c*,*q*). This generalizes natural deduction. - Reasoning and communicating in context permits taking only limited phenomena into account. Treating contexts as objects permits stating the limitations explicitly within the formalism.
- Statements about contexts are themselves in contexts.
- There is no universal context. This is a fact of epistemology (both of the physical world and the mathematical world). It is always possible to generalize the concepts one has used up to the present. Attempts at ultimate definitions always fail--and usually in uninteresting ways. Humans and machines must start at middle levels of the conceptual world and both specialize and generalize.
- We can deal with this phenomenon in our formalism by
ensuring that it is always possible to
*transcend*the outermost context used so far. Thus a robot designed in this way is not stuck with the concepts it has been given. - Because of the possibility of transcendence, the use of contexts
as objects is not just a matter of efficiency. Any given set of
sentences including contexts can always be
*flattened*(at the cost of lengthening) to eliminate explicit contexts. However, the resulting flat theory can no longer be transcended within the formalism, because it is not an object that can be referred to as a whole. - There is often a theory associated with a context--the set of sentences true in the context. However, two contexts with the same theory need not be the same, because they may have different relations with other contexts. Not all useful contexts will be closed under logical inference.
- We advocate using
*propositions*as discussed in [McC79] for the objects true in contexts rather than logical or natural language sentences. This has the advantage that the set of propositions true in a context may be finite when the set of sentences that can express these propositions will be infinite. However, our present applications of context would work equally well if sentences were used. Buvac and Mason [BBM95] treat*ist*(*c*,*p*) as a modal logic formula in a propositional theory. - Besides the truth of propositions in contexts, we consider the
value
*value*(*c*,*exp*) of a term*exp*representing an*individual concept*in a context*c*as discussed in [McC79]. This presents problems beyond those presented by propositions, because in general the space of values of individual concepts will depend on some outer context.

Wed Feb 28 22:47:51 PST 1996