Features of the Formalism

Here are some features of our formalizations.

1. 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.
2. 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

   

3. 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.
4. Reasoning and communicating in context permits taking only limited phenomena into account. Treating contexts as objects permits stating the limitations explicitly within the formalism.
5. Statements about contexts are themselves in contexts.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.