Human intelligence involves an ability that no-one has yet undertaken to put in computer programs--namely the ability to transcend the context of one's beliefs.
That objects fall would be expected to be as thoroughly built into human mental structure as any belief could be. Nevertheless, long before space travel became possible, the possibility of weightlessness was contemplated. It wasn't easy, and Jules Verne got it wrong when he thought that there would be a turn-over point on the way to the moon when the travelers, who had been experiencing a pull towards the earth would suddenly experience a pull towards the moon.
In fact, this ability is required for something less than full intelligence. We need it to be able to comprehend someone else's discovery even if we can't make the discovery ourselves. To use the terminology of , it is needed for the epistemological part of intelligence, leaving aside the heuristic.
We want to regard the system as being at any time within an implicit outer context; we have used c0 in this paper. Thus a sentence p that the program believes without qualification is regarded as equivalent to , and the program can therefore infer from p, thus transcending the context c0. Performing this operation again should give us a new outer context, call it . This process can be continued indefinitely. We might even consider continuing the process transfinitely, for example, in order to have sentences that refer to the process of successive transcendence. However, I have no present use for that.
However, if the only mechanism we had is the one described in the previous paragraph, transcendence would be pointless. The new sentences would just be more elaborate versions of the old. The point of transcendence arises when we want the transcending context to relax or change some assumptions of the old. For example, our language of adjacency of physical objects may implicitly assume a gravitational field, e.g. by having relations of and . We may not have encapsulated these relations in a context. One use of transcendence is to permit relaxing such implicit assumptions.
The formalism might be further extended to provide so that in the whole set of sentences true in is an object .
Transcendence in this formalism is an approach to formalizing something that is done in science and philosophy whenever it is necessary to go from a language that makes certain assumptions to one that does not. It also provides a way of formalizing some of the human ability to make assertions about one's own thoughts.
The usefulness of transcendence will depend on there being a suitable collection of nonmonotonic rules for lifting sentences to the higher level contexts.
As long as we stay within a fixed outer context, it seems that our logic could remain ordinary first order logic. Transcending the outermost context seems to require a changed logic with what Tarski and Montague call reflexion principles. They use them for sentences like , e.g `` `Snow is white.' is true if and only if snow is white.''
The above discussion concerns the epistemology of transcending contexts. The heuristics of transcendence, i.e. when a system should transcend its outer context and how, is entirely an open subject.