Nonmonotonic reasoning

Elaboration tolerance clearly requires nonmonotonic reasoning. For example, elaborating MCP0 with a requirement for oars adds preconditions to the action of going somewhere in the boat. If oars are not mentioned, nonmonotonic reasoning prevents such additional preconditions.

However, it is still not clear how to formulate the nonmonotonic reasoning so as to obtain tolerance of a wide class of elaborations, such as those of section 7. We propose to use some variant of circumscription, but this still leaves open what is to be circumscribed.

[McC80] discusses several nonmonotonic aspects of the human understanding of MCP0. See 2. They all have a Gricean [Gri89] character. They all concern the non-existence of features of the problem that should have been mentioned, were they supposed to exist. What can be inferred from such contexts includes the Gricean implicatures. Very likely, the formal theory of contexts [McC93] can be used, but that is beyond the scope of this article.

Here are a few nonmonotonic inferences that come up. Each of them seems to present its own formal problems.

• If there were a bridge, it should have been mentioned. A puzzle problem like MCP is given in a context.

• The river can't be forded, and there isn't an extra boat.

• There isn't a requirement for a permit or visa to cross the river.

• There is nothing wrong with the boat. In general, when a tool is mentioned, it is supposed to be usable in the normal way.

• The group of missionaries and cannibals is minimized. Mentioning that one of the missionaries is Jesus Christ will include him in the number otherwise inferrable rather than adding him as an additional missionary. Of course, assertions about him in the general database should have no effect unless he is mentioned in the problem.

• If you keep transporting cannibals to the island (in the variant with an island) they will eventually all be at the island.

These kinds of nonmonotonic reasoning were anticipated in [McC80] and have been accomodated in situation calculus based nonmonotonic formalisms, although the Yale shooting problem and others have forced some of the axiomatizations into unintuitive forms.

The elaborations discussed in this article mainly require the same kinds of nonmonotonic reasoning.