We propose the formula to assert that the proposition p holds in context c. It expresses explicitly how the truth of an assertion depends on context. The relation asserts that the context c2 is more general than the context c1.
Formalizing common-sense reasoning needs contexts as objects, in order to match human ability to consider context explicitly. The proposed database of general common-sense knowledge will make assertions in a general context called C0. However, C0 cannot be maximally general, because it will surely involve unstated presuppositions. Indeed we claim that there can be no maximally general context. Every context involves unstated presuppositions, both linguistic and factual.
Sometimes the reasoning system will have to transcend C0, and tools will have to be provided to do this. For example, if Boyle's law of the dependence of the volume of a sample of gas on pressure were built into C0, discovery of its dependence on temperature would have to trigger a process of generalization that might lead to the perfect gas law.
The following ideas about how the formalization might proceed are tentative. Moreover, they appeal to recent logical innovations in the formalization of nonmonotonic reasoning. In particular, there will be nonmonotonic ``inheritance rules'' that allow default inference from to , where c' is either more general or less general than c.
Almost all previous discussion of context has been in connection with natural language, and the present paper relies heavily on examples from natural language. However, I believe the main AI uses of formalized context will not be in connection with communication but in connection with reasoning about the effects of actions directed to achieving goals. It's just that natural language examples come to mind more readily.
As an example of intended usage, consider
Suppose that this sentence is intended to assert that a particular person is in a particular car on a particular occasion, i.e. the sentence is not just being used as a linguistic example but is meant seriously. A corresponding English sentence is ``He's in the car'' where who he is and which car and when is determined by the context in which the sentence is uttered. Suppose, for simplicity, that the sentence is said by one person to another in a situation in which the car is visible to the speaker but not to the hearer and the time at which the the subject is asserted to be in the car is the same time at which the sentence is uttered.
In our formal language c17 has to carry the information about who he is, which car and when.
Now suppose that the same fact is to be conveyed as in example 1, but the context is a certain Stanford Computer Science Department 1980s context. Thus familiarity with cars is presupposed, but no particular person, car or occasion is presupposed. The meanings of certain names is presupposed, however. We can call that context (say) c5. This more general context requires a more explicit proposition; thus, we would have
A yet more general context might not identify a specific John McCarthy, so that even this more explicit sentence would need more information. What would constitute an adequate identification might also be context dependent.
Here are some of the properties formalized contexts might have.
1. In the above example, we will have , i.e. c5 is more general than c17. There will be nonmonotonic rules like
Thus there is nonmonotonic inheritance both up and down in the generality hierarchy.
2. There are functions forming new contexts by specialization. We could have something like
We will have .
3. Besides , we may have , where is a term. The domain in which takes values is defined in some outer context.
4. Some presuppositions of a context are linguistic and some are factual. In the above example, it is a linguistic matter who the names refer to. The properties of people and cars are factual, e.g. it is presumed that people fit into cars.
5. We may want meanings as abstract objects. Thus we might have
6. Contexts are ``rich'' entities not to be fully described. Thus the ``normal English language context'' contains factual assumptions and linguistic conventions that a particular English speaker may not know. Moreover, even assumptions and conventions in a context that may be individually accessible cannot be exhaustively listed. A person or machine may know facts about a context without ``knowing the context''.
7. Contexts should not be confused with the situations of the situation calculus of (McCarthy and Hayes 1969). Propositions about situations can hold in a context. For example, we may have
This can be interpreted as asserting that under the assumptions embodied in context c1, a plan of walking to the car and then driving to the airport would get the robot to the airport starting in situation S0.
8. The context language can be made more like natural language and more extensible if we introduce notions of entering and leaving a context. These will be analogous to the notions of making and discharging assumptions in natural deduction systems, but the notion seems to be more general. Suppose we have . We then write
This enables us to write p instead of . If we subsequently infer q, we can replace it by and leave the context c. Then will itself hold in the outer context in which holds. When a context is entered, there need to be restrictions analogous to those that apply in natural deduction when an assumption is made.
One way in which this notion of entering and leaving contexts is more general than natural deduction is that formulas like and (say) behave differently from and which are their natural deduction analogs. For example, if c1 is associated with the time 5pm and c2 is associated with the time 6pm and p is , then might be used to infer that I left the office between 5pm and 6pm. cannot be used in this way; in fact it is equivalent to .
9. The expression (note the caps) represents the proposition that p holds in c. Since it is a proposition, we can assert .
10. Propositions will be combined by functional analogs of the Boolean operators as discussed in (McCarthy 1979b). Treating propositions involving quantification is necessary, but it is difficult to determine the right formalization.
11. The major goals of research into formalizing context should be to determine the rules that relate contexts to their generalizations and specializations. Many of these rules will involve nonmonotonic reasoning.