Introduction

These notes contain some of the reasoning behind the proposals of [McCarthy, 1987] to introduce contexts as formal objects. The present proposals are incomplete and tentative. In particular the formulas are not what we will eventually want, and I will feel free to use formulas in discussions of different applications that aren't always compatible with each other. [While I dithered, R.V. Guha wrote his dissertation.]

Our object is to introduce contexts as abstract mathematical entities with properties useful in artificial intelligence. Our attitude is therefore a computer science or engineering attitude. If one takes a psychological or philosophical attitude, one can examine the phenomenon of contextual dependence of an utterance or a belief. However, it seems to me unlikely that this study will result in a unique conclusion about what context is. Instead, as is usual in AI, various notions will be found useful.

One major AI goal of this formalization is to allow simple axioms for common sense phenomena, e.g. axioms for static blocks world situations, to be lifted to contexts involving fewer assumptions, e.g. to contexts in which situations change. This is necessary if the axioms are to be included in general common sense databases that can be used by any programs needing to know about the phenomenon covered but which may be concerned with other mattters as well. Rules for lifting are described in section and an example is given.

A second goal is to treat the context associated with a particular circumstance, e.g. the context of a conversation in which terms have particular meanings that they wouldn't have in the language in general.

The most ambitious goal is to make AI systems which are never permanently stuck with the concepts they use at a given time because they can always transcend the context they are in--if they are smart enough or are told how to do so. To this end, formulas $ist(c,p)$ are always considered as themselves asserted within a context, i.e. we have something like $ist(c',ist(c,p))$. The regress is infinite, but we will show that it is harmless.

The main formulas are sentences of the form

$$c': \quad\quad\quad\quad ist(c,p),$$ (1)

which are to be taken as assertions that the proposition $p$ is true in the context $c$, itself asserted in an outer context $c'$. (I have adopted Guha's [Guha, 1991] notation rather than that of [McCarthy, 1987], because he built his into Cyc, and it was easy for me to change mine. For now, propositions may be identified with sentences in English or in various logical languages, but we may later take them in the sense of [McCarthy, 1979b] as abstractions with possibly different identity conditions. We will use both logical sentences and English sentences in the examples, according to whichever is more convenient.

Contexts are abstract objects. We don't offer a definition, but we will offer some examples. Some contexts will be rich objects, like situations in situation calculus. For example, the context associated with a conversation is rich; we cannot list all the common assumptions of the participants. Thus we don't purport to describe such contexts completely; we only say something about them. On the other hand, the contexts associated with certain microtheories are poor and can be completely described.

Here are some examples.

$$\begin{eqnarray*} \begin{array}{l} c0:\quad ist(context\hbox{-}of(\hbox{\lq\lq Sherlo... ...ies''}),\\ \ \ \ \ \hbox{\lq\lq Holmes is a detective''}) \end{array}\end{eqnarray*}$$

asserts that it is true in the context of the Sherlock Holmes stories that Holmes is a detective. We use English quotations here, because the formal notation is still undecided. Here $c0$ is considered to be an outer context. In the context $context\hbox{-}of(\hbox{\lq\lq Sherlock Holmes stories''})$, Holmes's mother's maiden name does not have a value. We also have

$$\begin{eqnarray*} \begin{array}{l} c0: \quad\quad ist(context\hbox{-}of(\hbox{\lq\lq ... ... \ \ \ \hbox{\lq\lq Holmes is a Supreme Court Justice''}). \end{array}\end{eqnarray*}$$

Since the outer context is taken to be the same as above, we will omit it in subsequent formulas until it becomes relevant again. In this context, Holmes's mother's maiden name has a value, namely Jackson, and it would still have that value even if no-one today knew it.

$ist(c1,at(jmc, Stanford))$ is the assertion that John McCarthy is at Stanford University in a context in which it is given that $jmc$ stands for the author of this paper and that $Stanford$ stands for Stanford University. The context $c1$ may be one in which the symbol $at$ is taken in the sense of being regularly at a place, rather than meaning momentarily at the place. In another context $c2$, $at(jmc,Stanford)$ may mean physical presence at Stanford at a certain instant. Programs based on the theory should use the appropriate meaning automatically.

Besides the sentence $ist(c,p)$, we also want the term $value(c,term)$ where $term$ is a term. For example, we may need $value(c,time)$, when $c$ is a context that has a time, e.g. a context usable for making assertions about a particular situation. The interpretation of $value(c,term)$ involves a problem that doesn't arise with $ist(c,p)$. Namely, the space in which terms take values may itself be context dependent. However, many applications will not require this generality and will allow the domain of terms to be regarded as fixed.

Here's another example of the value of a term depending on context:

$$\begin{array}{l} c0: \quad\quad value(context\hbox{-}of(\hbox{\lq\lq S... ...s stories''}), \\ \ \ \ \ \hbox{\lq\lq number of Holmes's wives''}) = 0 \end{array}$$

whereas

$$\begin{array}{l} c0: \quad\quad value(context\hbox{-}of(\hbox{\lq\lq U... ... history''}), \\ \ \ \ \ \hbox{\lq\lq number of Holmes's wives''}) = 1. \end{array}$$

We can consider $setof\hbox{-}wives(Holmes)$ as a term for which the set of possible values depends on context. In the case of the Supreme Court justice, the set consists of real women, whereas in the Sherlock Holmes case, it consists of fictitious women.