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A LOGICAL AI APPROACH TO CONTEXT
Computer Science Department
Stanford, CA 94305
1996 Feb 6, 12:09 p.m.
Logical AI develops computer programs that represent what they
know about the world primarily by logical formulas and decide what
to do primarily by logical reasoning—including nonmonotonic logical
reasoning. It is convenient to use logical sentences and terms whose
meaning depends on context. The reasons for this are similar to what
causes human language to use context dependent meanings. This note
gives elements of some of the formalisms to which we have been led.
Fuller treatments are in [McC93], [Guh91] and [MB94] and the refer-
ences cited in the Web page [Buv95]. The ﬁrst main idea is to make
contexts ﬁrst class objects in the logic and use the formula
to assert that the proposition p is true in the context c. A second
idea is to formalize how propositions true in one context transform
when they are moved to diﬀerent but related contexts. An ability to
transcend the outermost context is needed to give computer programs
the ability to reason about the totality of all they have thought about
so far [McC96].
As requested by Johan van Benthem, this is a brief introduction to the logical
formalism for context being explored by John McCarthy and Saˇsa Buvaˇc at
Stanford University. It is motivated by the need to use contexts as ﬁrst order
objects for artiﬁcial intelligence. I hope the description is suitable for com-
parison with other approaches to context that often have other motivations.
2 Features of the Formalism
Here are some features of our formalizations.
what a context is is like asking what a group element is. See section 4
for more on this.
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
ist(c, q). This generalizes natural deduction.
the limitations explicitly within the formalism.
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.
Thus a robot designed in this way is not stuck with the concepts it has
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.
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. Buvaˇc 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.
Here are some applications of the logical theory of contexts.
useful for more complex anaphora. For example, we need to relate
the surgeon’s “Scalpel” to the sentence “Please hand me a number 3
scalpel”. See [Buv96]. These applications require associating contexts
with sentences or parts of sentences.
simple theory of
above(x, y) as the transitive closure of
on(x, y) to an
outer situation calculus context that uses
on(x, y, s) and
above(x, y, s).
A key formula of that paper is
on(x, y, s) ≡
ist(context-of -situation(s), on(x, y))),
which relates the three argument situation calculus predicate
on(x, y, s)
and the two element predicate
on(x, y) of the specialized theory of on
and above. The use of contexts to implement “microtheories” in Cyc
is described in [Guh91]. This allowed people entering knowledge about
some phenomenon, e.g. automobiles, to do it in a limited context, but
leave open the ability to use the knowledge in a larger context.
sibilities. For example, in formulating the missionaries and cannibals
problem a person or program must take a number of common sense
facts into account, but ends up with a 32 state space, because all that
is relevant in this context is the numbers of missionaries, cannibals and
boats on each bank of the river.
the Airforce and the General Electric Company have databases both
of which include prices for the jet engines that the company sells the
Airforce. However, suppose the databases don’t agree on what the
price covers, e.g. spare parts. We use one context cAF for the Air Force
database, another cGE for the GE database, and a third context c0 that
needs to relate information from both. Lifting formulas in the context
true in c0 relate information in the diﬀerent databases to the context
in which reasoning is done, , e.g. they tell about the relation of the
prices listed in cAF and cGE to the inclusion or not of spare parts.
to work together—or plans originally intended to work together but
which have drifted apart in the course of separate development.
4 Desiderata for a Mathematical Logic of Con-text
The simplest approach to a logic of context is to treat
ist(c, p) as a modal
operator with p quantiﬁer free. Saˇsa Buvaˇc and Ian Mason [BBM95] did
this. However, the applications to natural language, to databases and to
formalizing common sense knowledge and reasoning require a lot more. Here
are some desiderata for a formal theory.1
truths(c) is the set of p such that
ist(c, p). In some formalizations it
will be a ﬁrst class object. In any case we can think about it in the
• The simplest possibility for
truths(c) for a particular context c is that
it is an arbitrary set of propositions, i.e. not required to be closed
under some logical operations.
• The second possibility is that
truths(c) is closed under deduction in
some logical system—perhaps the theory of contexts.
truths(c) may be the set of propositions true about some subject mat-
ter. We can assert propositions about this set of proposition without
knowing what sentences are in it.
• Associated with at least some contexts is a domain
domain(c). As with
domain(c) may be an object, presumably in a higher level
context, or it may be only in the metalanguage.
The variety of potential applications of contexts as objects suggests look-
ing at contexts as mathematics looks at group elements. Groups were ﬁrst
identiﬁed as sets of transformations closed under certain operations. How-
ever, it was noticed that the integers with addition as an operation, the
non-zero rationals with multiplication as an operation and many others had
the same algebraic property. This motivated the deﬁnition of abstract group
around the turn of the century. In such a theory, formulas express relations
among contexts would be primary rather than the propositions true in the
contexts. Thus the theory would emphasize
specializes(c1, c2, time) rather
1Just so Johan doesn’t get oﬀ too easily in keeping his promise to make one.
Johan van Benthem asked for the following in soliciting this essay and John
My proposal is the following. I would like to invite the two
Johns to send me a rough outline of their contribution. It would
be good if you could bring out (1) what the notion of context is
and what it does according to you:
in both cases, I think you
want it to achieve ’eﬃciency’ and ’portability’ of information,
(2) what is involved in the dynamics of changing contexts,
perhaps with attendant changes in linguistic formulation (add or
drop variables, etcetera). I would then like to comment on this,
adding some thoughts on possible logical formalizations, empha-
sizing the interplay between what is said in a formula and what
remains implicit in the models where it gets evaluated.
I have rejected the idea of deﬁning what a context is, but I hope I have
given some idea of what they do. The example relating the three argument
on and the two argument on should provide a basis for comments. In the
formulation of the ideas, the ability to combine formulas arising in diﬀerent
contexts has been more important than computational eﬃciency.
[McC93] and [MB94] have additional references. Also Saˇsa Buvaˇc has sev-
eral other papers on context on his Web page http://www-formal.stanford.edu/buvac/.References
[BBM95] Saˇsa Buvaˇc, Vanja Buvaˇc, and Ian A. Mason. Metamathematics
of contexts. Fundamenta Informaticae, 23(3), 1995.
[Buv95] Saˇsa Buvaˇc.
Saˇsa buvaˇc’s web page, 1995.
[Buv96] Saˇsa Buvaˇc. Resolving lexical ambiguity using a formal theory
of context. In Semantic Ambiguity and Underspeciﬁcation. CSLI
Lecture Notes, Center for Studies in Language and Information,
Stanford, CA, 1996.
[Guh91] R. V. Guha. Contexts: A Formalization and Some Applications.
PhD thesis, Stanford University, 1991. Also published as techni-
cal report STAN-CS-91-1399-Thesis, and MCC Technical Report
John McCarthy and Saˇsa Buvaˇc. Formalizing Context (Expanded
Notes). Technical Note STAN-CS-TN-94-13, Stanford University,
[McC79] John McCarthy. First order theories of individual concepts and
propositions. In Donald Michie, editor, Machine Intelligence, vol-
ume 9. Edinburgh University Press, Edinburgh, 1979. Reprinted
[McC90] John McCarthy. Formalizing Common Sense: Papers by John Mc-
Carthy. Ablex Publishing Corporation, 355 Chestnut Street, Nor-
wood, NJ 07648, 1990.
[McC93] John McCarthy. Notes on formalizing context. In IJCAI-93, 1993.
Available on http://www-formal.stanford.edu/jmc/.
[McC96] John McCarthy. Making robots conscious of their mental states.
In Stephen Muggleton, editor, Machine Intelligence 15. Oxford
University Press, 1996.
to appear, available on http://www-
/@steam.stanford.edu:/u/jmc/f95/context.tex: begun 1995 Sep 22, latexed 1996 Feb 6 at 12:09 p.m.