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NOTES ON FORMALIZING CONTEXT ∗
Computer Science Department
Stanford, CA 94305
These notes discuss formalizing contexts as ﬁrst class objects. The
basic relation is
ist(c, p). It asserts that the proposition p is true in the
context c. The most important formulas relate the propositions true in
diﬀerent contexts. Introducing contexts as formal objects will permit ax-
iomatizations in limited contexts to be expanded to transcend the original
limitations. This seems necessary to provide AI programs using logic with
certain capabilities that human fact representation and human reasoning
possess. Fully implementing transcendence seems to require further ex-
tensions to mathematical logic, i.e. beyond the nonmonotonic inference
methods ﬁrst invented in AI and now studied as a new domain of logic.
Various notations are considered, but these notes are tentative in not
proposing a single language with all the desired capabilities.
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
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.
∗This work was partly supported by DARPA contract NAG2-703.
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 4 and an example is given.
A second goal is to treat the context associated with a particular circum-
stance, e.g. the context of a conversation in which terms have particular mean-
ings 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 tran-
scend 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(cid:48), ist(c, p)). The regress is
inﬁnite, but we will show that it is harmless.
The main formulas are sentences of the form
which are to be taken as assertions that the proposition p is true in the context
c, itself asserted in an outer context
c(cid:48). (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.
ist(context-of (“Sherlock Holmes stories”),
“Holmes is a detective”)
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-of (“Sherlock Holmes stories”), Holmes’s mother’s maiden name does
not have a value. We also have
ist(context-of (“U.S. legal history”),
“Holmes is a Supreme Court Justice”).
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, Stanf ord)) is the assertion that John McCarthy is at Stan-
ford University in a context in which it is given that jmc stands for the author
of this paper and that Stanf ord 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, Stanf ord) may mean physical presence at Stanford at a certain
instant. Programs based on the theory should use the appropriate meaning
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
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:
value(context-of (“Sherlock Holmes stories”),
“number of Holmes’s wives”) = 0
value(context-of (“U.S. legal history”),
“number of Holmes’s wives”) = 1.
We can consider setof -
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
2 Relations among Contexts
There are many useful relations among contexts and also context valued func-
tions. Here are some.
time(t, c)is a context related to c in which the time is special- ized to have the value t. We may have the relation
time(t, c), at(jmc, Stanf ord))
≡ ist(c, at-time(t, at(jmc, Stanf ord))).
time(t, p) is the assertion that the proposition p holds at time t. We call
this a lifting relation. It is convenient to write at-time(t, f
oo(x, y, z)) rather than
oo(x, y, z, t), because this lets us drop t in certain contexts. Many expressions
are also better represented using modiﬁers expressed by functions rather than by
using predicates and functions with many arguments. Actions give immediate
examples, e.g. slowly(on-f
oot(go)) rather than go(on-f oot, slowly).
Instead of using the function specialize-time, it may be convenient to use a
predicate specializes-time and an axiom
time(t, c1, c2) ∧
ist(c2, at-time(t, p)).
This would permit diﬀerent contexts c1 all of which specialize c2 to a particular
There are also relations concerned with specializing places and with special-
izing speakers and hearers. Such relations permit lifting sentences containing
pronouns to contexts not presuming speciﬁc places and persons.
assuming(p, c)is another context like c in which p is assumed, where “assumed” is taken in the natural
deduction sense of section 3.
specializes(c1, c2)when c2 involves no more assumptions than c1 and every proposition meaningful in c1 is translatable into one meaningful in c2. We have nonmonotonic relations
specializes(c1, c2) ∧ ¬
ab1(p, c1, c2) ∧
ist(c1, p) ⊃
specializes(c1, c2) ∧ ¬
ab2(p, c1, c2) ∧
ist(c2, p) ⊃
This gives nonmonotonic inheritance of ist in both from the subcontext to the
supercontext and vice versa. More useful is the case when the sentences must
change when lifted. See below for an example.
the situation in which the conversation took place. References to persons and
objects are decontextualized in the report, and sentences like those given above
can be used to express their relations.
Suppose at time 331, the value of this signal is 0. We can write this
ist(cwire117(331), signal = 0).
Suppose the meaning of the signal is that the door of the microwave oven is
open or closed according to whether the signal on wire117 is 0 or 1. We can
then write the lifting relation
ist(cwire117(t), signal = 0) ≡ door-
The idea is that we can introduce contexts associated with particular parts of a
circuit or other system, each with its special language, and lift sentences from
this context to sentences meaningful for the system as a whole.
3 Entering and Leaving Contexts
Suppose we have the sentence
ist(c, p). We can then enter the context c and infer
the sentence p. We can regard
ist(c, p) as analogous to c ⊃ p, and the operation
of entering c as analogous to assuming c in a system of natural deduction as
invented by Gentzen and described in many logic texts. Indeed a context is a
generalization of a collection of assumptions, but there are important diﬀerences.
For example, contexts contain linguistic assumptions as well as declarative and
a context may correspond to an inﬁnite and only partially known collection
of assumptions. Moreover, because relations among contexts are expressed as
sentences in the language,
ist(c, p) allows inferences within the language that
could only be done at the meta-level of the usual natural deduction systems.
There are various ways of handling the reasoning step of entering a context.
The way most analogous to the usual natural deduction systems is to have an
operation enter c. Having done this, one could then write any p for which
one already had
ist(c, p). However, it seems more convenient in an interactive
theorem proving to use the style of Jussi Ketonen’s EKL interactive theorem
prover [Ketonen and Weening, 1984].
In the style of that system, if one had
ist(c, p), one could immediately write p, and the system would keep track of
the dependence on c. To avoid ambiguity as to where an occurrence of ist( , p)
came from, one might have to refer to a line number in the derivation. Having
obtained p by entering c and then inferring some sentence q, one can leave c
In natural deduction, this would be called discharging the
Human natural language risks ambiguity by not always specifying such as-
sumptions, relying on the hearer or reader to guess what contexts makes sense.
The hearer employs a principle of charity and chooses an interpretation that
assumes the speaker is making sense. In AI usage we probably don’t usually
want computers to make assertions that depend on principles of charity for their
Another application of entering a context has to do with quantiﬁers.
involves a distinguished predicate
present(c, exp), where exp names an object.
If we have
present(c, x) ⊃ P (x)),
then when we enter c, then a special inference rule associated with the predicate
Likewise if we have shown
within the context c, we can infer
present(c, x) ∧ P (x)).
We could get similar eﬀects by associating a domain (call it
each context c.
I’m presently doubtful that the reasoning we will want our programs to do
on their own will correspond closely to using an interactive theorem prover.
Therefore, it isn’t clear whether the above ideas for implementing entering and
leaving contexts will be what we want.
Sentences of the form
ist(c, p) can themselves be true in contexts, e.g. we
ist(c0, ist(c1, p)). In this draft, we will ignore the fact that if we want
to stay in ﬁrst order logic, we should reify assertions and write something like
ist(c0, Ist(c1, p)), where Ist(c, p) is a term rather than a wﬀ. We plan to ﬁx
this up in some way later, either by introducing terms like Ist(c, p) or by using
a modiﬁed logic. Actually the same problem arises for p itself; the occurrence
of p in
ist(c, p) might have to be syntactically distinct from the occurence of p
standing by itself.
4 Rules for Lifting
Consider a context above-theory, which expresses a static theory of the blocks
world predicates on and above. In reasoning about the predicates themselves it
is convenient not to make them depend on situations or on a time parameter.
However, we need to lift the results of above-theory to outer contexts that do
involve situations or times.
To describe above-theory, we may write informally
on(x, y) ⊃
above(x, y) ∧
above(y, z) ⊃
which stands for
on(x, y) ⊃
We want to apply above-theory in a context c in which on and above have
a third argument denoting a situation. In the following formulas, we put the
context in which the formula is true to the left followed by a colon. c0 denotes
an outer context in which formulas not otherwise qualiﬁed are true. The next
section has more about c0. Suppose that in context c we have
on(x, y, s) ≡
ist(c1(s), on(x, y))),
above(x, y, s) ≡
ist(c1(s), above(x, y))),
thus associating a context
c1(s) with each situation s. We also need
ist(c, (∀p s)(
which abbreviates to
ist(above-theory, p) ⊃
and asserts that the facts of above-theory all hold in the contexts associated with
situations. Mike Genesereth points out that this necessarily involves quantifying
into an ist. Now suppose we have the speciﬁc fact
ist(c, on(A, B, S0))
asserting that block A is on block B in a speciﬁc situation S0, and we want to
ist(c, above(A, B, S0)). We proceed as follows.
First use (5) to get
ist(c1(S0), on(A, B)).
Now we enter
c1(S0) and get
From (4) and (8) we conclude
on(x, y) ⊃
from which entering
on(x, y) ⊃
(11) and (13) give
holding in context
c1(S0). We can now either continue reasoning in
c1(S0) and get
ist(c1(S0), above(A, B))
and using (6)
above(A, B, S0)
ist(c, above(A, B, S0)).
In this derivation we used a function giving a context
c1(s) depending on the
situation parameter s. Contexts depending on parameters will surely present
problems requiring more study.
Besides that, the careful reader of the derivation will wonder what system of
logic permits the manipulations involved, especially the substitution of sentences
for variables followed by the immediate use of the results of the substitution.
There are various systems that can be used, e.g. quasi-quotation as used in the
Lisp or KIF, use of back-quotes, or the notation of [Buvac and Mason, 1993] or
the ideas of [McCarthy, 1979b], but all have disadvantages. At present we are
more attached to the derivation than to any speciﬁc logical system and consider
preferable a system in which the above derivation is preserved with as little
change as possible.
As a further example, consider rules for lifting statements like those of section
1 to one in which we can express statements about Justice Holmes’s opinion of
the Sherlock Holmes stories.
5 Transcending Contexts
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
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 travellers, who had been
experiencing a pull towards the earth would suddenly experience a pull towards
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 [McCarthy and Hayes, 1969],
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
ist(c0, p), and the
program can therefore infer
ist(c0, p) from p, thus transcending the context c0.
Performing this operation again should give us a new outer context, call it c−1.
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 on and above. 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 c−1 the whole
set of sentences true in c0 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 asumptions to one that does not. It also provides
a way of formalizing some of the human ability to make assertions about one’s
The usefulness of transcendence will depend on there being a suitable col-
lection 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 true(p∗) ≡ p, 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.
6 Relative Decontextualization
Quine  uses a notion of “eternal sentence”, essentially one that doesn’t
depend on context. This seems a doubtful idea and perhaps incompatible with
some of Quine’s other ideas, because there isn’t any language in which eternal
sentences could be expressed that doesn’t involve contexts of some sort. We
want to modify Quine’s idea into something we can use.
The usefulness of eternal sentences comes from the fact that ordinary speech
or writing involves many contexts, some of which, like pronoun reference, are
valid only for parts of sentences. Consider, “Yes, John McCarthy is at Stanford
University, but he’s not at Stanford today”. The phrase “at Stanford” is used
in two senses in the same sentence. If the information is to be put (say) in a
book to be read years later by people who don’t know McCarthy or Stanford,
then the information has to be decontextualized to the extent of replacing some
of the phrases by less contextual ones.
The way we propose to do the work of “eternal sentences” is called relative
decontextualization. The idea is that when several contexts occur in a discussion,
there is a common context above all of them into which all terms and predi-
cates can be lifted. Sentences in this context are “relatively eternal”, but more
thinking or adaptation to people or programs with diﬀerent presuppositions may
result in this context being transcended.
7 Mental States as Outer Contexts
A person’s state of mind cannot be adequately regarded as the set of propositions
that he believes—at least not if we regard the propositions as sentences that
he would give as answers to questions. For example, as I write this I believe
that George Bush is the President of the United States, and if I were entering
information in a database, I might write
president(U.S.A) = George.Bush.
However, my state of mind includes, besides the asertion itself, my reasons for
believing it, e.g. he has been referred to as President in today’s news, and I
regard his death or incapacitation in such a short interval as improbable. The
idea of a TMS or reason maintenance system is to keep track of the pedigrees
of all the sentences in the database and keep this information in an auxiliary
database, usually not in the form of sentences.
Our proposal is to use a database consisting entirely of outer sentences where
the pedigree of an inner sentence is an auxiliary parameter of a kind of modal
operator surrounding the sentence. Thus we might have the outer sentence
believe(president(U.S.A.) = George.Bush, because . . .),
where the dots represent the reasons for believing that Bush is President.
The use of formalized contexts provides a convenient way of realizing this
idea. In an outer context, the sentence with reasons is asserted. However, once
the system has committed itself to reasoning with the proposition that Bush is
President, it enters an inner context with the simpler assertion
president(U.S.A.) = GeorgeBush.
If the system then uses the assertion that Bush is President to reach a fur-
ther conclusion, then when it leaves the inner context, this conclusion needs to
acquire a suitable pedigree.
Consider a belief revision system that revises a database of beliefs solely as
a function of the new belief being introduced and the old beliefs in the system.
Such systems seem inadequate even to take into account the information used
by TMS’s to revise beliefs. However, it might turn out that such a system used
on the outer beliefs might be adequate, because the consequent revision of inner
beliefs would take reasons into account.
8 Short Term Applications
We see the use of formalized contexts as one of the essential tools for reaching
human level intelligence by logic based methods. However, we see formalized
contexts as having shorter term applications.
• Guha has put contexts into Cyc, largely in the form of microtheories. The
above − theory example is a microtheory. See [Guha, 1991] for some of
• Suppose the Air Force and General Electric Co.
each have databases
that include prices of jet engines and associated equipment. The items
overlap in that jet engines that General Electric sells to the Air Force are
included. Suppose further that the databases are not entirely compatible
because the prices are based on diﬀerent assumptions about spare parts
and warranty conditions. Now suppose that the databases are to be used
together by a program that must check whether the Air Force database is
up-to-date on General Electric prices.
Our idea is that corresponding to each database is a context, e.g. context-
GE-engine-prices and context-AF-engine-prices. The program, however,
must work with a context we may call context-GE-AF-engine-prices. Its
language allows statements with auxiliary information about what is in-
cluded in the price of an item. Suitable lifting rules allow translating the
sentences of the two other databases into this more comprehensive context.
ing abbreviations, by specializing the subject matter (e.g. to U.S. legal
history), by making assumptions and by specializing to the context of a
conversation. These are all specializations of one kind or another. Get-
ting a new context by transcending an old context, e.g. by dropping the
assumption of a gravitational ﬁeld, gives rise to a whole new class of ways
of getting new contexts.
These are too many ways of getting new contexts to be treated separately.
context in the ways we propose may be useful in studying the semantics
of natural language. Natural language exhibits the striking phenomenon
that context may vary on a very small scale; several contexts may occur
in a single sentence.
Consider the context of an operation in which the surgeon says, “Scalpel”.
In context, this may be equivalent to the sentence, “Please give me the
number 3 scalpel”.
ist(c, p)can be considered a modal operator dependent on c applied to p. This was explored in [Shoham, 1991].
useful for AI systems using logic for representation. This is likely to be an
approximate theory in the sense described in [McCarthy, 1979a]. That is,
the term “context” will appear in useful axioms and other sentences but
will not have a deﬁnition involving “if and only if”.
The development of these ideas has beneﬁtted from discussions with Saˇsa Buvaˇc,
Tom Costello, Mike Genesereth, Fausto Giunchiglia and R. V. Guha. Guha
wrote his thesis [Guha, 1991] while this article was going through many versions
as the ideas developed, and the mutual inﬂuences cannot be speciﬁed.
[Buvac and Mason, 1993] Saˇsa Buvaˇc and Ian A. Mason.: “Propositional
logic of context”, in Proceedings of the Eleventh National Conference on
Artiﬁcial Intelligence, 1993. To appear.
[Guha, 1991] Guha, R. V.: Contexts: A Formalization and Some Applica-
tions, Stanford PhD Thesis, 1991.
[Ketonen and Weening, 1984] Ketonen, Jussi and Joseph S. Weening:
EKL—An Interactive Proof Checker: User’s Reference Manual, Com-
puter Science Department, Stanford University, Stanford, California,
[McCarthy and Hayes, 1969] McCarthy, John and P.J. Hayes:
Philosophical Problems from the Standpoint of Artiﬁcial Intelligence”,
in D. Michie (ed), Machine Intelligence 4, American Elsevier, New York,
NY, 1969. Reprinted in [McCarthy, 1990].
[McCarthy, 1979a] McCarthy, John (1979a):
“Ascribing Mental Quali-
ties to Machines” in Philosophical Perspectives in Artiﬁcial Intelli-
gence, Ringle, Martin (ed.), Harvester Press, July 1979. Reprinted in
[McCarthy, 1979b] McCarthy, John (1979b): “First Order Theories of In-
dividual Concepts and Propositions”, in Michie, Donald (ed.) Machine
Intelligence 9, (University of Edinburgh Press, Edinburgh). Reprinted
in [McCarthy, 1990]..
[McCarthy, 1987] McCarthy, John (1987): “Generality in Artiﬁcial Intelli-
gence”, Communications of the ACM. Vol. 30, No. 12, pp. 1030-1035.
Also in ACM Turing Award Lectures, The First Twenty Years, ACM
Press, 1987. Reprinted in [McCarthy, 1990]..
[McCarthy, 1990] McCarthy, John: Formalizing Common Sense, Ablex, Nor-
wood, New Jersey, 1990.
[Quine, 1969] Quine, W. V. O.: “Propositional Objects”, in Ontological Rel-
ativity and other Essays, Columbia University Press, New York, 1969.
[Shoham, 1991] Shoham, Y.: “Varieties of Context”, in Artiﬁcial Intelligence
and Mathematical Theories of Computation, Academic Press, San Diego
and London, 1991.
/@sail.stanford.edu:/u/ftp/pub/jmc/context-2.tex: created 1991 winter, latexed 1996 Mar 24 at 10:32 a.m.