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NOTES ON FORMALIZING CONTEXT ∗

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

U.S.A.

(email: jmc@cs.stanford.edu)

Abstract

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.

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.

∗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

c(cid:48) :

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(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.

c0 :

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

c0 :

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

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:

c0 :

value(context-of (“Sherlock Holmes stories”),

“number of Holmes’s wives”) = 0

whereas

c0 :

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

fictitious women.

2 Relations among Contexts

There are many useful relations among contexts and also context valued func-

tions. Here are some.

  1. specialize-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

    c0 :

    ist(specialize-time(t, c), at(jmc, Stanf ord))

    ≡ ist(c, at-time(t, at(jmc, Stanf ord))).

    Here at-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

    f 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

    c0 :

    specializes-time(t, c1, c2)ist(p, c1)

    ist(c2, at-time(t, p)).

    This would permit different contexts c1 all of which specialize c2 to a particular

    time.

    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.

  2. If q is a proposition and c is a context, then 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.

  3. There is a general relation specializes between contexts. We say 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)ist(c2, p).

    and

    specializes(c1, c2) ∧ ¬ab2(p, c1, c2)ist(c2, p)ist(c1, 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.

  4. A major set of relations that need to be expressed are those between the context of a particular conversation and a subsequent written report about

    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.

  5. Consider a wire with a signal on it which may have the value 0 or 1. We can associate a context with this wire that depends on time. Call it cwire117(t).

    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

    ∀ t(ist(cwire117(t), signal = 0) ≡ door-open(t).

    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

    and get ist(c, q).

    In natural deduction, this would be called discharging the

    assumption c.

    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

    interpretation.

    Another application of entering a context has to do with quantifiers.

    It

    involves a distinguished predicate present(c, exp), where exp names an object.

    If we have

    ∀x(present(c, x) ⊃ P (x)),

    then when we enter c, then a special inference rule associated with the predicate

    present gives

    Likewise if we have shown

    within the context c, we can infer

    ∀xP (x).

    ∃xP (x)

    ∃x(present(c, x) ∧ P (x)).

    We could get similar effects by associating a domain (call it domain(c)) with

    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

    can have 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

    (∀xy)(on(x, y)above(x, y))

    (∀xyz)(above(x, y)above(y, z)above(x, z))

    (2)

    (3)

    above-theory :

    which stands for

    c0 :

    ist(above-theory, (∀xy)(on(x, y)above(x, y)))

    (4)

    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

    etc.

    etc.

    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

    c :

    (∀xys)(on(x, y, s)ist(c1(s), on(x, y))),

    and

    c :

    (∀xys)(above(x, y, s)ist(c1(s), above(x, y))),

    thus associating a context c1(s) with each situation s. We also need

    etc.,

    c0 :

    ist(c, (∀p s)(ist(above-theory, p)

    ist(c1(s), p))),

    which abbreviates to

    c :

    (∀p s)(ist(above-theory, p)ist(c1(s), p)),

    (8)

    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

    c0 :

    ist(c, on(A, B, S0))

    (9)

    asserting that block A is on block B in a specific situation S0, and we want to

    derive ist(c, above(A, B, S0)). We proceed as follows.

    First use (5) to get

    c :

    ist(c1(S0), on(A, B)).

    Now we enter c1(S0) and get

    From (4) and (8) we conclude

    c1(S0) :

    on(A, B).

    c :

    ist(c1(S0), (∀xy)(on(x, y)above(x, y))),

    (12)

    from which entering c1(S0) gives

    c1(S0) :

    (∀xy)(on(x, y)above(x, y)).

    (11) and (13) give

    c1(S0) :

    above(A, B),

    holding in context c1(S0). We can now either continue reasoning in c1(S0) or

    leave c1(S0) and get

    (5)

    (6)

    (7)

    (10)

    (11)

    (13)

    (14)

    (15)

    c :

    ist(c1(S0), above(A, B))

    and using (6)

    and finally

    c :

    above(A, B, S0)

    c0 :

    ist(c, above(A, B, S0)).

    (16)

    (17)

    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

    beliefs.

    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

    the moon.

    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 truths(c0).

    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

    own thoughts.

    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 [1969] 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

    the details.

    • 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.

    9 Remarks

  6. We have mentioned various ways of getting new contexts from old ones: by specializing the time or place, by specializing the situation, by mak-

    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.

  7. We have used natural language examples in this article, although natural language is not our main concern. Nevertheless, I hope that formalizing

    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”.

  8. ist(c, p) can be considered a modal operator dependent on c applied to p. This was explored in [Shoham, 1991].
  9. It would be useful to have a formal theory of the natural phenomenon of in human life, as distinct from inventing a form of context

    context, e.g.

    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”.

  10. Useful nonmonotonic rules for lifting will surely be more complex than the examples given.

    Acknowledgments

    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.

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    /@sail.stanford.edu:/u/ftp/pub/jmc/context-2.tex: created 1991 winter, latexed 1996 Mar 24 at 10:32 a.m.