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FIRST ORDER THEORIES OF

INDIVIDUAL CONCEPTS AND

PROPOSITIONS

John McCarthy, Stanford University

2000 Oct 31, 10:36 a.m.

Abstract

We discuss first order theories in which individual concepts are

admitted as mathematical objects along with the things that reify

them. This allows very straightforward formalizations of knowledge,

belief, wanting, and necessity in ordinary first order logic without

modal operators. Applications are given in philosophy and in artificial

intelligence. We do not treat general concepts, and we do not present

any full axiomatizations but rather show how various facts can be

expressed.

Introduction

“...it seems that hardly anybody proposes to use different variables for

propositions and for truth-values, or different variables for individuals

and individual concepts.”—(Carnap 1956, p. 113).

Admitting individual concepts as objects—with concept-valued con-

stants, variables, functions and expressions— allows ordinary first or-

der theories of necessity, knowledge, belief and wanting without modal

operators or quotation marks and without the restrictions on substi-

tuting equals for equals that either device makes necessary.

In this paper we will show how various individual concepts and

propositions can be expressed. We are not yet ready to present a full

collection of axioms. Moreover, our purpose is not to explicate what

concepts are in a philosophical sense but rather to develop a language

of concepts for representing facts about knowledge, belief, etc. in the

memory of a computer.

Frege (1892) discussed the need to distinguish direct and indirect

use of words. According to one interpretation of Frege’s ideas, the

meaning of the phrase “Mike’s telephone number” in the sentence “Pat

knows Mike’s telephone number” is the concept of Mike’s telephone

number, whereas its meaning in the sentence “Pat dialed Mike’s tele-

phone number” is the number itself. Thus if we also have “Mary’s tele-

phone number = Mike’s telephone number”, then “Pat dialed Mary’s

telephone number” follows, but “Pat knows Mary’s telephone number

does not.

It was further proposed that a phrase has a sense which is a con-

cept and is its meaning in oblique contexts like knowing and wanting,

and a denotation which is its meaningin direct contexts like dialing.

Denotations are the basis of the semantics of first order logic and

model theory and are well understood, but sense has given more trou-

ble, and the modal treatment of oblique contexts avoids the idea. On

the other hand, logicians such as Carnap (1947 and 1956), Church

(1951) and Montague (1974) see a need for concepts and have pro-

posed formalizations. All these formalizations involve modifying the

logic used; ours doesn’t modify the logic and is more powerful, be-

cause it includes mappings from objects to concepts. Robert Moore’s

forthcoming dissertation also uses concepts in first order logic.

The problem identified by Frege—of suitably limiting the appli-

cation of the substitutitivity of equals for equals—arises in artificial

intelligence as well as in philosophy and linguistics for any system that

must represent information about beliefs, knowledge, desires, or logi-

cal necessity—regardless of whether the representation is declarative

or procedural (as in PLANNER and other AI formalisms).

Our approach involves treating concepts as one kind of object in

an ordinary first order theory. We shall have one term that denotes

Mike’s telephone number and a different term denoting the concept

of Mike’s telephone number instead of having a single term whose

denotation is the number and whose sense is a concept of it. The

relations among concepts and between concepts and other entities are

expressed by formulas of first order logic. Ordinary model theory can

then be used to study what spaces of concepts satisfy various sets of

axioms.

We treat primarily what Carnap calls individual concepts

like

Mike’s telephone number or Pegasus and not general concepts like tele-

phone or unicorn. Extension to general concepts seems feasible, but

individual concepts provide enough food for thought for the present.

This is a preliminary paper in that we don’t give a comprehensive

set of axioms for concepts. Instead we merely translate some English

sentences into our formalism to give an idea of the possibilities.

2 Knowing What and Knowing That

To assert that Pat knows Mike’s telephone number we write

true Know(P at, T elephone M ike)

(1)

with the following conventions:

  1. Parentheses are often omitted for one argument functions and predicates. This purely syntactic convention is not important.

    Another convention is to capitalize the first letter of a constant,

    variable or function name when its value is a concept. (We con-

    sidered also capitalizing the last letter when the arguments are

    concepts, but it made the formulas ugly).

  2. M ike is the concept of Mike; i.e. it is the sense of the expression “Mike”. mike is Mike himself.
  3. T elephone is a function that takes a concept of a person into a concept of his telephone number. We will also use telephone

    which takes the person himself into the telephone number itself.

    We do not propose to identify the function T elephone with the

    general concept of a person’s telephone number.

  4. If P is a person concept and X is another concept, then Know(P, X) is an assertion concept or proposition meaning that P knows the

    value of X. Thus in (1) Know(P at, T elephoneM ike) is a propo-

    sition and not a truth value. Note that we are formalizing know-

    ing what rather than knowing that or knowing how . For AI and

    for other practical purposes, knowing what seems to be the most

    useful notion of the three. In English, knowing what is written

    knowing whether when the “knowand” is a proposition.

  5. It is often convenient to write know(pat, T elephone M ike) in- stead of

    true Know(P at, T elephoneM ike)

    when we don’t intend to iterate knowledge further. know is a

    predicate in the logic, so we cannot apply any knowledge opera-

    tors to it. We will have

    know(pat, T elephone M ike) ≡ true Know(P at, T elephone M ike).(2)6. We expect that the proposition Know(P at, T elphone M ike) will

    be useful accompanied by axioms that allow inferring that Pat

    will use this knowledge under appropriate circumstances, i.e. he

    will dial it or retell it when appropriate. There will also be

    axioms asserting that he will know it after being told it or looking

    it up in the telephone book.

  6. While the sentence “Pat knows Mike” is in common use, it is harder to see how Know(P at, M ike) is to be used and axiom-

    atized.

    I suspect that new methods will be required to treat

    knowing a person.

  7. true Q is the truth value, t or f , of the proposition Q, and we must write true Q in order to assert Q. Later we will consider

    formalisms in which true has a another argument—a situation,

    a story, a possible world, or even a partial possible world (a

    notion we suspect will eventually be found necessary).

  8. The formulas are in a sorted first order logic with functions and equality. Knowledge, necessity, etc. will be discussed without

    extending the logic in any way—solely by the introduction of

    predicate and function symbols subject to suitable axioms. In

    the present informal treatment, we will not be explicit about

    sorts, but we will use different letters for variables of different

    sorts.

    The reader may be nervous about what is meant by concept. He

    will have to remain nervous; no final commitment will be made in this

    paper. The formalism is compatible with many possibilities, and these

    can be compared using the models of their first order theories. Actu-

    ally, this paper isn’t much motivated by the philosophical question of

    what concepts really are. The goal is more to make a formal structure

    that can be used to represent facts about knowledge and belief so that

    a computer program can reason about who has what knowledge in

    order to solve problems. From either the philosophical or the AI point

    of view, however, if (1) is to be reasonable, it must not follow from

    (1) and the fact that Mary’s telephone number is the same as Mike’s,

    that Pat knows Mary’s telephone number.

    The proposition that Joe knows whether Pat knows Mike’s tele-

    phone number, is written

    Know(J oe, Know(P at, T elephone M ike))

    and asserting it requires writing

    true Know(J oe, Know(P at, T elephone M ike))

    while the proposition that Joe knows that Pat knows Mike’s telephone

    number is written

    K(J oe, Know(P at, T elephone M ike)

    where K(P, Q) is the proposition that P knows that Q. English does

    not treat knowing a proposition and knowing an individual concept

    uniformly; knowing an individual concept means knowing its value

    while knowing a proposition means knowing that it has a particular

    value, namely t. There is no reason to impose this infirmity on robots.

    We first consider systems in which corresponding to each concept

    X, there is a thing x of which X is a concept. Then there is a function

    denot such that

    x = denot X.

    (3)

    (4)

    (5)

    (6)

    We call denotX the denotation of the concept X, and (7) asserts that

    the denotation of the concept of P ’s telephone number depends only

    on the denotation of the concept P . The variables in (7) range over

    concepts of persons, and we regard (7) as asserting that T elephone

    is extensional with respect to denot. Note that our denot operates

    on concepts rather than on expressions; a theory of expressions will

    also need a denotation function. From (7) and suitable logical axioms

    follows the existence of a function telephone satisfying

    (∀P )(denot T elephone P = telephone denot P ).

    (8)

    Functions like T elephone are then related to denot by equations like

    (∀P 1 P 2)(denot P 1 = denot P 2 ⊃ denot T elephone P 1 = denot T elephone P 2).(7)Know is extensional with respect to denot in its first argument,

    and this is expressed by

    (∀P 1 P 2)(denot P 1 = denot P 2 ⊃ denot Know(P 1, X) = denot Know(P 2, X)),(9)(∀P 1 P 2 x u)

    (denotes(P 1, x) ∧ denotes(P 2, x) ∧ denotes(T elephone P 1, u)⊃ denotes(T elephone P 2, u))

    (12)but it is Not extensional in its second argument. We can therefore

    define a predicate know(p, X) satisfying

    (∀P X)(true Know(P, X) ≡ know(denot P, X)).

    (10)

    (Note that all these predicates and functions are entirely extensional

    in the underlying logic, and the notion of extensionality presented here

    is relative to denot.)

    The predicate true and the function denot are related by

    (∀Q)(true Q ≡ (denot Q = t))

    (11)

    provided truth values are in the range of denot, and denot could also

    be provided with a (partial) possible world argument.

    When we don’t assume that all concepts have denotations, we use

    a predicate denotes(X, x) instead of a function. The extensionality of

    T elephone would then be written

    We now introduce the function Exists satisfying

    (∀X)(true Exists X ≡ (∃x)denotes(X, x))

    (13)

    Suppose we want to assert that Pegasus is a horse without asserting

    that Pegasus exists. We can do this by introducing the predicate

    Ishorse and writing

    true Ishorse P egasus

    (14)

    which is related to the predicate ishorse by

    (∀X x)(denotes(X, x) ⊃ (ishorse x ≡ true Ishorse X))

    (15)

    In this way, we assert extensionality without assuming that all con-

    cepts have denotations. Exists is extensional in this sense, but the

    corresponding predicate exists is identically true and therefore dis-

    pensable.

    In order to combine concepts propositionally, we need analogs of

    the propositional operators such as ∧, etc. which we will write And,

    etc., write as infixes, and axiomatize by

    (true(Q1 And Q2) ≡ true Q1 ∧ true Q2),

    etc. The corresponding formulas for Or, N ot, Implies, and Equiv are

    (∀Q1 Q2)(true(Q1 Or Q2) ≡ true Q1 ∨ true Q2),

    (true(N ot Q) ≡ ¬true Q),

    (true(Q1 Implies Q2) ≡ trueQ1 ⊃ true Q2)

    (true(Q1 Equiv Q2) ≡ (true Q1 ≡ true Q2)).

    (20)

    The equality symbol “=” is part of the logic so that X = Y asserts

    that X and Y are the same concept. To write propositions express-

    ing equality of the denotations of concepts, we introduce Equal(X, Y )

    which is the proposition that X and Y denote the same thing if any-

    thing. We shall want axioms 1

    (∀X)(true Equal(X, X)),

    (∀X Y )(true Equal(X, Y ) ≡ true Equal(Y, X))

    (16)

    (17)

    (18)

    (19)

    (21)

    (22)

    and

    and

    (∀X Y Z)(true Equal(X, Y )∧true Equal(Y, Z) ⊃ true Equal(X, Z)),(23)making true Equal(X, Y ) an equivalence relation, and

    (∀X Y x)(true Equal(X, Y )∧denotes(X, x)denotes(Y, x)),(24)

    which relates it to equality in the logic.

    We can make the concept of equality essentially symmetric by

    replacing (22) by

    (∀X Y )(Equal(X, Y ) = Equal(Y, X)),

    (25)

    i.e. making the two expressions denote the same concept.

    11995: I should have used an infixed Equal here.

    The statement that Mary has the same telephone as Mike is as-

    serted by

    true Equal(T elephone M ary, T elephone M ike)

    (26)

    and it obviously doesn’t follow from this and (1) that

    true Know(P at, T elephone M ary)

    (27)

    To draw this conclusion we need something like

    true K(P at, Equal(T elephone M ary, T elephone M ike))

    (28)

    and suitable axioms about knowledge.

    If we were to adopt the convention that a proposition appearing

    at the outer level of a sentence is asserted and were to regard the

    denotation-valued function as standing for the sense-valued function

    when it appears as the second argument of Know, we would have a

    notation that resembles ordinary language in handling obliquity en-

    tirely by context. There is no guarantee that general statements could

    be expressed unambiguously without circumlocution; the fact that the

    principles of intensional reasoning haven’t yet been stated is evidence

    against the suitability of ordinary language for stating them.

    3 Functions from Things to Concepts

    of them

    While the relation denotes(X, x) between concepts and things is many-

    one, functions going from things to certain concepts of them seem use-

    ful. Some things such as numbers can be regarded as having standard

    concepts. Suppose that Concept1 n gives a standard concept of the

    number n, so that

    (∀n)(denot Concept1 n = n)

    We can then have simultaneously

    true N ot Knew(Kepler, N umber P lanets)

    (29)

    (30)

    and

    true Knew(Kepler, Composite Concept1 denot N umber P lanets).(31)

    (We have bravely used Knew instead of Know, because we are not

    now concerned with formalizing tense.)

    (31) can be condensed us-

    ing Composite1 which takes a number into the proposition that it is

    composite, i.e.

    (∀n)(Composite1 n = Composite Concept1 n),

    (32)

    getting

    true Knew(Kepler, Composite1 denot N umber P lanets). (33)

    A further condensation can be achieved using Composite2 defined by

    (∀N )(Composite2 N = Composite Concept1 denot N ),

    (34)

    letting us write

    which is true even though

    true Knew(Kepler, Composite2 N umber P lanets),

    (35)

    true Knew(Kepler, Composite N umber P lanets)

    (36)

    is false. (36) is our formal expression of “Kepler knew that the number

    of planets is composite”, while (31), (33), and (35) each expresses a

    proposition that can only be stated awkwardly in English, e.g. as

    “Kepler knew that a certain number is composite, where this number

    (perhaps unbeknownst to Kepler) is the number of planets”.

    We may also want a map from things to concepts of them in or-

    der to formalize a sentence like, “Lassie knows the location of all her

    puppies”. We write this

    Here Conceptd takes a puppy into a dog’s concept of it, and Locationd

    takes a dog’s concept of a puppy into a dog’s concept of its location.

    The axioms satisfied by Knowd, Locationd and Conceptd can be tai-

    lored to our ideas of what dogs know.

    A suitable collection of functions from things to concepts might

    permit a language that omitted some individual concepts like M ike

    (replacing it with Conceptx mike) and wrote many sentences with

    quantifiers over things rather than over concepts. However, it is still

    premature to apply Occam’s razor. It may be possible to avoid con-

    cepts as objects in expressing particular facts but impossible to avoid

    them in stating general principles.

    (∀x)(ispuppy(x, lassie) ⊃ true Knowd(Lassie, Locationd Conceptd x)).(37)4 Relations between Knowing What

    and Knowing That

    As mentioned before, “Pat knows Mike’s telephone number” is written

    true Know(P at, T elephone M ike).

    (38)

    We can write “Pat knows Mike’s telephone number is 333-3333”

    trueK(P at, Equal(T elephone M ike, Concept1 “333−3333(cid:48)(cid:48))),(39)

    where K(P, Q) is the proposition that denot(P ) knows the proposi-

    tion Q and Concept1(“333 − 3333(cid:48)(cid:48)) is some standard concept of that

    telephone number.

    The two ways of expressing knowledge are somewhat interdefin-

    able, since we can write

    (∀P Q)(K(P, Q) = (Q And Know(P, Q)))

    (40)

    and

    (41) by

    (∀P X)(true Know(P, X) ≡ (∃A)(constant A∧true K(P, Equal(X, A)))).(41)Here constant A asserts that A is a constant, i.e. a concept such that

    we are willing to say that P knows X if he knows it equals A. This is

    clear enough for some domains like integers, but it is not obvious how

    to treat knowing a person.

    Using the standard concept function Concept1, we might replace

    (∀P X(true Know(P, X) ≡ (∃a)(true K(P, Equal(X, Concept1 a))))(42)with similar meaning.2

    (41) and (42) express a denotational definition of Know in terms

    of K. A conceptual definition seems to require something like

    (∀P X)(Know(P, X) = Exists X And K(P, Equal(X, Concept2 denot X))),(43)where Concept2 is a suitable function from things to concepts and

    may not be available for all sorts of objects.3

    21995: This idea is used in my Elephant 2000 paper to discuss the notion of a responsive

    answer to a question.

    31995: At present I don’t see why Concept2 needs to be different from Concept1.

    5 Replacing Modal Operators by Modal

    Functions

    Using concepts we can translate the content of modal logic into or-

    dinary logic. We need only replace the modal operators by modal

    functions. The axioms of modal logic then translate into ordinary

    first order axioms.

    In this section we will treat only unquantified

    modal logic. The arguments of the modal functions will not involve

    quantification although quantification occurs in the outer logic.

    N ec Q is the proposition that the proposition Q is necessary, and

    P oss Q is the proposition that it is possible. To assert necessity or

    possibility we must write true N ec Q or true P oss Q. This can be

    abbreviated by defining nec Q ≡ true N ec Q and poss Q correspond-

    ingly. However, since nec is a predicate in the logic with t and f as

    values, nec Q cannot be an argument of nec or N ec.

    Before we even get to modal logic proper we have a decision to

    make—shall N ot N ot Q be considered the same proposition as Q, or

    is it merely extensionally equivalent? The first is written

    (∀Q)(N ot N ot Q = Q)

    and the second

    (∀Q)(true N ot N ot Q ≡ true Q).

    (44)

    (45)

    The second follows from the first by substitution of equals for equals,

    but the converse needn’t hold.

    In Meaning and Necessity, Carnap takes what amounts to the first

    alternative, regarding concepts as L-equivalence classes of expressions.

    This works nicely for discussing necessity, but when he wants to discuss

    knowledge without assuming that everyone knows Fermat’s last theo-

    rem if it is true, he introduces the notion of intensional isomorphism

    and has knowledge operate on the equivalence classes of this relation.

    If we choose the first alternative, then we may go on to identify any

    two propositions that can be transformed into each other by Boolean

    identities. This can be assured by a small collection of propositional

    identities like (44) including associative and distributive laws for con-

    junction and disjunction, De Morgan’s law, and the laws governing

    the propositions T and F . In the second alternative we will want the

    extensional forms of the same laws. When we get to quantification

    a similar choice will arise, but if we choose the first alternative, it

    will be undecideable whether two expressions denote the same con-

    cept. I doubt that considerations of linguistic usage or usefulness in

    AI will unequivocally recommend one alternative, so both will have to

    be studied.

    Actually there are more than two alternatives. Let M be the free

    algebra built up from the “atomic” concepts by the concept forming

    function symbols. If ≡≡ is an equivalence relation on M such that

    (∀X1 X2)((X1 ≡≡ X2) ⊃ (true X1 ≡ true X2)),

    (46)

    then the set of equivalence classes under ≡≡ may be taken as the set

    of concepts.

    Similar possibilities arise in modal logic. We can choose between

    the conceptual identity

    (∀W )(P oss Q = N ot N ec N ot Q)

    and the weaker extensional axiom

    (∀Q)(true P oss Q ≡ true N ot N ec N ot Q).

    We will write the rest of our modal axioms in extensional form.

    We have

    and

    (∀Q)(true N ec Q ⊃ true Q)

    (∀Q1 Q2)(true N ec Q1∧true N ec(Q1 Implies Q2) ⊃ true N ec Q2)(50)yielding a system equivalent to von Wright’s T.4

    S4 is given by adding

    (∀Q)(true N ec Q ≡ true N ec N ec Q)

    and S5 by adding

    (∀Q)(true P oss Q ≡ true N ec P oss Q).

    (47)

    (48)

    (49)

    (51)

    (52)

    4It seems that something to replace necessitation is needed to get T and likewise for

    S4 and S5.

    Actually, there may be no need to commit ourselves to a particular

    modal system. We can simultaneously have the functions N ecT , N ec4

    and N ec5, related by axioms such as

    (∀Q)(true N ec4 Q ⊃ true N ec5 Q),

    (53)

    which would seem plausible if we regard S4 as corresponding to prov-

    ability in some system and S5 as truth in the intended model of the

    system.

    Presumably we shall want to relate necessity and equality by the

    axiom

    (∀X)(true N ec Equal(X, X)).

    Certain of Carnap’s proposals translate to the stronger relation

    (∀X Y )(X = Y ≡ true N ec Equal(X, Y )),

    which asserts that two concepts are the same if and only if the equality

    of what they may denote is necessary.

    (54)

    (55)

    6 More Philosophical Examples—Mostly

    Well Known

    Some sentences that recur as examples in the philosophical literature

    will be expressed in our notation so the treatments can be compared.

    First we have “The number of planets = 9” and “Necessarily 9 =

    9” from which one doesn’t want to deduce “Necessarily the number of

    planets = 9”. This example is discussed by Quine (1961) and (Kaplan

    1969). Consider the sentences

    ¬nec Equal(N umber P lanets, Concept1 9)

    (56)

    and

    nec Equal(Concept1 number planets, Concept1 9)

    (57)

    Both are true. (56) asserts that it is not necessary that the number

    of planets be 9, and (57) asserts that the number of planets, once

    determined, is a number that is necessarily equal to 9. It is a major

    virtue of our formalism that both meanings can be expressed and

    are readily distinguished. Substitutivity of equals holds in the logic

    but causes no trouble, because “The number of planets = 9” may be

    written

    number(planets) = 9,

    or, using concepts, as

    true Equal(N umber P lanets, Concept1 9),

    and “Necessarily 9=9” is written

    nec Equal(Concept1 9, Concept1 9),

    and these don’t yield the unwanted conclusion.

    Ryle used the sentences “Baldwin is a statesman” and “Pickwick

    is a fiction” to illustrate that parallel sentence construction does not

    always give parallel sense. The first can be rendered in four ways,

    namely true Statesman Baldwin or statesman denot Baldwin or

    statesman baldwin or statesman1 Baldwin where the last asserts

    that the concept of Baldwin is one of a statesman. The second can be

    rendered only as as true F iction P ickwick or f iction1 P ickwick.

    Quine (1961) considers illegitimate the sentence

    (∃x)(Philip is unaware that x denounced Catiline)

    (61)

    obtained from “Philip is unaware that Tully denounced Catiline” by

    existential generalization. In the example, we are also supposing the

    truth of “Philip is aware that Cicero denounced Catiline”. These sen-

    tences are related to (perhaps even explicated by) several sentences

    in our system. T ully and Cicero are taken as distinct concepts. The

    person is called tully or cicero in our language, and we have

    tully = cicero,

    denot T ully = cicero

    and

    denot Cicero = cicero.

    We can discuss Philip’s concept of the person Tully by introducing

    a function Concept2(p1, p2) giving for some persons p1 and p2, p1’s

    (58)

    (59)

    (60)

    (62)

    (63)

    (64)

    (∃P )(true Denounced(P, Catiline) And N ot K(P hilip, Denounced(P, Catiline)))(68)concept of p2. Such a function need not be unique or always defined,

    but in the present case, some of our information may be conveniently

    expressed by

    Concept2(philip, tully) = Cicero,

    (65)

    asserting that Philip’s concept of the person Tully is Cicero. The

    basic assumptions of Quine’s example also include

    true K(P hilip, Denounced(Cicero, Catiline))

    (66)

    ¬true K(P hilip, Denounced(T ully, Catiline)).

    (67)

    5 From (63), . . ., (67) we can deduce

    from (67) and others, and

    ¬(∃p)(denounced(p, catiline)

    ¬true K(P hilip, Denounced(Concept2(philip, p), Catiline))),(69)

    using the additional hypotheses

    (∀p)(denounced(p, catiline) ⊃ p = cicero),

    denot Catiline = catiline

    (70)

    (71)

    and

    and

    (∀P 1 P 2)(denot Denounced(P 1, P 2) ≡ denounced(denot P 1, denot P 2)).(72)Presumably (68) is always true, because we can always construct a

    concept whose denotation is Cicero unbeknownst to Philip. The truth

    of (69) depends on Philip’s knowing that someone denounced Catiline

    and on the map Concept2(p1, p2) that gives one person’s concept of

    another. If we refrain from using a silly map that gives something like

    51995: Quine would also want true N ot K(P hilip, Denounced(T ully, Catiline)).

    Denouncer(Catiline) as its value, we can get results that correspond

    to intuition.

    The following sentence attributed to Russell is is discussed by Ka-

    plan: “I thought that your yacht was longer than it is”. We can write

    it

    true Believed(I, Greater(Length Y ouryacht,

    Concept1 denot Length Y ouryacht)),

    (73)

    where we are not analyzing the pronouns or the tense, but are us-

    ing denot to get the actual length of the yacht and Concept1 to get

    back a concept of this true length so as to end up with a proposition

    that the length of the yacht is greater than that number. This looks

    problematical, but if it is consistent, it is probably useful.

    In order to express “Your yacht is longer than Peter thinks it is.”,

    we need the expression Denot(P eter, X) giving a concept of what

    Peter thinks the value of X is. We now write

    longer(youryacht, denot Denot(P eter, Length Y ouryacht)),(74)

    but I am not certain this is a correct translation.

    Quine (1956) discusses an example in which Ralph sees Bernard J.

    Ortcutt skulking about and concludes that he is a spy, and also sees

    him on the beach, but doesn’t recognize him as the same person. The

    facts can be expresed in our formalism by equations

    trueBelieve(Ralph, Isspy P 1)

    and

    true Believe(Ralph, N ot Issp P 2)

    (75)

    (76)

    where P 1 and P 2 are concepts satisfying denotP 1 = ortcutt and

    denotP 2 = ortcutt. P 1 and P 2 are further described by sentences

    relating them to the circumstances under which Ralph formed them.

    We can still consider a simple sentence involving the persons as

    things—write it believespy(ralph, ortcutt), where we define

    (∀p1 p2)(believespy(p1, p2) ≡ true Believe(Concept1 p1, Isspy Concept7 p2))(77)using suitable mappings Concept1 and Concept7 from persons to con-

    cepts of persons. We might also choose to define believespy in such

    a way that it requires true Believe(Concept1 p1, Isspy P ) for several

    concepts P of p2, e.g. the concepts arising from all p1’s encounters

    with p2 or his name. In this case

    believespy(ralph, ortcutt)

    will be false and so would a corresponding

    notbelievespy(ralph, ortcutt)

    . However, the simple-minded predicate believespy, suitably defined,

    may be quite useful for expressing the facts necessary to predict some-

    one’s behavior in simpler circumstances.

    Regarded as an attempt to explicate the sentence “Ralph believes

    Ortcutt is a spy”, the above may be considered rather tenuous. How-

    ever, we are proposing it as a notation for expressing Ralph’s beliefs

    about Ortcutt so that correct conclusions may be drawn about Ralph’s

    future actions. For this it seems to be adequate.

    7 Propositions Expressing Quantifica-

    tion

    As the examples of the previous sections have shown, admitting con-

    cepts as objects and introducing standard concept functions makes

    “quantifying in” rather easy. However, forming propositions and in-

    dividual concepts by quantification requires new ideas and additional

    formalism. We are not very confident of the approach presented here.

    We want to continue describing concepts within first order logic

    with no logical extensions. Therefore, in order to form new concepts

    by quantification and description, we introduce functions All, Exist,

    and T he such that All(V, P ) is (approximately) the proposition that

    “for all values of V , P is true”, Exist(V, P ) is the corresponding

    existential proposition, and T he(V, P ) is the concept of “the V such

    that P ”.

    Since All is to be a function, V and P must be objects in the logic.

    However, V is semantically a variable in the formation of All(V, P ),

    etc., and we will call such objects inner variables so as to distinguish

    them from variables in the logic. We will use V , sometimes with

    subscripts, for a logical variable ranging over inner variables. We also

    need some constant symbols for inner variables (got that?), and we

    will use doubled letters, sometimes with subscripts, for these. XX

    will be used for individual concepts, P P for persons, and QQ for

    propositions.

    The second argument of All and friends is a “proposition with

    variables in it”, but remember that these variables are inner variables

    which are constants in the logic. Got that? We won’t introduce a

    special term for them, but will generally allow concepts to include

    inner variables. Thus concepts now include inner variables like XX

    and P P , and concept forming functions like T elephone and Know

    take as arguments concepts containing internal variables in addition

    to the usual concepts.

    Thus

    Child(M ike, P P ) Implies Equal(T elephone P P, T elephone M ike)(78)is a proposition with the inner variable P P in it to the effect that

    if P P is a child of Mike, then his telephone number is the same as

    Mike’s, and

    All(P P, Child(M ike, P P ) Implies Equal(T elephone P P, T elephone M ike))(79)is the proposition that all Mike’s children have the same telephone

    number as Mike. Existential propositions are formed similarly to uni-

    versal ones, but the function Exist introduced here should not be

    confused with the function Exists applied to individual concepts in-

    troduced earlier.

    In forming individual concepts by the description function T he, it

    doesn’t matter whether the object described exists. Thus

    T he(P P, Child(M ike, P P ))

    (80)

    is the concept of Mike’s only child. Exists T he(P P, Child(M ike, P P ))

    is the proposition that the described child exists. We have

    true Exists T he(P P, Child(M ike, P P ))

    ≡ true Exist(P P, Child(M ike, P P )

    And All(P P 1, Child(M ike, P P 1) Implies Equal(P P, P P 1)))),(81)but we may want the equality of the two propositions, i.e.

    Exists T he(P P, Child(M ike, P P ))

    = Exist(P P, Child(M ike, P P )

    And All(P P 1, Child(M ike, P P 1) Implies Equal(P P, P P 1))).(82)

    This is part of general problem of when two logically equivalent con-

    cepts are to be regarded as the same.

    In order to discuss the truth of propositions and the denotation

    of descriptions, we introduce possible worlds reluctantly and with an

    important difference from the usual treatment. We need them to give

    values to the inner variables, and we can also use them for axioma-

    tizing the modal operators, knowledge, belief and tense. However, for

    axiomatizing quantification, we also need a function α such that

    π(cid:48) = α(V, x, π)

    (83)

    is the possible world that is the same as the world π except that

    the inner variable V has the value x instead of the value it has in

    π. In this respect our possible worlds resemble the state vectors or

    environments of computer science more than the possible worlds of the

    Kripke treatment of modal logic. This Cartesian product structure on

    the space of possible worlds can also be used to treat counterfactual

    conditional sentences. 6

    Let π0 be the actual world. Let true(P, π) mean that the proposi-

    tion P is true in the possible world π. Then

    (∀P )(true P ≡ true(P, π0)).

    (84)

    Let denotes(X, x, π) mean that X denotes x in π, and let denot(X, π)

    mean the denotation of X in π when that is defined.

    The truth condition for All(V, P ) is then given by

    (∀πV P )(true(All(V, P ), π) ≡ (∀x)true(P, α(V, x, π)).

    (85)

    Here V ranges over inner variables, P ranges over propositions, and x

    ranges over things. There seems to be no harm in making the domain

    of x depend on π. Similarly

    (∀πV P )(true(Exist(V, P ), π) ≡ (∃x)true(P, α(V, x, π)).

    (86)

    61995: (McCarthy 1979) treats “Cartesian counterfactuals”.

    The meaning of T he(V, P ) is given by

    (∀πV P x)(true(P, α(V, x, π)) ∧ (∀y)(true(P, α(V, y, π)) ⊃ y = x)

    (87)

    ⊃ denotes(T he(V, P ), x, π))

    (∀π V P )(¬(∃x)(true(P, α(V, x, π)) ⊃ ¬true Exists T he(V, P ))).(88)

    We also have the following syntactic rules governing propositions

    involving quantification:

    (∀π Q1 Q2 V )(absent(V, Q1) ∧ true(All(V, Q1ImpliesQ2), π)

    (89)

    ⊃ true(Q1ImpliesAll(V, Q2), π))

    and

    and

    (∀π V Q X)(true(All(V, Q), π) ⊃ true(Subst(X, V, Q), π)) (90)

    where absent(V, X) means that the variable V is not present in the

    concept X, and Subst(X, V, Y ) is the concept that results from sub-

    stituting the concept X for the variable V in the concept Y . absent

    and Subst are characterized by the following axioms:

    (∀V 1 V 2)(absent(V 1, V 2) ≡ V 1 (cid:54)= V 2),

    (91)

    (∀ V P X)(absent(V, Know(P, X)) ≡ absent(V, P )absent(V, X)),(92)

    axioms similar to (92) for other conceptual functions,

    (∀V Q)absent(V, All(V, Q)),

    (∀V Q)absent(V, Exist(V, Q)),

    (∀V Q)absent(V, T he(V, Q)),

    (∀V X)(Subst(V, V, X) = X),

    (∀X V )(Subst(X, V, V ) = X)

    (∀X V P Y )(Subst(X, V, Know(P, Y ))

    = Know(Subst(X, V, P ), Subst(X, V, Y ))),

    (93)

    (94)

    (95)

    (96)

    (97)

    (98)

    axioms similar to (98) for other functions,

    (∀X V Q)(absent(V, Y ) ⊃ Subst(X, V, Y ) = Y ),

    (99)

    (∀X V 1 V 2 Q)(V 1 (cid:54)= V 2 ∧ absent(V 2, X)

    ⊃ Subst(X, V 1, All(V 2, Q)) = All(V 2, Subst(X, V 1, Q)))

    (100)and corresponding axioms to (100) for Exist and T he.

    Along with these comes an axiom corresponding to α-conversion,

    (∀V 1 V 2 Q)(All(V 1, Q) = All(V 2, Subst(V 2, V 1, Q))).

    (101)

    The functions absent and Subst play a “syntactic” role in describ-

    ing the rules of reasoning and don’t appear in the concepts themselves.

    It seems likely that this is harmless until we want to form concepts of

    the laws of reasoning.

    We used the Greek letter π for possible worlds, because we did not

    want to consider a possible world as a thing and introduce concepts of

    possible worlds. Reasoning about reasoning may require such concepts

    or else a formulation that doesn’t use possible worlds.

    Martin Davis (in conversation) pointed out the advantages of an

    alternate treatment avoiding possible worlds in case there is a single

    domain of individuals each of which has a standard concept. Then we

    can write

    (∀V Q)(true All(V, Q) ≡ (∀x)(true Subst(Concept1x, V, Q)).(102)

    8 Possible Applications to Artificial In-

    telligence

    The foregoing discussion of concepts has been mainly concerned with

    how to translate into a suitable formal language certain sentences of

    ordinary language. The success of the formalization is measured by

    the extent to which the logical consequences of these sentences in the

    formal system agree with our intuitions of what these consequences

    should be. Another goal of the formalization is to develop an idea

    of what concepts really are, but the possible formalizations have not

    been explored enough to draw even tentative conclusions about that.

    For artificial intelligence, the study of concepts has yet a different

    motivation. Our success in making computer programs with general

    intelligence has been extremely limited, and one source of the limita-

    tion is our inability to formalize what the world is like in general. We

    can try to separate the problem of describing the general aspects of

    the world from the problem of using such a description and the facts

    of a situation to discover a strategy for achieving a goal. This is called

    separating the epistemological and the heuristic parts of the artificial

    intelligence problem and is discussed in (McCarthy and Hayes 1969).

    We see the following potential uses for facts about knowledge:

  9. A computer program that wants to telephone someone must rea- son about who knows the number. More generally, it must reason

    about what actions will obtain needed knowledge. Knowledge in

    books and computer files must be treated in a parallel way to

    knowledge held by persons.

  10. A program must often determine that it does not know some- thing or that someone else doesn’t. This has been neglected

    in the usual formalizations of knowledge, and methods of prov-

    ing possibility have been neglected in modal logic. Christopher

    Goad (to be published) has shown how to prove ignorance by

    proving the existence of possible worlds in which the sentence

    to be proved unknown is false. Presumably proving one’s own

    ignorance is a stimulus to looking outside for the information.

    In competitive situations, it may be important to show that a

    certain course of action will leave competitors ignorant.

  11. Prediction of the behavior of others depends on determining what they believe and what they want.

    It seems to me that AI applications will especially benefit from

    first order formalisms of the kind described above. First, many of

    the present problem solvers are based on first order logic. Morgan

    (1976) in discussing theorem proving in modal logic also translates

    modal logic into first order logic. Second, our formalisms leaves the

    syntax and semantics of statements not involving concepts entirely

    unchanged, so that if knowledge or wanting is only a small part of

    a problem, its presence doesn’t affect the formalization of the other

    parts.

    9 Abstract Languages

    The way we have treated concepts in this paper, especially when we

    put variables in them, suggests trying to identify them with terms in

    some language. It seems to me that this can be done provided we use

    a suitable notion of abstract language.

    Ordinarily a language is identified with a set of strings of symbols

    taken from some alphabet. McCarthy (1963) introduces the idea of

    abstract syntax, the idea being that it doesn’t matter whether sums

    are represented a + b or +ab or ab+ or by the integer 2a3b or by the

    LISP S-expression (PLUS A B), so long as there are predicates for de-

    ciding whether an expression is a sum and functions for forming sums

    from summands and functions for extracting the summands from the

    sum. In particular, abstract syntax facilitates defining the semantics

    of programming languages, and proving the properties of interpreters

    and compilers. From that point of view, one can refrain from specify-

    ing any concrete representation of the “expressions” of the language

    and consider it merely a collection of abstract synthetic and analytic

    functions and predicates for forming, discriminating and taking apart

    abstract expressions. However, the languages considered at that time

    always admitted representations as strings of symbols.

    If we consider concepts as a free algebra on basic concepts, then

    we can regard them as strings of symbols on some alphabet if we

    want to, assuming that we don’t object to a non-denumerable alpha-

    bet or infinitely long expressions if we want standard concepts for all

    the real numbers. However, if we want to regard Equal(X, Y ) and

    Equal(Y, X) as the same concept, and hence as the same “expres-

    sion” in our language, and we want to regard expressions related by

    renaming bound variables as denoting the same concept, then the al-

    gebra is no longer free, and regarding concepts as strings of symbols

    becomes awkward even if possible.

    It seems better to accept the notion of abstract language defined by

    the collection of functions and predicates that form, discriminate, and

    extract the parts of its “expressions”. In that case it would seem that

    concepts can be identified with expressions in an abstract language.

    10 Remarks and Acknowledgements

    The treatment given here should be compared with that in (Church

    1951b) and in (Morgan 1976). Church introduces what might be called

    a two-dimensional type structure. One dimension permits higher or-

    der functions and predicates as in the usual higher order logics. The

    second dimension permits concepts of concepts, etc. No examples or

    applications are given. It seems to me that concepts of concepts will

    be eventually required, but this can still be done within first order

    logic.

    Morgan’s motivation is to use first order logic theorem proving pro-

    grams to treat modal logic. He gives two approaches. The syntactic

    approach—which he applies only to systems without quantifiers—uses

    operations like our And to form compound propositions from elemen-

    tary ones. Provability is then axiomatized in the outer logic. His

    semantic approach uses axiomatizations of the Kripke accessibility re-

    lation between possible worlds.

    It seems to me that our treatment

    can be used to combine both of Morgan’s methods, and has two fur-

    ther advantages. First, concepts and individuals can be separately

    quantified. Second, functions from things to concepts of them per-

    mit relations between concepts of things that could not otherwise be

    expressed.

    Although the formalism leads in almost the opposite direction, the

    present paper is much in the spirit of (Carnap 1956). We appeal to

    his ontological tolerance in introducing concepts as objects, and his

    section on intensions for robots expresses just the attitude required

    for artificial intelligence applications.

    We have not yet investigated the matter, but plausible axioms for

    necessity or knowledge expressed in terms of concepts may lead to

    the paradoxes discussed in (Kaplan and Montague 1960) and (Mon-

    tague 1963). Our intuition is that the paradoxes can be avoided by

    restricting the axioms concerning knowledge of facts about knowledge

    and necessity of statements about necessity. The restrictions will be

    somewhat unintuitive as are the restrictions necessary to avoid the

    paradoxes of naive set theory.

    Chee K. Yap (1977) proposes virtual semantics for intensional log-

    ics as a generalization of Carnap’s individual concepts. Apart from

    the fact that Yap does not stay within conventional first order logic,

    we don’t yet know the relation between his work and that described

    here.

    I am indebted to Lewis Creary, Patrick Hayes, Donald Michie,

    Barbara Partee and Peter Suzman for discussion of a draft of this

    paper. Creary in particular has shown the inadequacy of the formalism

    for expressing all readings of the ambiguous sentence “Pat knows that

    Mike knows what Joan last asserted”. There has not been time to

    modify the formalism to fix this inadequacy, but it seems likely that

    concepts of concepts are required for an adequate treatment.

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