FIRST ORDER THEORIES OF INDIVIDUAL CONCEPTS AND

PROPOSITIONS

John McCarthy, Stanford University

2003 Apr 2, 10:50 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.

1 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 constants, variables, functions and expressions— allows ordinary first order theories of necessity, knowledge, belief and wanting without modal operators or quotation marks and without the restrictions on substituting 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 telephone number” is the number itself. Thus if we also have “Mary’s telephone 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 concept and is its meaning in oblique contexts like knowing and wanting, and a denotation which is its meaning in 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 trouble, 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 proposed formalizations. All these formalizations involve modifying the logic used; ours doesn’t modify the logic and is more powerful, because 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 application 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 logical 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 telephone 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 Mike) (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 considered also capitalizing the last letter when the arguments are concepts, but it made the formulas ugly).

2. Mike 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 elephoneMike) is a proposition and not a truth value. Note that we are formalizing knowing 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 Mike) instead of

true Know(P at, T elephoneMike)

when we don’t intend to iterate knowledge further. know is a predicate in the logic, so we cannot apply any knowledge operators to it. We will have

know(pat, T elephone Mike) ≡ true Know(P at, T elephone Mike).(2)

6. We expect that the proposition Know(P at, T elphone Mike) 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.

7. While the sentence “Pat knows Mike” is in common use, it is harder to see how Know(P at, Mike) is to be used and axiomatized. I suspect that new methods will be required to treat knowing a person.

8. 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).

9. 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. Actually, 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 telephone number, is written

Know(Joe, Know(P at, T elephone Mike)) (3)

and asserting it requires writing

true Know(Joe, Know(P at, T elephone Mike)) (4)

while the proposition that Joe knows that Pat knows Mike’s telephone number is written

K(Joe, Know(P at, T elephone Mike) (5)

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. (6)

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)

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)

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)

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

(∀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)

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 concepts have denotations. Exists is extensional in this sense, but the corresponding predicate exists is identically true and therefore dispensable.

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), (16)

etc. The corresponding formulas for Or, Not, Implies, and Equiv are

(∀Q1 Q2)(true(Q1 Or Q2) ≡ true Q1 ∨ true Q2), (17)

(true(Not Q) ≡ ¬true Q), (18)

(true(Q1 Implies Q2) ≡ trueQ1 ⊃ true Q2) (19)

and

(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 expressing equality of the denotations of concepts, we introduce Equal(X, Y) which is the proposition that X and Y denote the same thing if anything. We shall want axioms ^{1}

(∀X)(true Equal(X, X)), (21)

(∀X Y)(true Equal(X, Y) ≡ true Equal(Y, X)) (22)

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 asserted by

true Equal(T elephone Mary, T elephone Mike) (26)

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

true Know(P at, T elephone Mary) (27)

To draw this conclusion we need something like

true K(P at, Equal(T elephone Mary, T elephone Mike)) (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 entirely 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 manyone, functions going from things to certain concepts of them seem useful. 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) (29)

We can then have simultaneously

true Not Knew(Kepler, Number P lanets) (30)

and

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

(We have bravely used Knew instead of Know, because we are not now concerned with formalizing tense.) (31) can be condensed using 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 Number P lanets). (33)

A further condensation can be achieved using Composite2 defined by

(∀N)(Composite2 N = Composite Concept1 denot N), (34)

letting us write

true Knew(Kepler, Composite2 Number P lanets), (35)

which is true even though

true Knew(Kepler, Composite Number 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 order to formalize a sentence like, “Lassie knows the location of all her puppies”. We write this

(∀x)(ispuppy(x, lassie) ⊃ true Knowd(Lassie, Locationd Conceptd x)).(37)

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 tailored 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 Mike (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 concepts as objects in expressing particular facts but impossible to avoid them in stating general principles.

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 Mike). (38)

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

trueK(P at, Equal(T elephone Mike, Concept1 “333−3333′′)),(39)

where K(P, Q) is the proposition that denot(P) knows the proposition Q and Concept1(“333 − 3333′′) is some standard concept of that telephone number.

The two ways of expressing knowledge are somewhat interdefinable, since we can write

(∀P Q)(K(P, Q) = (Q And Know(P, Q))) (40)

and

(∀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

(41) by

(∀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 ordinary 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.

Nec 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 Nec Q or true P oss Q. This can be abbreviated by defining nec Q ≡ true Nec Q and poss Q correspondingly. However, since nec is a predicate in the logic with t and f as values, nec Q cannot be an argument of nec or Nec.

Before we even get to modal logic proper we have a decision to make—shall Not Not Q be considered the same proposition as Q, or is it merely extensionally equivalent? The first is written

(∀Q)(Not Not Q = Q) (44)

and the second

(∀Q)(true Not Not Q ≡ true Q). (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 theorem 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 conjunction 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 concept. 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 = Not Nec Not Q) (47)

and the weaker extensional axiom

(∀Q)(true P oss Q ≡ true Not Nec Not Q). (48)

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

(∀Q)(true Nec Q ⊃ true Q) (49)

and

(∀Q1 Q2)(true Nec Q1∧true Nec(Q1 Implies Q2) ⊃ true Nec Q2)(50)

yielding a system equivalent to von Wright’s T.^{4} S4 is given by adding

(∀Q)(true Nec Q ≡ true Nec Nec Q) (51)

and S5 by adding

(∀Q)(true P oss Q ≡ true Nec P oss Q). (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 NecT, Nec4 and Nec5, related by axioms such as

(∀Q)(true Nec4 Q ⊃ true Nec5 Q), (53)

which would seem plausible if we regard S4 as corresponding to provability 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 Nec Equal(X, X)). (54)

Certain of Carnap’s proposals translate to the stronger relation

(∀X Y)(X = Y ≡ true Nec Equal(X, Y)), (55)

which asserts that two concepts are the same if and only if the equality of what they may denote is necessary.

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(Number 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, (58)

or, using concepts, as

true Equal(Number P lanets, Concept1 9), (59)

and “Necessarily 9=9” is written

nec Equal(Concept1 9, Concept1 9), (60)

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 fiction1 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 sentences 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, (62)

denot T ully = cicero (63)

and

denot Cicero = cicero. (64)

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 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)

and

¬true K(P hilip, Denounced(T ully, Catiline)). (67)

5 From (63), …, (67) we can deduce

(∃P)(true Denounced(P, Catiline) And Not K(P hilip, Denounced(P, Catiline)))(68)

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), (70)

denot Catiline = catiline (71)

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

51995: Quine would also want true Not K(P hilip, Denounced(T ully, Catiline)).

The following sentence attributed to Russell is is discussed by Kaplan: “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 using 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) (75)

and

true Believe(Ralph, Not Issp P 2) (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 concepts 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 someone’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. However, 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 Quantification

As the examples of the previous sections have shown, admitting concepts as objects and introducing standard concept functions makes “quantifying in” rather easy. However, forming propositions and individual 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(Mike, P P) Implies Equal(T elephone P P, T elephone Mike)(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(Mike, P P) Implies Equal(T elephone P P, T elephone Mike))(79)

is the proposition that all Mike’s children have the same telephone number as Mike. Existential propositions are formed similarly to universal ones, but the function Exist introduced here should not be confused with the function Exists applied to individual concepts introduced 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(Mike, P P)) (80)

is the concept of Mike’s only child. Exists T he(P P, Child(Mike, P P)) is the proposition that the described child exists. We have

true Exists T he(P P, Child(Mike, P P))
≡ true Exist(P P, Child(Mike, P P)

And All(P P 1, Child(Mike, 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(Mike, P P))
= Exist(P P, Child(Mike, P P)

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

This is part of general problem of when two logically equivalent concepts 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 axiomatizing the modal operators, knowledge, belief and tense. However, for axiomatizing quantification, we also need a function α such that

π′ = 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 π_{0} be the actual world. Let true(P, π) mean that the proposition P is true in the possible world π. Then

(∀P)(true P ≡ true(P, π_{0})). (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

\begin{matrix}(∀πV P x)(true(P, α(V, x, π)) ∧ (∀y)(true(P, α(V, y, π)) ⊃ y = x) \\ ⊃ denotes(T he(V, P), x, π)) \end{matrix} (87)

and

(∀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:

\begin{matrix}(∀π Q1 Q2 V)(absent(V, Q1) ∧ true(All(V, Q1ImpliesQ2), π) \\ ⊃ true(Q1ImpliesAll(V, Q2), π)) \end{matrix} (89)

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 substituting 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 ≠ 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)), (93)

(∀V Q)absent(V, Exist(V, Q)), (94)

(∀V Q)absent(V, T he(V, Q)), (95)

(∀V X)(Subst(V, V, X) = X), (96)

(∀X V)(Subst(X, V, V) = X) (97)

\begin{matrix}(∀X V P Y)(Subst(X, V, Know(P, Y)) \\ = Know(Subst(X, V, P), Subst(X, V, Y))), \end{matrix}  (98)

axioms similar to (98) for other functions,

(∀X V Q)(absent(V, Y) ⊃ Subst(X, V, Y) = Y), (99)

\begin{matrix}(∀X V 1 V 2 Q)(V 1 ≠ V 2 ∧ absent(V 2, X) \\ ⊃ Subst(X, V 1, All(V 2, Q)) = All(V 2, Subst(X, V 1, Q))) \end{matrix} (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 describing 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 Intelligence

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 limitation 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:

1. A computer program that wants to telephone someone must reason 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.

2. A program must often determine that it does not know something or that someone else doesn’t. This has been neglected in the usual formalizations of knowledge, and methods of proving 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.

3. 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 2^{a}3^{b} or by the LISP S-expression (PLUS A B), so long as there are predicates for deciding 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 specifying 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 alphabet 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 “expression” in our language, and we want to regard expressions related by renaming bound variables as denoting the same concept, then the algebra 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 order 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 programs 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 elementary ones. Provability is then axiomatized in the outer logic. His semantic approach uses axiomatizations of the Kripke accessibility relation between possible worlds. It seems to me that our treatment can be used to combine both of Morgan’s methods, and has two further advantages. First, concepts and individuals can be separately quantified. Second, functions from things to concepts of them permit 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 (Montague 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 logics 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.

This article was published as McCarthy, John (1979): “First Order Theories of Individual Concepts and Propositions”, in Michie, Donald (ed.) Machine Intelligence 9, (University of Edinburgh Press, Edinburgh).

11 References

Carnap, Rudolf (1956)). Meaning and Necessity). University of Chicago Press.

Church, Alonzo (1951a). The Need for Abstract Entities in Semantic Analysis, in Contributions to the Analysis and Synthesis of Knowledge). Proceedings of the American Academy of Arts and Sciences, 80). No. 1 (July 1951), 100–112. Reprinted in The Structure of Language). edited by Jerry A. Fodor and Jerrold Katz, Prentice-Hall 1964

Church, Alonzo (1951b). A formulation of the logic of sense and denotation. In: P. Henle (ed.), Essays in honor of Henry Sheffer). pp. 3–24. New York.

Frege, Gottlob (1892). Über Sinn und Bedeutung. Zeitschrift für Philosophie und Philosophische Kritik 100:25–50. Translated by H. Feigl under the title “On Sense and Nominatum” in H. Feigl and W. Sellars (eds.) Readings in Philosophical Analysis). New York 1949. Translated by M. Black under the title “On Sense and Reference” in P. Geach and M. Black, Translations from the Philosophical Writings of Gottlob Frege). Oxford, 1952.

Kaplan, David (1969). Quantifying In, from Words and Objections: Essays on the Work of W.V. Quine). edited by D. Davidson and J. Hintikka, (Dordrecht-Holland: D. Reidel Publishing Co.), pp. 178–214. Reprinted in (Linsky 1971).

Kaplan, David and Montague, Richard (1960). A Paradox Regained, Notre Dame Journal of Formal Logic 1:79–90. Reprinted in (Montague 1974).

Linsky, Leonard, ed.(1971) Reference and Modality). Oxford Readings in Philosophy, Oxford University Press.

McCarthy, J. (1963). Towards a Mathematical Science of Computation, in Proceedings of IFIP Congress 1962). North-Holland Publishing Co., Amsterdam.

McCarthy, J. and Hayes, P.J. (1969). Some Philosophical Problems from the Standpoint of Artificial Intelligence. Machine Intelligence 4). pp. 463–502 (eds Meltzer, B. and Michie, D.). Edinburgh: Edinburgh University Press. (Reprinted in B. L. Webber and N. J. Nilsson (eds.), Readings in Artificial Intelligence, Tioga, 1981, pp. 431–450; also in M. J. Ginsberg (ed.), Readings in Nonmonotonic Reasoning, Morgan Kaufmann, 1987, pp. 26–45; also in this volume, pp. 000–000.)

McCarthy, John (1979): “Ascribing Mental Qualities to Machines” in Philosophical Perspectives in Artificial Intelligence, Ringle, Martin (ed.), Harvester Press, July 1979. Reprinted in (McCarthy 1990).

McCarthy, John (1990): Formalizing Common Sense, Ablex, Norwood, New Jersey

Montague, Richard (1963). Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability, Acta Philosophica Fennica 16:153–167. Reprinted in (Montague 1974).

Montague, Richard (1974). Formal Philosophy). Yale University Press

Morgan, Charles G. (1976). Methods for Automated Theorem Proving in Nonclassical Logics, IEEE Transactions on Computers). vol. C-25, No. 8, August 1976

Quine, W.V.O. (1956). Quantifiers and Propositional Attitudes, Journal of Philosophy). 53. Reprinted in (Linsky 1971).

Quine, W.V.O. (1961). From a Logical Point of View). Harper and Row.

Yap, Chee K. (1977). A Semantical Analysis of Intensional Logics). Research Report, IBM Thomas J. Watson Research Center, Yorktown Heights, New York. RC 6893 (#29538).