Intensions as Such: An Outline

ABS'IRACI'

The formal expression of propositional attitudes, espe­ cially when nested (iterated), is an important problem for AI. An interesting first-order extensional logical system for such expression has been proposed by Creary. In this sys­ tem concepts (and concepts of concepts, etc.) are made explicit. The system includes "concept functions", which are special functions which act on and deliver concepts. We point out a difficulty with these functions. A alterna­ tive system is proposed, in which there is a concept,. forming function corresponding roughly to complex­ concept formers (especially the phrase "the proposition that") implicit in English sentences. The resulting system has a more primitive and natural notional base than Creary's has. We avoid problems with quantification inside propositions which are the objects of propositional attitudes by recasting quantified expressions into variable-free form by means of certain functions ("com binators" ).

INTENSIONS AS SUCH: AN OU1LINE

John A. Barnden Computer Science Department Indiana University Bloomington, Indiana 47 405

concept function is a function which takes a concept of a man and delivers a concept of the man's wife ; another con­ cept function might deliver the proposition that the man's wife is French. Concept functions in { Creary, 1979) are based on similar functions in McCarthy's system (McCarthy, 1979), and are akin to the "characterizing functions" in {Church, 1951, 1973, 1974). We shall claim that the use of concept functions does not allow satisfactory formalisation of nested propositional attitudes {beliefs about beliefs , beliefs about beliefs about beliefs, etc.) . Our system avoids concept functions by introducing a special function which takes some concepts and delivers a com­ plex concept similar to a definite description. Such definite-descriptional concepts include propositions, which are regarded as concepts whose extensions are truth-values (cf. the view of propositions in {Church , 1951, 1973, 1974)).

I INTRODUCTION

In the past few years AI research has included some attacks on the important problem of representing beliefs, concepts, intensions, referential opacity, and the like [see e.g. {Brachman, 1979), {Hofstadter et al, 1980) , {Konolige, 1982), (Maida and Shapiro, 1982), {McCarthy, 1979), {Moore, 1977), {Shapiro, 1979), (Weyrauch, 1977, 1980)]. We propose an alternative to Creary's system as described in {Creary, 1979) (which we shall call "system C"); that system is an extension of a system of {McCarthy, 1979) and is loosely based on the work of Frege { Geach and Black, 1952) . Like system C, our own system keeps to first-order extensional logic, makes concepts explicit, has an infinite hierarchy of orders of concepts {concepts of con­ cepts, concepts of concepts of concepts, etc.r, and takes propositions to be a sort of concept. With Creary, we regard concepts as abstractions from mental entities. We claim, however, that a basic feature in system C - concept functions - is not well conceived. An example of a We shall use the term "denotation'' exclusively for the relationship between expressions in English or in a logical formalism and entities in an interpretation of the system of expressions. This relationship is to be carefully dis­ tinguished from the relationship which holds between a concept and its extension (which is often also called denp­ tation) . We shall say that a concept extends to its exten­ sion. Thus the phrase "the concept of Mary" denotes a c·oncept which extends to the person Mary.

This paper describes work done in the Department of Com­ puter Science at Indiana University. The research was supported by the National Science Foundation under Grant MCS-8102291.

· Although there have been objections to such hierarchies, e.g. Carnap [2].

We do not attempt to specify how to deduce extra beliefs that a cognitive agent holds on the basis of beliefs already ascribed to it. Such deductions are left to arise from particular axioms that some user of ou·r formalism may choose to include in a theory. Similarly, we do not legislate about the connections between knowledge and belief - such matters are again left to the whim of the user.

Our work differs from that of { Konolige 1979) in that we avoid the casting of beliefs as expressions in some language so that statements about- agents' beliefs are for­ malised in a metalanguage which talks about that language. The work differs from that of {Konolige 1979) and that of (Maida and Shapiro, 1982) by the inclusion of the above­ mentioned conceplrconstructing function and concepts of it. It differs also from {Maida and Shapiro, 1982) in not insisting that items in the formalism cannot denote "exten­ sions". The work differs from that of (Moore, 1977) in avoiding a possible-world approach to propositional atti­

tudes.

II INFORMAL CONSIDERATIONS

A. A Problem with Concept Functions

Consider the sentences

( ( la)) Mary is French

and

((lb)) Mike believes that Mary is French.

In system C these could be formalised as french(mary) and believe( mike, French(Mary)). Here mary and french have as their intended interpretations a particular person Mary and the predicate of being French, whereas Mary and French have as their intended interpretations a certain con­ cept of Mary and a "whether-French-concept-of" function whose value on a concept of a person is the propositional concept of that person being French! We call such a func­ tion a concept function. The term French(Mary) denotes the proposition (a type of concept) that Mary is French .

Let us paraphrase sentence ( ( 1 b)) as

((2)) Mike believes the propoeition that Mary is French.

We can view the concept-function technique in system C as taking the phrase "the proposition that Mary is French" as denoting the same proposition that "the whether-French­ concept-of the concept of Ma.ry" denotes. Thus (( lb))/((2)) is equivalent·· to

• ((3)) Mike believes the whether-French-concept-of the concept of Mary.

We have no objection to the equivalence of ((2)) and ((3)). The trouble arises when we go to second-order pro­ positional attitudes, such as in the sentence

• (( 4)) Pat believes that

Mike believes that Mary is French.

We take this to be merely an abbreviation for

• ((5)) Pat believes (the proposition) that

Mike believes the proposition that Mary is French.

The point is that the equivalence of ((2)) and ((3)) does not sanction an equivalence of ( ( 5)) with

f(6)) Pat believes (the proposition) that Mike believes the whether-French-ooncept-of the ooncept of Mary

because the outer "the proposition that" sets up an opaque context. Deducing ((6)) from ((5)) is precisely the same mistake as deducing "Pat believes (the proposition) that Mike is married to the mother of Mary" from "Pat believes (the proposition) that Mike is married to the eld­ est sister of Jim" on the basis of an equivalence between "Mike is married to the mother of Mary" and "Mike is married to the eldest sister of Jim" (this equivalence iteslf resulting validly from an identification of Mary's mother and Jim's eldest sister). Now system C provides a formula which formalises ((6)), namely

It appears that the symbol Mary is to be interpreted as some sort or special, "standard" concept or the person Mary, but the correctness or this appearance is not crucial to the discussion .

·· This equivalence results from ordinary substitution or co­ denoting expressions in a transparent context: noting that the "believe" in both ( ( 2)) and ( ( 3)) is extensional with respect to its object- it is the phrase "the proposition that" which creates an opaque context.

((7)) believe( pat, Believe(Mike, French$(Mary$)))

where Mary$ and French$ are respectively an individual constant and a concept-function sym hoi bearing much the same relationship to Mary and French as these do to mary and french . The machinery of system C does not allow formulae which are any closer to ((5)) than ((6)) is . Unfortunately, ( ( 5)) /( ( 4)) is precisely the sort of sentence we would like to be able to formalise , and which system C was set up to formalise .

B. Making Proposition- Construction Explicit

Our main claim is that the satisfactory formalisation of ((4))/((5)) requires explicit symbols to play the proposition-constructing role of the phrase "(the proposi­ tion) that". Our prst step is to impose a further step of paraphrase : ((2)) is now regarded as a paraphrase of

• ((8)) Mike believes the proposition-constructed-from:

the concept of being French, and the concept of Mary.

Here "the proposition-constructed-from" is like a two­ place function. We then partially paraphrase ( ( 5)) as

• ((9)) Pat believes the propoeition that: Mike believes the proposition-constructed-from: the concept of being French, and

the concept of Mary.

The point is that the function "the proposition constructed from" itself now enters explicitly into Pat's belief, in just the way that the two-place function "the children of" enters explicitly into Pat's belief in

• ((10)) Pat believes that

Mike is a member of the children of Jack and Jill.

Note that the phrase " the proposition-constructed-from" is extensional (transparent) in both its arguments.

C. Definite-Description&! Concepts

Consider now

((11)) Mike believes that the wife of Jim is French.

Using our paraphrases, one subsidiary meaning of this 1s rendered as

((12)} Mike believes the propOBition-constructed-from: the concept of being French, and the concept of X

where X stands for the person who happens to be Jim's wife. (( 12)) is a reading of (( 11 )) in which the characteri­ zation of that person as Jim's wife does not enter into Mike's belief; if Jim's wife is Mary then ((12)) is equivalent to "Mike believes that Mary is French". To capture the more direct meaning of ( ( 11)), in which the wife characterization does enter into Mike 's be lief, we pro­ pose

((13)) Mike believes the propo~~ition-constructed-from:

the concept of being French, and

(the concept-constructed-from:

the concept of the wife function, and

the concept of Jim)

where the parentheses are used to effect correct grouping. We have no general right to deduce from ((13)) that Mike believes the proposition constructed from the concept of being French and the concept of Mary. The concept c referred to by the parenthesized expression in (( 13)) is (in general) a different one from the standard concept of Mary. We call c a "definite-descriptional concept" . The observa­ tion now is that the concept (i .e . the proposition) that Mike believes in ( ( 13)) is also a definite-descriptional concept. We may regard "the proposition-constructed-from" as just a convenient rewriting of "the concept-constructed-from" in certain contexts.

D. Standard Concepts and Absoluteness

For simplicity, we have been ass uming and shall con­ tinue to assume that for every individual, such as Mary, there is a special concept extending to that individual and which is deemed to be the "standard" concept of the indi­ vidual. A fur ther simplifying assumption is that concepts are ab~o/ute, i.e. not rel ative to the cognitive agents which entertain them . Thus, for a given individual x, each cogni­ tive agent has the same standard concept of x, and the function "the concept constructed from" is not parametrized by a particular cognitive agent. It is emphasized that both absoluteness and the postulation of standard concepts could be abandoned from our considera­ tions, at the price of making the presentation of the for­ malism in Section III more difficult.

A most important point is that it is possible for a stan­ dard concept to be a definite-descriptional concept. For instance, the (standard) concept of Mary might actually be the definite-descriptional concept constructed from the con­ cept of the wife function and the concept of Jim. In that case, ((13)) would be equivalent to ((12)) with 'X' replaced by 'Mary'. We do not develop such possibilities in this paper. They are not a result of our particular approach similar things could be done in system C.

Consider now

• ((14)) Mike holds-in-mind the concept of Mary

(as a paraphrase of "Mike thinks about Mary"). We wonder how we would approach the formalisation of (the most direct meaning of)

• ((15)) Pat believes that Mike holds-in-mind the concept of Ma.ry.

We propose that this should be paraphrased as

• ( ( 16)} Pat believes the-concept-constructed-from:

the concept of holds-in-mind,

the concept of Mike, and

(the concept constructed from: the concept of the-concept-of, and the concept of Mary)

(where "the concept of" is short for "the standard concept of" ) . This is exactly parallel to paraphrasing

• ((15A)) Pat believes that

Mike is-married-to the mother of Mary

as

( ( 16A)) Pat believes the-concept-constructed-from: the concept of is-married-to, the concept of Mike, and (the concept constructed from: the concept of the-mother-of, and the concept of Mary).

III FORMALISATION

A. Preliminaries

We replace all functions and predicates by individual constants wife, french , $, etc. which we call " functions" or " functional individuals" for convenience . We introduce the genuine function a (read "apply"), so that a( wife, jim) and a(plus, 11 2) replace wife(jim) and plus( 1, 2) respectively.~ The atomic formul ae french(john) and meet(mike, pat) are replaced by the term8 a( french, john)and a( meet, mike, pat). Such terms denote truth values, and the truth values TRUE and FALSE are required to be in any interpretation of a theory in our logic.

For simplicity, we abbreviate an expression a(E 0 , E 1 , ··· E 0 ) to E 0 [E 1 , ··· E 0 ]. Thus , for instance , a( wife, jim) is abbreviated to wife[jim] .

B. Construction of Complex Concepts

In any intended interpretation of a theory in our for­ malism we assume there exists the concept-forming func­ tional individuals 'the-( standard- )-concept-of' and 'the­ concept-constructed-from '. These are denoted by the individual constants $ and ¢ respectively. We shall often abbreviate $(x] (which is already an abbreviation) to $x when x is a single sym hoi. Thus a($, wife)and a($, $) can be abbreviated to $wife and $$ . (The $ is inspired by Creary's notation. The ability to "apply" "functions" to themselves is commented on later.)

· Technically, we need different functions a 0 of different arities n. Also, our logic should be sorted, but for brevity we do not discuss this matter.

Let us look at some examples. We take hold-in­ mind(mike, $(mary]] to denote TRUE iff Mike thinks about Mary (using the standard concept of her). If this term and the term equal[mary, wife(jim]J denote TRUE then we can deduce that hold-in-mind(mike·, $(wife(jim]]J denotes TRUE. On the other hand

hold-in-mind(mike, ¢($(wife], $(jim]]J

denotes TRUE iff Mike thinks about {the wife of Jim}As· Here we use the AS subscript (meaning "as such") to indi­ cate that the woman concerned is conceived of as the wife of Jim by Mike . This term replaces system C's hold-in­ mind(mike, "Wife(Jim)) where "Wife denotes a concept function and Jim denotes a (standard) concept of Jim. The term

hold-in-mind[mike, ¢($wife, ¢($father, $jim]]]

denotes TRUE iff Mike thinks about {the wife of the father of Jim }AS . (Note here that we are using the $x abbrevia­ tion for $(x] .) The term

hold-in-mind[mike, ¢($[wife], $[father(jim]]]]

denotes TRUE iff Mike thinks about {the wife of u}AS' where u is the father of Jim. The formula is equivalent to the one derived by replacing father(jim] by bill, if indeed we have that equal[bill, father(jim]J denotes TRUE.

We have been using hold-in-mind for illustrative pur­ poses. To get back to the question of belief, the term

believe(pat, ¢($is-married-to, $mike, $jane]]

denotes TRUE iff Pat believes that Mike is married to Jane (where the standard concepts of being married to, Mike and Jane are used in that belief). The term

believe[pat, ¢($is-married-to, $mike, ¢($mother, $jim]]J

denotes TRUE iii Pat believes that Mike is married to the mother of Jim, where now that lady is characterized in Pat's belief as the mother of Jim.

The term

believe[mike, ¢($equal, ¢($tel-num-of, $jim], $1234]]

denotes TRUE iff Mike believes that Jim's telephone number is 1234. The denotation of this term is indepen­ dent of that of

believe[mike, ¢($equal, $1234, $1234]].

C. Iterated Propositional Attitudes

Notice the " lifting" transformation applied to

is-married-to[mike, mother(jim]J

to get the outer ¢term in

believe(pa.t, ¢($is-married-to, $mike, ¢($mother, $jim]]].

This transformation proceeds as follows: (i) remove all square- bracket abbreviation, by replacing every express ion of form f( ... J by a(f, ... ); (ii) replace each individual con­ stant x by $x; and (iii) replace each expression of form a( ·.. ) by ¢[ ... ]. It is therefore in accord with previous examples to lift((l)) to form ((2)) in:

• ((1)) hold-in-mind(mike, $(mary]]

( (2)) believe[pa.t, ¢($hold-in-mind, $mike, ¢($$, $mary]]]

((2)) denotes TRUE iii Pat believes the proposition that Mike thinks about Mary, where Mike is supposed by Pat to use her standard co nee pt. Similarly, the term

• ( ( 3)) believe[pa.t,

¢($hold-in-mind, $mike,

¢[$¢, ¢[)~, $wife], ¢($$, $jim]J]J

denotes TRUE iii Pat believes that Mike thinks about the wife of Jim , where Mike is supposed by Pat to use the characterization of her as the wife of Jim. ( ( 3)) is derived from hold-in-mind( mike, ¢($wife, $jim]] by "lifting". ( ( 3)) can be read as "Pat believes that: {Mike holds-in­ mind the-concept-constructed-from the concept of wife-of and the co nee pt of Jim} AS".

The reader is invited to try writing a formula for a third-order propositional attitude ("George believes that Pat believes that Mike believes that .... "). We get terms whose length increases exponentially with number of lifting steps. However, if we introduce for each constant symbol x the abbreviations ¢:Sx, ¢2:$x, ¢3:$x, etc., where: ¢:$x stands for the lifted version of S[x] (i.e. ¢($$, $x]); ~:$x stands for the lifted version of ¢:$x (i.e. ¢($¢, ¢($$, $$], ¢($$, $x]]) ; and so on, then we get a approximately linear increase in the size of terms.

We introduce a special functional individual "ext-of" which delivers the extension of a concept. So, for instance, ext-of($(mary]J denotes Mary, and ext-of($(wife(jim]J] and ext-of(¢($wife, $jim]] both denote Jim's wife. The term ext-of[¢($french, $mary]] denotes whatever truth-value french(mary] denotes. We assume that there are axiom schemata*

equal[ext-of[$[x]], x]

and

· To say tbat tbese terms are axiom schemata is to say tbat Cor an interpretation of a tbeory in our logical system to be a model it must make tbese terms denote TRUE. Also, we assume that equal is always interpreted in a special way and that suitable axioms for it are provided.

D. Connectives

The first part of our treatment of connectives is to cast them as "functional" constants whose only distinctive quality is that in any interpretation they must denote cer­ tain truth-value-returning " functions". For instance, the term and[hit[john, judy], hit[judy, john]] denotes TRUE iff both hit[john, judy]and hit[judy, john] denote TRUE. We assume the existence of "truth-functional" constants not, and, or and implies. We allow and and or to have an arbitrary num her of arguments .

The treatment of the inclusion of connectives in pro­ positions which are the objects of propositional attitudes now falls out immediately from our preceding considera­ tions . For example, we could have

believe[ mike,

¢[$and, ¢{$hit, $john, $judy], ¢{$hit, $judy, $johnllJ

by lifting the and[ ... ] term above.

E. Quantifiers

To express quantification inside propositions which are themselves the objects of propositional attitudes, (Creary, 1979) uses special variable-binding operators, e .g. Exist. (These operators replace a somewhat cumbersome tech­ nique used by (McCarthy, 1979) .) In contrast, our treat­ ment avoids variables in the first place, and therefore avoids the need for special variable-binding operators. The avoidance arises from two hypothetical moves: 1) casting quantification temporarily in terms of functional abstraction (>.-abstraction), and 2) casting functional abstraction into variable-free form by means of special "functions" called com binators. The first step is essentially a technique used in (Church, 1951) . We regard

(Vx)(implies[man[x], mortal[x]])

as an abbreviation of

all[>.x. implies[man[x], mortal[xJ]J.

Here the >.-expression is a term denoting a "functional" individual. The all is a truth-valued "function" which, like a connective, must be given a special meaning in any interpretation. That is, all[f') for any one-place truth-valued "function" r denotes TRUE iff for each X in the domain of interpretation the "application" of f to x yields TRUE. Similarly,

c::lx)(or[man[x], woman[x]]} is abbreviated to some[>.x. or[man[x], woman[x]]].

The second step takes >.-expressions to be rewntmgs of terms which use com binators (certain "functional" con­ stants) but contain no variables. The com binators we use are I (the identity "function") and denumerably many symbols 1rfJ where superscript fJ is a non-empty slashseparated list of (possibly empty) comma-separated lists of non-negative integers (e .g. ( 1,2)/2/(1,4,6); or (0,2) ; or()). If fJ has k components then 'll'fJ denotes , so to speak, >.­ abstraction of a certain mode: each element of the list fJ corresponds to a bound variable in a >.-abstraction, and every integer within that element specifies the positions in which the bound variable is used in the body of the >.­ abstraction .

In detail, if fJ has one element then (rrfl[E 0 , ... , ED]) [x] equals F 0 [F 1 , ... , FD] where: Fi is E;[x] if i isin the list of integers which is the only element of {J, and is otherwise just Ei . As an example, the expression >.x. implies[human[x], mortal[x]] is considered merely a re-notation of rr< 1 · 2 l[implies, human, mortal].

If fJ has k > 1 elements, then (rrfl[E 0 , ... , ED]) [x 1 , ··· , xk] {J' equals (rr [F 0 , ... , FD]) [a 2 , ... , a,] where: Fi is Ei[a 1 ] if i is in the first list of integers in {J , and is otherwise just Ei ; and {J' is the tail of fl. For example , the expression >.x.>.y. or[p[x], q[x,y)] is a re-notation of 7r (1,2)/(2l[or, p, 7r (1)/(2l[q, I, I)].

The point now is that we can easily "lift" expressions involving quantification up by one intensional level , because we have replaced all conventional quantificational constructs by ordinary applicative expressions. The only new factor is that we have the new " functions" rrfJ, some and all. For instance , we can obtain

believe[mike, ¢{$all, ¢[$rr< 1 · 2 l, $implies, $man, $morta1Jll,

where the belief deploys the standard concepts of universal quantification, of implication, of being human, of being mortal , and of" >.-abstraction in mode {J" .·

The form of "function" abstraction we have introdu ced via the 1rl combinators allows the definition of self-contradictory "functions" , e.g. 1r (I) [tr, 1r (I) [equal, 1r (O,ll[I, I], 0), 1, 0) where lf[true, x, y) equals x and lf[falee, x, y) equals y. We get a con­ tradiction from applying this "function" to itself. Something must be done to avoid this classic problem, and several ap­ proaches are being considered. One is to use a type hierarchy similar to that in (Church, 1951, 1973, 1974). However, the one which is receiving closest attention is based on the relativization device used in Zermelo-Fraenkel set theory to avoid the Russell paradox (which is akin to the self-application paradox just men­ tioned) . That is, any set definition must be made relative to some set A, as in {x EA 1 ... } Let us regard "functions" as tuple-sets in the domain or interpretation, so that in particular a 1r '[ ... ] term denotes a set of tuples. (We depart from standard set theory in allowing cycles in the set..membership relation, so that self-application of "functions" is possible .) Instead of re­ garding 1r '[ ... ] terms as re-notations of >.-expressions, we now re­ gard them as re-notations of set..definition expressions of form {(x,y, ... z) E PI <some condition on x,y, ... z> }. Here P denotes some set or tuples. The replacement of >.-expressions by such relativized set..definition expressions serves the purpose of avoiding inconsistent function definitions. A tr 1 [ ... ) term must now contain a tuple-set,.valued term (possibly a "function" term) P as its zeroth argument. In the following, we will ignore these modifications.

F. Some Examples

We may formalise "Pat knows Mike's telephone number" by

which is a rewriting of

(We might also have used a formalisation in terms of correct belief.) We could formalise "Pat wonders what Mike 's telephone number is" (adapted from (Maida and Shapiro, 1982)) by

curious(pat, ¢($tel-num, $mike]].

(where we assume curious denotes the " predicate" of wondering what the extension of some concept is.) Simi­ larly, "Mary wonders whether John is taller than Bill" ( adapted from (Maida and Shapiro 1982)) could be formal­ ised by

curious(mary, ¢($taller, $john, $bill]].

In (Creary, 1979) Creary gives three formul ae corresponding to different readings of "Pat believes Mike believes Jim's wife IS French". We have three corresponding terms, which, however , do not mean exactly the same thing. The differences arise from the formulation in terms of ¢rather than concept functions .

The first term is

where ¢:$x is an abbreviation for ¢($$,$x) and where the oute r ¢( ... ] is the lifted form of

believe(mike, ¢{$french, ¢{$wife, $jim]]].

• ((1)) means that Pat believes that Mike believes that Jim's wife is French, where Pat's belief involves the characteriza­ tion of (Mike's) characterization: wife of Jim. ((1)) expresses what is arguably the most natural reading of the above sentence.

The secorid term is

(transformed into combinator form) whert; now the con­ cept c of Jim's wife which is imputed to Mike is ~erely ~orne concept of her ( c could be the denotation of ¢{$mother, $sally] for instance), and the concept c' used by Pat is simply 6ome concept of that concept c (e .g. the one denoted by $(¢($mother, $sally]]) .

The third term is

where the oute r ¢[ ... ] is the lifted form of

transformed first into the combinator form :

• ( ( 3)) means that Pat believes that Mike believes that Jim 's wife is French, where part of Pat's belief is the imputation to Mike of 6ome concept of the person characterized in Pat's belief as Jim 's wife .

Consider the Creary formul a ((7)) and sentences ((5 , 6)) of Section II.A again; formula((7)) form alisesse ntence ((6)), which we contrasted with ((5)) . Our formul ations of ((5)) and ((6)) are :

• ( ( 4)) believe(pat, ¢{$believe, $mike, ¢($¢, ¢($$, $french], ¢($$, $mary]]]]

where the outer ¢( ... ] is the lifted form of

believe(mike, ($(french])[$(mary]]]]

so that we are paraphrasing the phrase "the whether­ French-concept-of" in ( ( 6)) of !LA as "the concept of being French" . We are also now postulating that the terms $french and ¢{$$,$french] denote " functions ".

We observe that if we adopt the axiom schema

then ( ( 5)) reduces to

believe(pat, $believe($ mike, (($$)($french])[($$) ($mary]]]]

which is analogous to the formula ((7)) of II .A. Our reconstruction of terms closely analogous to those of sys­ tem C by means of the above axiom schema serves the purpose of explicating the notional basis of that system .

IV CONCLUDING REMARKS

It is hoped that the discussion of Section II, which achieves more formal explication later in the paper , shows that the ¢ function is at least as natural as concept func­ tions are. In particular, the use of ¢results in formulae closer in form to natural-language sentences than the formulae of system C do, in that¢ often corresponds to the phrase "the proposition that" and therefore allows that phrase itself to be an explicit component of a conceptual characterization at a higher intensional level.

A number of semantic-network researchers [e.g. {Brachman, 1979), (Shapiro, 1979), (Maida and Shapiro, 1982)] have realized the importance of basing their formalisms on intensions (concepts, descriptions, proposi­ tions , ... ) rather than on "extensions". Now, semantic network formalisms are very similar to (and often just re­ expressions of) logical systems. If our analyses in this paper are correct, therefore, we must suspect that existing semantic network formalisms need some modification, in order to bring in entities corresponding to $ and ¢.

It is important to note that, although we have called ¢ a concept constructor, the concepts which are constructed are not necessarily to be regarded as expressions or some­ thing similar to expressions. (In fact, we make no claim about the particular psychological nature of constructed concepts. All we say is that constructed concepts are tran­ sparently determined by the concepts they are constructed from.) It would be possible in a theory in our formalism to have two ¢[ .·. ] terms denoting the same concept even though the arguments in the terms denote different con· cepts. For instance, it would be possible to have ¢{$odd, $x] denoting the same concept as ¢{$not, ¢($even, $x]] for every numerical term x. It is up to individual users of the formalism to decide whether such identities are appropriate. Similar points apply to Creary's formalism, but it is not clear whether he intends to allow such identities.

There is no claim that the system as sketched here is the last word on the issues dealt with. For instance, there is the question of whether it is plausible that beliefs should be couched in terms of the concepts denoted by the $li"P symbols (though we feel that our treatment of quantification is at least as plausible as those of Creary and McCarthy). Note that a person's entertaining the concept denoted by a ¢{$n·P, ··· ] term does l!'lt imply that the per­ son actually entertains the concepts denoted by the argu­ ments to the ¢{ ·.· ]term, in particular the $l!"P term.

A fuller paper which discusses in more detail the gen­ eral and technical issues raised by our considerations in this report is in preparation.

ACKNOWLEDGMENTS

I should like to thank Mitchell Wand for useful com­ ments on an initial draft of the paper. I am indebted to Robert Filman, who was the original director of the grant which supported this research.