Week 1 & 2 Application

British Journal of Psychology (19^2), 73, 261-216 Printed in Great Britain 267 [111]

Class inclusion and conclusions about Piaget's theory

Leslie Smith

An interpretation of Piaget's account of how a child understands class relationships that is compatible with the account put forward by recent critics is proposed. It is claimed that Piaget's account is distinctive because it is an investigation of logical competence and so seeks to explain a child's understanding of the necessity of certain class relationships and not just their correctness. Central to Piaget's account is his claim that a child who understands inclusion must be able to characterize and systematically interrelate the positive, observational properties of a subclass with its negative, inferential properties. By contrast, recent experimentally based accounts allow inclusion to be understood by means of the positive, observational properties of a subclass alone and so allow the correctness of that relation, but not its necessity, to be comprehended. It is claimed that the understanding of correctness may be developmentally prior to the understanding of necessity and in consequence that Piaget's account is compatible with the accounts of his critics. Compatibility is also guaranteed by the fact that the former states necessary, unlike the latter who state sufficient, conditions of understanding.

It is to Jean Piaget that is owed the tnajor psychological investigation of class relationships. In one of his first papers Piaget realized that the words 'some' and 'all' remain opaque to a child's understanding, as is shown by the inability of young children to deduce whether all, some or none of a collection of flowers are yellow if a portion of them are (Piaget, 1921, p. 450). His mature treatment of the topic is contained in three studies (Piaget, 1952, 1977 a; Inhelder & Piaget, 1964) which support the conclusion of the early study but considerably extend its explanation. Recent experimental studies have not always supported Piaget's conclusion and have tended to suggest that children (might) attain an understanding of class relationships at an age earlier than that allowed by Piaget. The main aim of the present paper is to clarify one central feature in Piaget's account which seems to have been overlooked by others, namely the importance that the account accords to the presence of necessity in a child's understanding. In what follows, the discussion will (1) review Piaget's account of a child's understanding of class relationships, (2) identify and contrast some of the findings of recent experimental studies of this problem and (3) attempt to evaluate the dispute by showing how Piaget's account is importantly different from that of his critics and so compatible with, since distinct from, their accounts.

(1) Piaget's account Piaget's account of a child's understanding of class relationships may be reviewed under three headings since that account is stated to be (a) an account of a child's logical competence with respect to his understanding of class relationships, (b) the members of which are to be identified by their negative, inferential - and not merely positive, observational - properties, (c) which condition is a necessary condition of understanding the necessity of certain class relationships.

(a) Piaget has clearly stated his interest in a child's deductive capacities both in early and in recent writings, for example:

one can think what one wants of formal logic... But what nobody must dispute is that formal thought can be studied as a psychological fact.. .1 call formal reasoning that reasoning which, from one or several propositions, draws a conclusion to which the mind assents with certainty, without thereby having recourse to observation. What is beyond dispute is that such reasoning exists (Piaget, 1922, p. 222 - translation and emphasis mine).

0007-1269/82/020267-10 $02.00/0 © 1982 The British Psychological Society

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Two poitits should be noticed here. Firstly, a distitiction is drawn between a child who understands that a conclusion has to follow from certain premises and a child who understands that a conclusion is true (correct) on the basis of observational evidence. This distinction is a well-founded one since someone who finds out by careful measurement that the interior angles of a Euclidean triangle are equal to 180 degrees does not thereby understand that this same conclusion is a deductive consequence of certain geometrical propositions; again, a young child who sees that his square jigsaw has four corner pieces might fail to see that his jigsaw has to have four such pieces if it is to be a square one. It is, therefore, one thing to make a correct claim about the observed properties of an object; it is quite another to realize that a certain claim is a necessarily true one. Piaget's interest resides primarily in the latter of these and any interest in the former is dependent upon this interest. Secondly, Piaget in early investigations tried to find instances of a child's understanding of necessity in his conscious thought and the verbal reports based upon that thought. In later works, he rejected such an approach on the grounds that a child's logical competence is not restricted to his ability to make formal, deductive inferences. It is beyond the scope of the present paper to discuss Piaget's structuralist theory in detail. It is sufficient to note that his structuralist theory requires the use of different logical systems at different developmental points, corresponding to a child's possession of cognitive structures such as schemes or groupings, and that the point behind this use of logical models of development is to chart the progress in a child's understanding of the necessity of some claim as the outcome of his previous understanding of the correctness of that claim.

The continuity of Piaget's interest in a child's logical competence, or capacity to understand deductive necessity, may now be documented because of its importance to the ensuing discussion. Piaget makes substantially the same point as the claim made, in the quotation just given, in his paper on the reasoning ability of formal operational children (Piaget, 1972, p. 159), since such children are taken by him to be capable of deducing the logical consequences of the hypotheses they form, whatever the truth-value of those hypotheses. In his account of infant development, Piaget points out that his commitment to there being a functional a priori does not require his commitment to there being a structural a priori at the outset of development (Piaget, 1953, p. 3). In his first, mature report on class-inclusion studies, Piaget stated that his aim was to show how a child becomes aware of the necessity that is displayed in his own operations (Piaget, 1952, p. 161). A child eventually understands not just that a whole has more members than one of its parts but that this is necessarily so:

an 'intensive' quantification necessarily intervenes in the relations of inclusion that are inherent in every additive composition. Indeed, from the additive point of view, there are necessarily 'more' elements in a whole than in one of its parts (Piaget & Szeminska, 1941, p. 199—my translation and emphasis).

It is unfortunate that the term 'necessarily' is omitted in the standard translation (Piaget, 1952, p. 162) of this claim with a consequential failure to make explicit in English what the precise modal status of the claim is. In general, a child possesses an operational structure when that structure is closed and the criterion of closure cited by Piaget is the presence of necessity in a child's understanding (Piaget, 1978, p. 124).

(b) Piaget's account of a child's understanding of class relationships makes particular use of the general distinction noted in (a). A child who uses an operational structure, namely a grouping* (Piaget, 1952, p. 181), is one who does understand the necessity of certain

• The third sentence ofthe last paragraph ofthe Introduction (Piaget, 1952, p. 162) might be better translated as: 'in other words, is not the additive composition of classes, which is the sole way to unite the latter in a coherent "grouping" of hierarchical inclusions and so to assign them a precise structure, the psychological counter-part of the additive composition of numbers...' (Piaget & Szeminska, 1941, p. 200).

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class relationships, unlike a pre-operational child who may merely understand the correctness of those relationships. The latter may correctly and consistently allocate individuals to classes or draw pictures of the classes so formed and he may have some conception of the difference between a whole and a part on the basis of an object's graphic or perceptual properties (Piaget, 1952, p. 171; Inhelder & Piaget, 1964, pp. 8, 98). A child who possesses an operational structure has a greater logical competence than his pre-operational counterpart. In his most recent statement of position, Piaget takes there to be two conditions whose presence is required for a child to possess an operational structure that permits an understanding of inclusion-relations:

inclusion is correctly understood, and so quantifiable in the form nA < nB only if two conditiotis are met. . . (1) it is necessary that subclass A (for example daisies) forms a part of a total class B which is resistant and permanent enough to conserve its extension when the subject centres his attention on its subdivisions... (2) it is further necessary to subdivide the whole B into subclasses A and A' which are explicitly characterized by partial negations: A' = the B which are not A and A = the B which are not A' (Piaget, 1977a, p. 88 - translation and emphasis mine; see also Inhelder & Piaget, 1964, p. 106).

The first of these conditions states that a child should be able to compare a subclass with its total class such that decomposition of the latter does not preclude its reformation. The second condition is more complex and states that a child should be able to understand both that A = B-A' and that A' = B-A.

To see the complexity of this condition, consider a case where a child is presented with nine flowers (class B), seven of which are daisies (class A) and two of which are roses (class A'). Here the classes are characterized by their positive, observational properties. But these classes may be differently characterized since either of the subclasses may be identified by subtraction of the other subclass from their total class. Thus the class of daisies {A) is the class of flowers (fi) minus the class of roses {A') and %o A = B-A'\ and the class of roses {A') is the class of flowers {B) minus the class of daisies {A) and so A'= B—A. Thus both class A and class A' can be characterized negatively and inferentially - negatively because members of class A are the members of class B which are not members of class A' and inferentially because the formation of class A is inferred on the basis of the subtraction of class A' from class B. Now it is this latter type of characterization, in terms of the negative and inferential properties of a class, that is appealed to by Piaget as the second condition cited. Subclasses A and A' may be characterized in two alternative ways which are extensionally equivalent, since they pick out the same objects, but which are not psychologically interchangeable, since they are intensionally distinct. Only a child who can systematically relate the extensions of these subclasses with their intensions,* both positive and observational as well as negative and inferential, will understand inclusion-relations.

It is evident that Piaget is attributing to an operational child some understanding of complementation. Piaget is not claiming that the possessor of a grouping will, when he understands that class A is included in class B, take class A' to be the class of any object whatsoever that is not a member of class A; if that were Piaget's claim, then he would be attributing to such a child an understanding that class A together with class A', which is strictly the logical complement of A, exhaust the universe. Certainly, Piaget does use the standard notation of the theory of classes, for example when he claims that subclass A and its complement A' are disjoint and so that A)<A' = <d (Inhelder & Piaget, 1964, p. 7). But what Piaget does claim is that the user of a grouping will have a partial understanding of complementation: given three classes, where A and A' are subclasses of B, a child will understand that these classes are interderivable since, relative to a universe consisting of class B, class A remains when class A' is subtracted from class B and class A' * The last two occurrences of'comprehension' (Piaget, 1952, p. 161) should read 'intension'.

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remains when class A is subtracted from class B. Piaget does not deny that the user of a grouping understands that there are classes other than classes A, A' and B, for example the class of tulips, the class of animals, the class of birds, and so on. What he does deny is that such a child must use his understanding of those other classes for him to characterize classes A and A' by their negative and inferential properties. Indeed, children who do understand that a class B has more members than one of its subclasses. A, will not typically understand that the membership of class B', which is the complement of class B, is smaller than the membership of class A', which is the complement of class A (Inhelder & Piaget, 1964, p. 291; Piaget, 1977a, p. 82).

It may now be apparent why Piaget's claim that his two conditions are conditions for the correctness of a child's understanding of inclusion relations is also a claim that such an understanding will embody an understanding of necessity: it is necessarily the case that class A is formed by the subtraction of class A' from class B, given a universe consisting of class B, and so a child who correctly understands that this is so is also one who understands the necessity of this link. This claim is distinct from the false claim that such a

\ child necessarily understands the link. It is for this reason that Piaget contrasts the position of a child who has an intuitive but non-deductive understanding (Piaget & Szeminska, 1941, p. 215, 'non-deductive' is omitted from the translation; Piaget, 1952, p. 175) with that of the possessor of a grouping.

What is the criterion for attributing to a child an understanding of necessity? Piaget denies that a verbal criterion is reliable and so does not look for a child's use of words such as 'necessary'. Nor does he require that a child should be capable of reflecting on associated concepts. What is decisive is the inclusion-question itself. In its most general form, the question to ask is: are there more members of A or more members of BI (Piaget & Inhelder, 1969, p. 169). To give a correct answer to some instance of this question a child is required to count the members of ^ ; to count the members of B; and to subtract the former result from the latter. A child who correctly concludes that B has more members than A (when this is in fact the case) is one who must be able to identify the members of class A' by their possession of a negative, inferential property and so must be able to characterize members of class A' both by their being roses and by their being that which remains when the daisies are subtracted from the flowers {A' = B — A). Similarly, that child must also be able to see that class B has more members than its other subclass, A', and in consequence must be able to characterize class A by either mode of characterization {A = B—A'). A child's use of words such as 'as well' (Piaget, 1952, p. 176; Inhelder & Piaget, 1964, p. 108) is a confirmation that this is so. The criterion used by Piaget is, then, the selection of a question the correct answer to which requires a child to make the stipulated deductions as a condition of that answer's being the correct answer that it is.

(c) It is clear that the conditions stated by Piaget are necessary conditions. Since this is explicit in the quotation given in {b), no further comment is needed. A necessary condition is not, of course, a sufiScient condition.

In sum, Piaget's account is an account of a child's logical competence and the two conditions that he cites, namely the ability to form stable classes and the abihty to inter- relate a pair of subclasses and a total class in a deductive manner, are necessary conditions of a child's being able to understand the necessity of certain class relationships.

(2) Experimental studies

Discussion in the present section will be limited in aim and will review recent experimental studies of a child's understanding of class relationships. The main aim of the review will be that of showing the distinctiveness of Piaget's account of the understanding of class relationships.

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McGarrigle et al. (1978), whose study has been reviewed by Donaldson (1978), have an interest in performance factors relevant to the understanding of class relationships and are concerned with the perceptual salience of the classes under investigation. Their concern is with a child's ability to interrelate classes on the basis of positive, observational properties alone since a child is presented with an array consisting of four cows such that: class A = the three black (cows), class A' = the one white (cow), class B = the four cows, and class C = the four sleeping (cows). Evidently, class C has the same members as class B and so children can more easily answer a test-question 'Are there more black cows or more sleeping cows?' than a test-question 'Are there more black cows or more cows?'. To see why the former is an easier question, it may be noted that a child can correctly answer the former test-question by quantifying the membership of class A, by quantifying the membership of class C and by subtracting the result of the former count from that of the latter. Thus a correct answer is available to a child who does not understand that some members of class B are not members of class A and so does not understand that some A' = B—A. But if this is so, then such a child may have a correct understanding of the class-relationships in the array without an understanding of the necessity of certain of those relationships. By contrast, the latter test-question is one that is favoured by Piaget. To answer that question correctly, a child must quantify the membership of class A, quantify that of class B and then subtract the result of the former from that of the latter. It is a condition of his being able to do this successfully that he should be able to understand that A' = B—A and so any correct understanding possessed by such a child will be one that displays necessity. McGarrigle et al. (1978) do not specify the status of the conditions that they wish to establish but it is clear that their concern is diflerent from that of Piaget.

An essentially similar study is reported by Meadows (1977), who states that her interest is in both competence and performance and who wishes to replace Piaget's account with one that stresses performance factors. Children are presented with an array consisting of coloured spots such that: class A = the five pink (spots), class A' = the three blue (spots), class B = the eight spots, and class C = the eight round (spots). Classes B and C have the same membership and so children can answer the test-question ' If I took away the blue ones, what would be left, the pink ones or the blue ones or the round ones?' without having to understand that some members of class B are not members of class A. Once again, the use of class C, over and above classes A, A' and B, which has the same members as class B allows a child to have a correct understanding of the relationships but not one that displays necessity. Meadows (personal communication) claims that her test-question is taken from Piaget; yet the question cited has no equivalent in the list oflered in Piaget's main account (Inhelder & Piaget, 1964, p. 101), although it does have an equivalent in a later review (Piaget & Inhelder, 1969, p. 169). (See also Section 3 below.) Meadows also claims that the conditions that she discusses are both necessary and suflicient for the understanding of inclusion and so her account is diflerent from Piaget's account in this respect as well.

Markman & Seibert (1976) and Markman (1979) are interested in performance factors relevant to the understanding of inclusion and propose that a child's understanding of the inclusion-relation is facilitated by his understanding of certain types of part-whole relations. On this view, collections are distinguished from classes by possession of a natural organization, by being wholes in more than an abstract sense and by being referred to by singular nouns (Markman & Seibert, 1976). It is claimed that children can more easily give a correct answer to a collection-question, ' Who would have more pets, someone who owned the baby frogs or someone who owned the family?', than to a class-question, 'Are there more frogs or more baby frogs?' (Markman & Seibert, 1976) and so that an understanding of the former facihtates that of the latter (Markman, 1979). It is not

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stated whether this condition is to be taken as a necessary or as a suflicient condition. It seems clear, however, that a child may correctly answer the former question without understanding necessity. Let class A = the four baby (frogs), class A' = the two big (frogs), class B — the six frogs and collection/class C = the six members of the family. Since the membership of class B and collection/class C is the same, ai child who correctly quantifies the membership of A and C and who performs the requisite subtraction may correctly answer the collection-question. Yet such an answer does not require a child to understand that some members of class B are not members of class A and so such a child is not required to comprehend the necessity of the link. Thus this account is distinct from that of Piaget.

Trabasso et al. (1978) cite evidence that might constitute a different type of challenge to Piaget's account and confirmation of their case is provided by a different set of experiments carried out by McGarrigle et al. (1978). The previous studies cited in this section attempt to show that Piaget's account is too restrictive on the grounds that some types of class-inclusion question are more easily comprehended than others. The challenge now to be considered rests on the claim that Piaget's account is too restricted on the grounds that children find class relationships that do not involve inclusion to be just as diflicult to comprehend as those that do involve inclusion. Trabasso et al. (1978, pp. 157-160) present children with two systems of classes as follows: class A = the eight animals, class A^ = the six dogs, and class A^ = the two cats, whilst class B = the eight fruits, class ^j = the four apples, and class B^ = the four oranges. Children are invited to answer a standard inclusion- question, 'Are there more dogs or more animals?', and so make an AJA comparison and also to answer a between-class question, 'Are there more dogs or more fruits?', and so make an AJB comparison. Only the former ofthese, claim Trabasso et al. (1978, p. 159), has an inclusion-relation; yet the results show that children find these questions to be equally diflicult. What is suspect about their case is the claim that Piaget's theory would predict that children should find an AJB comparison to be easier than an AJA comparison. Firstly, exception can be taken to their contention that to count the membership of class A^, a child will count the membership of A^ and of class A2', add the results of the two counts to quantify the membership of class A; and then subtract the result of the count of class A^ from the previous addition (Trabasso et al., 1978, p. 159). The interpretation of Piaget's account presented in Section 1 (b) above diverges from this by allowing that a child will count the membership of class A^; count the membership of class A^; and, as a condition of his adding together the results ofthese counts, understand that ^1 is the subtraction of A^ from A. It is a condition of a child's being able to add together two subclasses to form a superordinate class that he should be able to subtract one of the subclasses from the superordinate class to yield the other subclass. Disagreement occurs, therefore, as to the interpretation of Piaget's account.

Secondly, even if the interpretation of Trabasso et al. is accepted in preference to that presented in Section 1 (b) above, it does not follow that Piaget's account predicts that an AJB comparison should be easier than an AJA comparison. To count the membership of class B, when making an AJB comparison, a child must count the membership of class 5j, count that of B^ and add the two results together. But it is clear that only a child who understands that class B is an including class, and so that class B^ and class B^ are included classes in that class B, can add the two results together. So an inclusion-relation is, pace Trabasso et al., involved in the quantification of class B, for a child who cannot understand inclusion cannot quantify class B. Moreover, on this same interpretation, a child who wishes to quantify class A^ is required to subtract class A^ from class A and so a child is, once again, required to understand that class A includes classes A^ and A^. It follows that the analysis of Piaget's account provided by Trabasso et al. does require that a

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child who makes an AJB comparison should be able to understand inclusion-relations as a condition of his being able correctly to quantify both class A^ and class B.

Thirdly, an inclusion-relation is still involved in an AJB comparison if the interpretation of Piaget's account that is presented in Section 1 {b) is accepted. An ability to understand inclusion would not be required for the quantification of class ^ i ; but it would be required for the quantification of class B, which is admitted to be a superordinate class in relation to B^ and B^ and which could not be correctly quantified by a child who did not understand that this is so. Thus, on either interpretation of Piaget's account, an inclusion- relation is involved in an AJB comparison, as well as in an AJA comparison, and it is presumably for this reason that children find both types of comparison to be equally diflicult. For both require an understanding of necessity, the absence of which excludes success in either case.

A similar conclusion applies to the study carried out by McGarrigle et al. (1978). Children are presented with an array consisting of four cows, two of which are black and two of which are white, and four horses, two of which are black and two of which are white. To give a correct answer to a between-class question, such as 'Are there more white cows or more horses?', a child must understand that the class of horses is an including class in relation to its two subclasses. Thus although the invited comparison does not embody an inclusion-relation, it is a condition of a child's being able to quantify the membership of one of the classes in the comparison that he should be able to understand inclusion. Once again, the absence of necessity in a child's understanding results in a failure correctly to answer the question asked.

Judd & Mervis (1979) have an interest in factors that influence a child's performance on class-inclusion problems and attempt to state necessary conditions that facihtate successful performance, namely counting the membership of the classes and awareness of contradiction between incorrect answers to an inclusion-question and correct counting of the classes. It is claimed that the second of these conditions is the important one, when a child is asked a standard inclusion-question. Judd & Mervis note that their evidence is consistent with Piaget's account and it is apparent why this is so. The interpretation of Piaget's account suggested here is that it requires a child to perform a deductive inference and so requires him to understand necessity. Let class B = the class of toys, class A = the class of balls and class A' = the class of teddy bears. A child who does understand inclusion is one who understands that B = A + A'; and that A = B-A'; and that A' = B — A. Further, a child's understanding of the latter pair in this trio is a deductive consequence of his understanding the first member of the trio. Thus a child who can correctly count the members of class B is committed to the first member of the trio, B = A + A'; hut if that child then claims that the membership of class A exceeds that of class B, he is committed to a denial of A = B—A'. Thus that child's total response is contradictory since his commitment to B = A + A' requires his commitment to A = B—A', yet the child's responses show that he accepts only the former of these and not the latter. It is clear that the elimination of contradictions is an essential task that faces any child who misunderstands inclusion-relations in this way. Thus the account suggested by Judd & Mervis is compatible with Piaget's account.

In conclusion, it is clear that Piaget's account is untouched by his critics. His account, unlike their accounts, is primarily concerned with the investigation of logical competence; his account, unlike their accounts, investigates a child's abihty to interrelate the negative, inferential properties of a subclass with its positive, observational properties; his account, unlike their accounts, attempts to state necessary conditions alone for the understanding of inclusion-relations. The most important difference arises from the fact that whilst his critics seek to investigate the correctness of a child's understanding of class relationships, Piaget

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seeks to investigate the extent to which a child understands the necessity (if any) of those relationships.

(3) Evaluation

The aim of the present section is to show that Piaget's account is compatible with, even though distinct from, that of his critics; to identify possible sources of confusion in Piaget's account; and, finally, to indicate why the inclusion-relation is centrally important to Piaget's theory of genetic epistemology.

There are two reasons why Piaget's account is compatible with that of his critics. Firstly, even though the latter investigate the correctness of a child's understanding of inclusion, in contrast to the former who investigates the necessity displayed in such understanding, it is clear that the type of investigation carried out by Piaget presupposes the type of investigation carried out by his critics. For a child who understands that an including class is one that, necessarily, has more members than a class that it includes is a child who understands the correctness of this claim. Thus correctness of understanding is presupposed for a child to have an understanding of necessity. Now this claim is unexceptional. Yet it hides an empirical question, for only empirical research can determine whether a child who understands necessity is one who gains that understanding concurrently with his understanding of correctness or consecutive to his understanding of correctness. It is apparent, from the research cited in Section 2, that a child may understand correctness even though he does not understand necessity and thus it may be argued that a child's understanding is consecutive rather than concurrent. Thus the distinction between correctness and necessity is crucial to this empirical question. The critics seem not to draw this distinction and so criticize Piaget on the grounds that a child can correctly understand inclusion-relations at an age earlier than that allowed by his account. In consequence they overlook the difference between themselves and Piaget. Yet Piaget seems to suppose that an understanding of necessity is concurrent with that of correctness and so fails to see that one is a developmental antecedent of the other. If the argument presented here is accepted, it can be claimed that Piaget's account is compatible with that of his critics.

Secondly, Piaget's account is one that presents necessary conditions of understanding, unlike his critics who (tend to) state sufficient conditions. But there is an interesting asymmetry here. That X^ is a sufficient condition of Y does not exclude there being some other factor, X^, from also being sufficient for Y and thus Y will be present when either X^ or X^ is present. Thus Y may be present, given A'̂ is present, even though X^ is absent and Y may be present, given ^2 is present, even though X^ is absent. By contrast, if Z^ and Zj are necessary conditions of Y, then the absence of either Z^ or Z^ results in the absence of Y. Thus Y will be absent, given the absence of Zj, even though Zj is present and Y will be absent, given the absence of Zj, even though Z^ is present. It follows from this that // Piaget's account does state necessary conditions for the understanding of inclusion, his account must be accepted as being more basic than any account that presents conditions that are (severally) sufficient for the understanding of inclusion. Thus an account that states necessary conditions must be compatible with any account stating sufficient conditions. So even if, contrary to the argument of the previous sections, Piaget's account was not distinct from that of his critics, there would still be good reason to retain his account just if it states necessary conditions of understanding rather than sufficient conditions.

Two possible sources of confusion in Piaget's account may be mentioned. Firstly, Piaget sometimes uses 'easy' versions of his class-inclusion question, 'Are there more wooden beads or more brown beads?' (Piaget, 1952, p. 164); sometimes includes such questions in

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reviews of his own work (Piaget & Inhelder, 1969, p. 169); sometimes uses collection questions, 'Are there more flowers or more daisies in this bunch?' (Piaget, 1977a, p. 84; see also Inhelder & Piaget, 1964, p. 101); and so makes it difficult for his reader to appreciate the importance ofthe 'hard' version such as 'Are there more primulas or more flowers?' (Inhelder & Piaget, 1964, p. 101). Secondly, Piaget does not, perhaps, signal clearly enough his use, with modification, of the notation taken from the theory of classes. Thus A' is to be taken both as a subclass that is observationally characterizable and as a complementary class that is inferentially characterizable (Piaget, 1952, pp. 163 and 172 respectively; Inhelder & Piaget, 1964, pp. 100 and 103 respectively). In this respect, however, the English reader is not helped by mistranslation of logical and structuralist concepts. Taken together, these two deficiencies suggest that the interpretation presented in Section 1 above is one to be extracted from the work of Piaget rather than one that is explicitly stated in that work.

Finally, brief reference may be made to the place of the inclusion-relation in Piaget's genetic epistemology. On this view, any biological system will display a functioning that is logical in nature. Thus instances of inclusion are cited by Piaget where one characteristic (and not just one class) is embedded in some other, as the characteristic of being a domestic cat is embedded in that of being a cat (Piaget, 1971, p. 159), or where one action-scheme is included in that of another (Piaget, 1953, p. 239). A child's understanding of the (correctness and necessity of) inclusion-relation arises out of what Piaget takes to be these more primitive biological counterparts and further claims that the equilibratory process that links them is one that requires a child systematically to interrelate an object's negative and affirmative characteristics (Piaget, 1978, pp. 10-11). Since processes of abstraction and generalization are allied to the process of equilibration, it is clear that a child's understanding of inclusion-relations is a striking instance ofthe claim that:

any generalization tied to empirical abstraction is only extensional and consists in refinding in new objects a property that already existed there... by contrast, reflective abstraction consists in the introduction into new objects of properties that they did not possess (Piaget 19776, p. 318 - my translation).

What the child who understands the necessity of an inclusion-relation can do is to introduce the property of necessity by applying it to instances of inclusion the understanding of which is necessary for his cognitive development to be completed.

In sum, Piaget's account is both distinctive and important. It has been the object of the present paper to discuss its distinctiveness and so support the conclusion suggested by Winer (1980, p. 325) that 'class inclusion at different ages represents different skills or processes'. It has been beyond the scope of the paper to discuss the wider implications of Piaget's account in the context of his theory. It may be noted, however, that the account of the understanding of inclusion is not independent of the theory of genetic epistemology which is an attempt to trace the developmental route whereby a child has the capacity to make deductive, and so logically necessary, inferences.

Acknowledgements I wish to thank Derek Wright and (anonymous) referees of this journal for their comments as a result of which this paper has been substantially improved. Any defects that remain are, of course, mine. Support for this study was provided by a SSRC postgraduate research award.

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Requests for reprints should be addressed to Leslie Smith, School of Education, University of Leicester, 21 University Road, Leicester LEI 7RF, UK.