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INTRODUCTION

Cognitive development is typically thought to be a process which involves the acquisition of new concepts. As a child progresses from infancy to maturity, he shows an increasing ability to deal with complex situations, and it is natural to account for this in terms of the addition of concepts. A child is understood to cope with tasks of greater complexity because he acquires new concepts which yield conceptual systems appropriate to the tasks. One example of this view is Piaget's theory, which characterizes development as proceeding in stages. Piaget distinguishes four basic stages of development, each having a certain kind of 'algebraic' structure. The first stage is reached towards the end of the second year when the child manifests behaviour which reflects an understanding of the nature of objects, e.g., their persistence and continuity through space and time. In the middle two stages the child progresses from a certain 'immediate' involvement with the contents of task domains (the preoperational stage) to a more abstract appreciation of their structure (the operational stage). The final stage occurs in adolescence with the acquisition of the capacity to reason with the full generality of first-order logic. In regard to their 'algebraic' structure adjacent stages are related in at least two significant respects: (i) the structure of the successor stage requires an interpretation wholly different from that of its predecessor; and (ii) the latter structure is in some sense properly included in the former. Since Piaget takes each stage to constitute a distinct conceptual system, i.e., one independent of the system at all the other stages, the result of moving from one stage to another can only be accounted for by resorting to a radically new interpretation. This will involve radical re-interpretation since an earlier stage will be incorporated as a whole into its successor. When it is integrated, the earlier stage becomes part of a larger system whose interpretation is fundamentally different from its predecessor. How this is to be spelled out in detail may be somewhat obscure, but the 'shape' of the theory seems plain enough: stages will have a unique linear ordering and hence development will consist of a Synthese 65 (1985) 33-64. 0039-7857/85.10 © 1985 by D. Reidel Publishing Company

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determinate sequence of conceptual systems, each different and richer than its predecessor. Although Piaget suggests that a stage must be interpreted holistically, he does not intend to imply that the whole is therefore seamless. A stage is made up of discriminable components, called schemas, which contribute to the content of the whole of which they are part. The particular contribution, however, can only be specified relative to the contributions of all the other schemas in the stage. Here one might detect a certain analogy between Piaget's view of schemas and Quine's view of concepts. For Quine concepts are always parts of wholes, where no part has an identity independent of the whole. A part can be discriminated only relative to the other parts because the contents of the parts are all interdependent. T o specify the content of a concept one must not only identify the relevant whole but also indicate the contents of the other concepts which constitute the whole. In effect there can be no bottom-up analysis of conceptual systems. However, such systems do have parts, interdependent though they are. Piaget takes schemas to be parts in a similar sense: their content is discrete but relative to their place in a stage. Each schema will have a distinct content relative to a stage and if it belongs to more than one stage, it will have more than one content. A schema will not typically have a unique reading because it must always be interpreted relative to a stage. This relativity suggests that a schema has properties which are rather similar to the properties that Quine attributes to the terms of natural language. Their content is not univocal but varies in a holistic way. Given Quine's views on the nature of meaning, one might anticipate an explanation of how stages can be ordered by inclusion and yet radically independent in regard to interpretation. When considering the problem of language translation, Quine argues that distinct conceptual systems can always be invoked to interpret a particular network of connections. In the case of language the envisaged network is determined by the inference relations holding among its sentences. 1 He claims that no such network admits of a unique interpretation. Since this obviously extends to sub-networks, there is a sense in which one network can include another, viz., syntactically, without necessitating that their interpretations be similarly inclusive. There will always be a way of reading the two networks in an incommensurable manner. If one takes a stage to be a network of connections among schemas, with

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schemas regarded as the analogues of sentences, Piaget's theory can be seen to inherit the coherence of Quine's account. From a 'syntactic' perspective one may discern a particular network as included in another, and yet semantically one may read the two networks in entirely different ways. In the circumstance one can foresee how it might be possible to elaborate Piaget's stage-theory of development. T o vindicate his theory, however, one must also explain the process of transition from one stage to another. Piaget takes the transition to be effected through a mechanism which invokes new schemas and integrates them, thereby giving rise to a new system. But what sort of mechanism could this be? It may be tempting to think that the mechanism is biological and hence that development is really a maturational phenomenon. In this event Piaget's stages would constitute a description of the effect of an underlying biological process. This, however, does not seem to be what Piaget has in mind. H e views development as a psychological p h e n o m e n o n whose biological foundations alone do not determine the course it will take. T h e course of development is determined by psychological factors whose effect must be appropriately explained. T h e mechanism he envisages is sometimes called reflective abstraction. He sees the process of transition as one which realizes a certain reflective awareness of the structure of the earlier stage, a result that emerges from the integration of this structure into its successor. T h e catalyst for transition is a sort of conceptual conflict which arises at the earlier stage and can only be resolved by invoking a more comprehensive conceptual system. Despite Piaget's attempt to elaborate on the nature of this conflict and the role it plays in reflective abstraction, it remains unclear what actually effects the transition from one stage to another. Since he insists that each stage is fundamentally impoverished relative to its successor, no stage can be sufficient by itself to bring about the transition. Whatever the mechanism may be, it seems that he can only regard it as stage independent. Nevertheless Piaget displays a certain ambivalence on the matter, which may partly explain why his theory is so obscure at this point. If the mechanism of transition is really independent, then no stage can by itself determine a particular state of cognitive development. T h a t part which constitutes the mechanism will have to be separately specified. One might wonder whether cognitive development actually does proceed through a sequence of radically isolated stages. T h e mechanism, if it is in-

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dependent, may suggest a 'graph' that looks more like a continuum. Piaget was unable to clarify the status of the mechanism and hence vindicate his view that development goes in stages. Until one knows something about the nature of the mechanism, its status must remain obscure and so too must the status of the stages. The task then is to see whether the mechanism can be articulated without undermining the basic idea of stage development. To this end we shall explore a somewhat extreme proposal, at least from an orthodox Piagetian point of view. On the assumption that development is not purely maturational, it is natural to suppose that the underlying mechanism is one of reasoning. Indeed there seems to be no choice in the matter: stage transition can only be effected through reasoning if the process is to be psychological. But what kind of reasoning, logical or nonlogical? Piaget regards the final stage of development to consist in the acquisition of the ability to reason logically, i.e., as given in classical first-order logic. This ability is seen as a matter of competence, not performance; that is, the adolescent does not merely refine his logical performance but acquires a new logical competence. How the emergence of such an ability can be explained is not easy to decipher, unless one supposes that the mechanism itself is logical. Since the third stage of development is pre-logical, it does not have the expressive power of any logic, first-order or otherwise. 2 Hence it cannot by itself explain how logical reasoning arises. The same applies to the mechanism of transition if it too is pre-logical. It must s u f e r an expressive impoverishment similar to the third stage and thus cannot account for the emergence of the ability to reason logically. We can only assume that this ability is at least partly constitutive of the mechanism of development. Which logic we should envisage to be involved, however, is a matter for speculation. It would seem natural to consider classical logic, if only because Piaget regards it as determining the fourth stage. He does, nevertheless, entertain another possibility, viz., relevance logic. We shall explore the hypothesis that both logics are actually involved in development. Given that the mechanism need not be determined by a single logic, we shall suppose that it is in a certain sense unstable, sometimes using classical logic and other times employing relevance logic, i.e., the first-degree fragment. The nature of this instability will be subsequently characterized as having considerable psychological significance.

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Since we take logical reasoning to belong to the mechanism of development, we are bound to say that Piaget cannot be right in his characterization of the fourth stage. This stage cannot be seen as unique in admitting logical reasoning and hence, both its content and status will have to be rethought. Whether this will lead to the conclusion that there really is no such stage is a matter we need not speculate about just now. Let it be enough to note the issue and affirm our assumption that there are at least three stages of development - the initial three adumbrated by Piaget. In fact we shall concentrate on just the first stage, the so-called object p e r m a n e n c e stage. Piaget describes it as breaking up into six substages, each related to its successor in much the same way as adjacent major stages. T h o u g h the leaps may have less catastrophic consequences, they are rather similar in kind. For this reason we can illustrate and explore the problems of stage development by focusing on the spectrum covered by the object concept. How this concept emerges will be paradigmatic of the whole of stage development. DESIDERATA

T o begin with let us identify certain desiderata which any adequate theory of development may be expected to satisfy. Fodor has recently argued that development must be a matter of concept learning and there can be but one suitable mechanism, viz., one which involves the projection and confirmation of hypotheses. T h e envisaged hypotheses will take the form of biconditionals which specify the conditions under which concepts apply. For example, if C is the concept to be learned and F specifies the conditions under which it applies, the appropriate hypothesis wilt be the following: (x) (x is C if and only if x is F). Fodor points out that the possibility of confirming such a hypothesis, and thereby learning C, presupposes that one already understands F. In effect to learn C one must know F. This would seem to suggest a certain developmental dilemma. If a child learns C in terms of F, viz., by projecting and confirming the above biconditional, it is not clear how his learning C can represent a significant conceptual advance. Since C is equivalent to F and he already knows F, he can hardly have enriched his conceptual resources. If, on the other hand, he does not learn C in terms of F, it is not at all clear how his learning could proceed, i.e., as a psychological process.

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Unless one can identify an alternative to the process of projecting and confirming biconditionals, one is faced with the prospect that learning cannot be seen as reflecting cognitive development, Plainly any adequate theory must resolve this dilemma. It must allow for genuine learning by yielding an account of conceptual enrichment. A related desideratum concerns the directionality of learning. Piaget claims that the order in which new concepts are acquired is uniquely fixed; that is, each stage of conceptual development admits of only one successor. Although his claim must remain somewhat speculative, given the lack of sufficient evidence, there is an underlying issue that must be reckoned with: why, at a particular stage and in a certain circumstance, does the child acquire one concept rather than another? It is not unreasonable to expect that a satisfactory theory of development will provide some insight into this problem. A third desideratum relates to the 'increments' of learning and how t h e y are to be understood. It has been noted that Piaget views development as proceeding through the addition and integration of schemas. It is to be measured, not just by the schemas added, but by the systems which arise from their integration. If schemas have the status of 'syntactic' objects, the effect of integration may be analogous to the formal SYstems of logic. Since the mechanism of development is to be logical reasoning, it seems natural to view its output as such a formal system, i.e., on the assumption that schemas can sustain logical relations. There is a certain puzzle, however, if one attempts to specify the process semantically. From a logical point of view formal systems may be semantically incommensurable; that is, the meaning of an expression may be specifiable only as part of the system to which it belongs. In the event there may be no system-independent characterisation of its meaning, a circumstance which Piaget takes to be the case for schemas. There is supposedly no way of specifying their content independent of the stages to which they belong and as a result, there is no way of identifying semantically what has been added to a stage to give rise to its successor. The content of a schema will not be the same both before and after integration. Does this mean that development can only be interpreted syntactically? It would seem to insofar as the mechanism of transition is concerned: while each stage may admit of an interpretation, the process of transition would appear to be semantically inscrutable. One would hope an adequate theory

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might clarify whether such a position is tenable, given the other desiderata, and thus whether it allows for an acceptable explanation of development. DEVELOPMENT

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LOGICS

We have indicated that the mechanism of development is to be understood in terms of two logics, classical logic and the logic of first-degree entailment. For our immediate purposes, which are circumscribed by the first stage of development, it will be sufficient to consider the propositional fragments of these logics and specify appropriate semantic characterizations. The semantic perspective will be adopted for reasons specific to the domain of consideration. The content of the two logics, and the differences between them, can be defined with equal clarity in either a semantic or a proof-theoretical way. From the point of view of development, however, the semantic approach reveals structure which is essential for understanding the mechanism of transition. Since we can identify no proof-theoretical counterpart, we are inclined to see semantics as playing an indefeasible role in the theory. In any event the semantics will play an essential role in the elaboration of the theory and this is significant, if only because psychologists have typically viewed logical reasoning from the perspective of formal deduction. To characterize the logics at issue we shall invoke a propositional language having just three basic connectives, and, or and not. Other connectives, e.g., if and only if, can be introduced by definition if they are required. We assume that the language has a nonempty set of atomic sentences, i.e., sentences which have no occurrences of the logical connectives. It is not necessary at this point to identify any particular examples since our semantic objectives can be served simply by the general assumption that there are some, Later when we apply the logics, examples will emerge. Given a nonempty set of atomic sentences, we can define the set of all possible sentences in this way: where A and B are any sentences, so too are (A and B), ( A or B) and not A. Since we propose to discuss this propositional language, particularly the entailment relations holding among its sentences, it will be useful to have an expression to represent entailment, viz., 7 . This expression is

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not to be taken as belonging to the language and hence not as a logical connective. Its function is solely metalinguistic, serving to refer to the relation of entailment. Thus, the sentence

(A and B) ~ B belongs to the metalanguage and is to be read as saying that (A and B) entails B. For each of the logics under consideration we shall indicate the extension of ~ ; that is, we shall specify what entails what. T o this end we invoke the semantic notion of a truth value. From~a classical point of view it is assumed that there are just two possible truth values, true and false, and that every sentence is either true or false. This assumption, however, is not uncontentious. Some logicians have argued that certain sentences may be neither true nor false and thus there is a third truth value. Whether the argument is sound is a matter we need not worry about. It is enough to say that the possibility of additional truth values is logically coherent. T r u t h and falsity may have a privileged place in our thinking but they do not determine the range of possibilities. T h e logic of first-degree entailment is a case in point. T o define it semantically four values are required: true, false, and two other values, one of which may be taken as neither true nor false. T h e fourth value is sometimes taken to be the paradoxical possibility, both true and false. Before defining the logic of first-degree entailment it is convenient to have specified classical entailment. As we have mentioned, this logic is based on the assumption that all sentences are either true or false. In effect classical languages are bivalent. We can define the classical meanings of the logical connectives and, or and not in terms of the following diagram.

T o find the value of a conjunction one reads the diagram downwards; that is, the truth value of (A and B) will be the lower of the truth values of A and B if they are different, and otherwise identical to them. For example, if A is false and B is true, (A and B) will be false; only if A and B are both true will the conjunction be true. In the case of disjunction

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one reads the diagram upwards; the truth value of ( A or B) will be the higher of the truth values of A and B when they are different, and otherwise identical to them. Thus if A is true and B is false, ( A or B) will be true; only if they are both false will it be false. T h e interpretation of negation is given by the double-headed arrow, which indicates that the truth value of not A is the opposite of the truth value of A. If A is true, not A is false, and vice versa. We can now define the relation of classical entailment. Let the two truth values be ordered such that f is lower than t. Then for any sentences A and B, A entails B (A ~ B) if the truth value of A is always lower than or equal to the truth value of B, no matter what the truth values of the atomic sentences in A and B. By way of example let us consider the principle of disjunctive syllogism: ((A or B) and not A ) ~ B. T o see that this is in fact a classical law we need only examine the case where ((A or B) and not A ) is true, for otherwise the definition will obviously be satisfied. If the sentence ((A or B) and not A ) is false, then its truth value must be lower than or equal to the truth value of B. Now let us suppose that both ( A or B) and not A are true. In the circumstance A must be false and thus B will have to be true. In effect the truth value of B will always be higher than or equal to the truth value of ((A or B) and not A). Although the principle of disjunctive syllogism holds in classical logic, it fails to obtain in the logic of first-degree entailment. T o explain the difference we must invoke two further truth values which are taken to be incommensurable; that is, neither can be characterized as lower than the other. From an algebraic perspective it is not important what these truth values are taken to be, provided they are rightly ordered with respect to each other and the other truth values. Nevertheless, it is highly suggestive and illuminating to regard the additional values as neither true nor false, and both true and false, represented respectively as 0 and (t, f). Although the latter may be paradoxical, it is not wholly incomprehensible, for it can be seen to play a coherent semantic role. T o indicate the meanings of the connectives in first-degree entailment we use the familiar Hasse diagram. t

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The method of interpretation is the same for the classical diagram: conjunctions are interpreted downwards, disjunctions upwards and negations by the arrows. Note that there are two additional arrows here, one for each of the new truth values. Thus, if A is ~, not A is also ~; and similarly if A is (t, ~), then not A is (t, f). The values t and f are related as in the classical case. Before discussing conjunctions and disjunctions we should note that the diagram suggests the intended ordering relation on the four truth values: f is lower than both ~3and (t, f) which are lower than t, but ~ and (t, f) are themselves incomparable. This structure is important when interpreting conjunctions and disjunctions. The truth value of (A and B) will be the lower of the truth values of A and B if they are different and comparable; if they are incomparable (A and 13) will be false. Where A and B have the same truth value, (A and B) will have that truth value. A similar situation holds with respect to (A or B). Its truth value will be the higher of the truth values of A and B if they are different and comparable; if they are incomparable, it will be true. When A and B agree in truth value, (A or 13) will have that truth value. We can illustrate these definitions by showing that disjunctive syllogism is not a law of first-degree entailment. The definition of entailment remains the same; that is, A entails B if the truth value of A is always lower than or equal to the truth value of B. Consider again the schema of disjunctive syllogism ((A or 13) and not A) ~ 13. Suppose that A is ~ and B is (t, ~. Then (A or B) must be t since ~ and (t, ~ are incomparable. Given that A is ~, so too is not A, and therefore ((A or B) and not A) must be ~ since ~ is lower than t. But ~ is incomparable with the assumed truth value of 13, viz., (t, f). Hence, the truth value of ((A or B) and not A) is not always lower than or equal to the truth value of/3 and thus, disjunctive syllogism does not hold in the logic of first-degree entailment. Having distinguished the two theories of entailment, we hasten to clarify one or two technical points. The logic of first-degree entailment is actually a sub-theory of classical logic; that is, if A ~ B holds with respect to the former, it also holds with respect to the latter. In effect classical entailment is strictly a stronger relation (has a larger extension) than first-degree entailment and wholly includes it. The strength of the two logics is thus inversely related to the size of the valuation space, i.e.,

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the number of possible truth values. As one reduces the number of truth values, one increases the number of entailments. 3 Significantly there is a logic which lies between first-degree entailment and classical entailment, which is sometimes known as Kalman logic. This is a three valued theory where the 'middle' value can be taken either as 0 or as (t, f). By invoking the intended ordering relation and defining the connectives according to the usual strategy, one can show that Kalman entailment is strictly stronger than first-degree entailment and strictly weaker than classical entailment. T h e difference between the first two emerges in the fact that (A and not A) ~ (B or not B) holds in Kalman logic but not in the logic of first-degree entailment. Disjunctive syllogism, however, fails to hold in Kalman logic, thereby distinguishing it from classical logic. 4 THE

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THEORY

Piaget claims to chart the course of development through a sequence of problem solving tasks which are given to mark conceptual growth. H e observes that children typically fail to solve certain puzzles at certain ages, and envisages that their success in finding solutions reflects an underlying conceptual advance. The experimental data over which he speculates is especially robust at the object-concept stage. Here a remarkably stable pattern of behaviour emerges from children's attempts to solve set puzzles. Success seems to occur in a definite order; children are able to solve one kind of puzzle only after they can solve another kind. M o r e o v e r their responses to particular tasks within a certain age range are predictable and general. In view of this we propose to accept Piaget's basic approach to the problem of mapping development, where the strategy is to identify tasks on which children display stable and uniform behaviour at fixed ages. Since this strategy is perhaps most clearly articulated at the object-concept stage, it seems appropriate to elaborate our theory in this environment. We assume Piaget to have characterized the relevant substages and we must now try to explain the mechanism of transition. Children will be assumed to have the capacity to reason according to the principles of both classical logic and the logic of first-degree entailment. When reasoning against the background of four possible truth values, viz., t, f, 0 and (t, f), children are able to distinguish the

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class of entailments that are first-degree valid; that is, they know (in some nonarticulated sense) what entails what in the logic of first-degree entailment. Similarly when reasoning with respect to a bivalent background, they are able to distinguish the instances of classically valid inferences. That children possess such logical acumen may seem an extraordinary thing to suppose, if only because their reasoning behaviour seems so limited in scope. And yet the assumption is arguably minimal. Children are only taken to understand the simple connectives and, or and not; the extent of their logical ability is exhausted by the meanings of these expressions, i.e., by their logical meanings. That this should be part of their biological endowment seems hardly an unreasonable hypothesis, especially since they are given to be rational beings. It is inconceivable that children could be rational if they do not have a basic grasp of the principles governing and, or and not. This is not to say that what they grasp must be univocal in content. What content they attribute to the connectives will depend upon which possible truth values they entertain. We suppose that they consider at least two possible valuation sets, the bivalent one and the four valued one. Accordingly they entertain two possible meanings for the connectives which are captured in the alternative logics. We must emphasize that this hypothesis is not entirely bereft of empirical content; as we shall indicate, the behaviour responses of children to developmental tasks can be interpreted as reflecting reasoning patterns distinctive of the two logics. Broadly speaking, we regard children as first conceiving the tasks relative to the logic of first-degree entailment. When presented with a puzzle, they reason initially in terms of the weaker logic, which explains why they cannot solve it. The logic is too weak to allow the solution to be deduced. The same does not apply to classical logic. If they reason classically, the solution follows logically from propositions they already accept. The question we must resolve is what motivates them to move from the weaker to the stronger theory, for it is this move which explains development. We should emphasize that this transition is not to be seen as once off; that is, each developmental task involves the same process of moving from the weaker to the stronger logic. Puzzles arise because children formulate the tasks initially in the logic of first-degree entailment and this theory is too weak to carry them to the solution. Subsequently they

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reconceive the tasks in a classical logic and are then able to deduce the desired result. We envisage this process to be uniform throughout development and thus as independent of any particular stage. In the circumstance there is a general answer to one of the deep questions of development, i.e., to the issue of what gets constructed. From our perspective it is basically a sequence of classical theories, each extending the range of application of its predecessor. T h e extensions result from the addition of new atomic sentences (and perhaps some generalizations, when the logic is extended to admit quantification). How the appropriate additions are generally to be deciphered is something we are not entirely clear about. Such vagueness, however, does not obscure the structure of the theory; it rather identifies a sensitive 'interface' with empirical research, where extensions to a theory may be suitably determined. Although we propose to indicate the framework within which this issue is to be approached, we must leave a detailed investigation to a more appropriate environment, viz., to a well-specified research project on cognitive development. Immediately we intend to focus on the broad outlines of the theory. One general problem that must be considered is the manner in which developmental tasks are formulated by the children. We have suggested that the course of development is to be seen in terms of a sequence of classical theories, each richer than its predecessor. These arise from puzzles which are initially formulated in a theory whose logic is that of first-degree entailment. Since the theories are specified in terms of the sentences of a language, we seem bound to regard children, even pre-linguistic children, as formulating the puzzles in a linguistic way. One may then ask whether we are envisaging a language of thought. T o some degree we must be; that is, we must see children as possessing a representational system at least equivalent, if not identical, to that embodied in a natural language. Whether this system is really a language of thought seems a fine point, and one we shall not speculate about. T o be sure, the form of representation may be psychologically important, but not for the issues we shall be considering. It is sufficient for us to assume that the system can express propositions and represent their logical relations. This assumption leads naturally to the idea that the content of a Piagetian schema can be characterized in terms of a set of sentences and their place in a theory. We suggest that this holds for the schemas which constitute substages of any of the major stages and of course for

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the m a j o r stages themselves. But not every schema can be so regarded, e.g., those which Piaget identifies as sensori-motor schemas appear to have a 'sub-cognitive' status which renders them inappropriate for the a p p r o a c h we shall adumbrate. Only the cognitive schemas have a role which can be articulated in terms of logical reasoning. Before looking m o r e closely at the substages, let us introduce some terminology that will be helpful in explaining the mechanism of d e v e l o p m e n t . Since we invoke the truth values ~ (neither true nor false) and (t, f) (both true and false), it is natural to ask what it would be to entertain a proposition as having one of these values. W h a t is it, for example, to entertain a proposition as neither true nor false? One thing seems reasonably plain: if one is attempting to solve a certain problem, any proposition which is neither true nor false can only be seen as irrelevant to the problem. Since the intention is to deduce a true proposition (the solution), given true premises (the problem), any proposition which is neither true nor false can be set aside as not bearing on the objective one way or the other. It cannot contribute to its being realized, nor can it contribute to its being shown to be unrealizable. T h e point here is really one of perspective, viz., the perspective of the reasoner, and this is captured in the following definition. A proposition is said to be irrelevant if it is held to be neither true nor false. W h a t is to be said of a proposition that is seen to be both true and false? Insofar as the reasoner is c o n c e r n e d it seems that he can only be ambivalent about the matter: when a proposition is both true and false, it will be paradoxical in that it will both satisfy and not satisfy the conditions of reasoning relevant to p r o b l e m solving. Given that the objective is to reason f r o m true premises to true conclusions, one can only be in two minds about a proposition that is both true and false. If the proposition is one of the premises, one will be ambivalent about whether all the premises are true; and if the proposition is a conclusion, one will be ambivalent a b o u t whether it follows truly. In this circumstance one will be in a state of mind that m a k e s reasoning pointless since no inference can yield a gain in knowledge. W h a t is not known to be univocally true is not known, whether that be the conclusion of an a r g u m e n t or one of its premises. T o reflect this situation we introduce a definition which is parallel to the one above.

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A proposition is said to be true and false.

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paradoxical if it is held to be both

In view of what has been said, there seems to be only one way to perceive a proposition as relevant to an argument, given that the aim is to argue from true premises to true conclusions: it must be seen as bivalent, i.e., as true or false. If the proposition is true, the premises of the argument should yield inferences which conform; that is, the premises should not allow the deduction of a contradictory proposition. On the other hand, if the proposition is false, the premises should obviously not permit it to be deduced. One might emphasize the general point by saying that only bivalent propositions can be verified or falsified and therefore, only these can be relevant to an argument. This is not so much a matter of the definition of validity but rather of the point of reasoning as a means of increasing knowledge. Accordingly we define the concept of relevance in the following way. A proposition is said to be

relevant if it is held to be bivalent.

With the above definitions in hand, we can now illustrate how our theory is meant to apply. In so doing we shall formulate the problem of transition and offer a possible solution.

SUBSTAGE

3

It is convenient to begin our analysis of the object concept with substage 3 which is typically located between 4 and 8 months. This is not to suggest that the earlier substages are in some way significantly different but only that the transitions between the later substages offer especially clear paradigms. As we have mentioned, Piaget discriminates a substage in terms of a behavioural pattern which is the characteristic response to a type of problem solving task. In the case of substage 3 the task is simple and the response surprising. T h e child is presented with some manipulable object, which attracts his attention. Within reaching distance is a c o v e r or some other occluder sufficient to hide the object from sight. In full view of the child the object is then hidden under the cover. One might expect that the child, since he is interested in the object, would attempt to recover it, perhaps by looking under the cover. But this is not what happens. Once the object is r e m o v e d from sight the child seems to lose interest; not only does his attention lapse, but so does any apparent awareness of what he was attending to. Some

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have interpreted this to indicate that once the object is out of sight it is also out of mind. Piaget does not regard the child's behaviour as reflecting just a deficit of performance, e.g., of memory. It is not merely that the child cannot r e m e m b e r the object after it has disappeared; it is that he does not yet have the concepts necessary to formulate the appropriate memory. Thus Piaget takes the child's behaviour to arise from a c o m p e t e n c e deficit which is cognitive in nature. T h e issue of whether the deficit is really one of c o m p e t e n c e or of performance is significant from the point of view of the theory of development. If the child's problem is one of performance, it would be inappropriate to regard development as proceeding through conceptual growth. It would be more a matter of maturation or practice, from which the relevant skills might emerge. If, however, the problem is one of competence, it would seem natural to understand development as a conceptual matter. We propose to adopt Piaget's perspective on the issue and attempt to show how development can be viewed conceptually. Having said this, we must emphasize that our approach is less than orthodox, i.e., relative to the theory as originally set out by Piaget. With this warning, let us now formulate the state of affairs at substage 3. Recall that the child is attending to an object that has been presented to him and then he watches it disappear under a cover, which happens to be within his reach. In response to the object's disappearance we conjecture that the child considers the following proposition. (1)

T h e object is there.

T h e deictic there refers to the location identified by the cover and of course the object refers to the object. H o w these semantic relations are realized cognitively is not a matter we need be concerned with. It is enough that we can interpret what the child entertains. 5 T h e issue is the m a n n e r in which he entertains it. Since he does not go to look under the cover, or even near about, we suppose that he must entertain (1) as neither true nor false. H e takes (1) to be irrelevant to the situation in which he finds himself. In effect the child's response is basically one of bewilderment. At one m o m e n t an object is there in front of him and then suddenly it disappears. What, if anything, can he do about it? Were he to know that objects which disappear can often be found near the place of disappearance, he could look round the cover, or perhaps even under it. But

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he does not seem to know this; he does not yet associate points of disappearance with places of hiding. Thus it is no wonder that he does not display any regular pattern of behaviour which could be interpreted as searching for something. H e is at a loss as to what to do. This is not to say that the child does not entertain the significant proposition, viz., the one expressed by (1). He projects this proposition but does not see it as relevant to his situation. It does not lead him to do anything that would bring about the return of the object he was attending to. We can represent his state of mind, viz., that of projecting (1) as irrelevant, by invoking the truth value 0: the child holds (1) to be neither true nor false. This is not the end of the matter, however; what he first takes to be irrelevant subsequently comes to be paradoxical. At a certain point it seems to occur to the child that (1) may in fact be significant but this does not emerge as an unambiguous hypothesis. From a behavioural point of view the child begins to show distress when presented with the substage 3 problem. When the object disappears, he seems to be frustrated rather than bewildered. Piaget interprets this as indicating that the child is actually in an ambivalent state of mind; he envisages simultaneously that the object is both there and not there. It is as if the child, while coming to suspect that (l) is significant and true, also suspects that it is false. This is not entirely unreasonable. Although he may suppose that (1) is perhaps true, he still has reason to suppose that it is false; after all, it is not yet evident that objects can typically be found in the region of their disappearance. What is curious, according to Piaget, is that the child seems to envisage something paradoxical, viz., that the object both is and is not there. This seems to be the only way to explain his behaviour. The child does not go and look to see if the object is under the cover; he just remains fixed in a state of frustration. One can explain the nature of his distress by noting the paradoxical status of what he is entertaining. If he takes (1) to be both true and false, then looking under the cover will be of no use to him; he can neither verify nor falsify what he entertains. W h a t e v e r the state of affairs turns out to be, it will coincide with what he envisages. Plainly frustration is an appropriate response to this situation. T h e r e is only one way out of this dilemma and that is to project (1) as either true or false, i.e., as bivalent. Once the child entertains the idea that (1) must be either true or false, he will have identified a clear course of action. He will realize that if he looks under the cover, he will either

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verify or falsify (1). From a cognitive point of view, therefore, the substage 3 problem will be solved by projecting (1) as bivalent. At this point the child will be rationally motivated to look under the cover (or in the immediate environs) to determine what the state of affairs really is. When he looks, he looks with good reason; and what he discovers, he can understand. The key to solving the substage 3 task is to envisage the bivalence of the projected proposition. Until the child does this he is without a clear motivation to do anything in particular. We can now formulate the problem of transition for substage 3. The child is said to project (1) first as 0, then as (t, f) and finally as bivalent. In the first two states he is unable to solve the puzzle but in the last he can (and does). But how is this progression to be explained? When considering the matter of transition from one stage to another, Piaget invokes a process which he calls reflective abstraction. This is characterized as involving the abstraction of structures from one stage and their integration into its successor. The issue which Piaget never resolves, however, is the mechanism by which this process occurs. We interpret this problem as one of explaining the successive attribution of truth values, in the case of substage 3 to the proposition (1). Here we are constrained by factors that are both cognitive and empirical. Initially the child is confronted with a puzzle that bewilders him, and this is because he does not see the relevance of what he entertains. Subsequently he comes to envisage that it might be relevant but he is frustrated because he entertains it as paradoxical. Finally he resolves the paradox by projecting the proposition as bivalent. It is not unnatural to think of this progression as involving a general problem-solving strategy which is directed towards transforming irrelevant propositions into relevant ones. To appreciate how this strategy works, two things must be remembered. First, the strategy concerns only those propositions which are currently entertained as irrelevant (or paradoxical), i.e., as neither true nor false (or both true and false). Second, it proceeds under the constraint of consistency: what is projected must be projected as consistent with what is already accepted. 6 It is the requirement of consistency which leads the child to an ambivalent state of mind. At a certain point he has as much reason to project (1) as true as he does to project it as false, and conversely. In an attempt to square the evidence he then entertains (1) as both true and false. But this assignment is vacuous and is subsequently seen to be so: it is consistent with

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absolutely anything that obtains. It is neither contradictory nor noncontradictory with any proposition whatever. This paradox is resolved by projecting (1) as bivalent, i.e., as either true or false, thereby making it relevant to a definite course of behaviour. As such it is seen to be answerable to the facts, to be either verifiable or falsifiable. In summary the strategy is to make relevant what is first entertained as irrelevant and to do so under the constraint of consistency. This is what underlies the process of transition. Basically the strategy involves an alteration of perspective which is reflected in the changing status of the proposition entertained. As we have indicated, the child works initially within a four valued 'perspective' consisting of t, f, ~ and (t, f), or to be more precise, within the lattice which characterizes the logic of first-degree entailment. Given this framework, the child responds to the substage 3 puzzle by projection (1) as 0, i.e., as irrelevant. T o solve the puzzle, however, he must change his perspective, and for this h e must somehow 'step back' from (1). His strategy, we conjecture, is to prune the lattice, or as we shall say, abstract from it. His first move is to abstract the following three valued lattice.

f This forces a reassignment of truth value to (1), which under consistency can only be (t, f). T h e next m o v e is to abstract the classical two valued lattice from the above one, which necessitates that (1) be projected as either t or )~. At this point the child has achieved a perspective from which he can solve the substage 3 problem. It does not matter whether he projects (1) as t or as [, or whether he remains agnostic. It is enough that he sees it as bivalent. The process which gives rise to this result might be aptly characterized as one of reflective abstraction. It should be emphasized that solving the substage 3 problem does not consist in the application of any particular rule of inference. Once the child realizes that (1) is bivalent he has sufficient reason to act; he is now in a position to verify or falsify the only proposition he projects. Although his action is not traced to any classical inference, the characteristic lattice for classical logic is crucial. In subsequent sub-

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stages, however, certain classical rules will also emerge as essential. T h e general process of transition is, in effect, a complex affair, involving both inferences and the semantic structure which justifies them. Sometimes the structure may be seen to contain the key to explanation but typically this will arise as a product of both inference and structure. According to Piaget the ability to solve the substage 3 problem marks the transition to substage 4. At this point a new structure emerges, reflecting a significant conceptual advance. Since we have set aside the earlier substages, we can regard this structure as the child's initial construction. Let us think of it as consisting of a classical language having at least one atomic sentence, viz., (1). When the child can solve the substage 3 problem, he adds (1) as an atomic sentence to a classical theory which is otherwise empty of empirical content, i.e., empty of any atomic sentences. It would be a mistake to suppose, however, that only one sentence is added, for the child is able to solve more than the immediate problem he is working on. Evidence warrants the conclusion that he acquires the c o m p e t e n c e to solve all problems of a similar sort. T h e difficulty is to specify precisely what problems these are. When does a sentence which is similar in form to (1) get added to the theory? Only those sentences which are projected in response to the substage 3 problem should be added. What is needed is some kind of generalization which defines the range of appropriate instances. Unfortunately it is not easy to identify the relevant factors and even if it were, it would be cumbersome to formulate them into a suitable generalization. In the circumstance we shall adopt a certain expedient. We shall identify the classical theory which gets constructed in terms of (1) and the following atomic schema. (2)

[The object is there]3.

T h e italics are to indicate that (2) is a schema with object and place parameters; and the subscript is to mark the schema as emerging from substage 3. Thus (1) is an instance of (2). We need not assume that all instances of (2) are specified from the start, only that the conditions of admissibility are fixed. As new substage 3 occasions arise, new atomic sentences will be added to the theory under schema (2). Since the admission of atomic sentences is strictly constrained by this schema, the theory will retain a uniform character, even though the number of atomic sentences continues to increase. All these sentences will be, as it

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were, substage 3 sentences. While they may figure in more elaborate patterns of inference, they will not take the theory to a more advanced stage, a state of affairs which will become clear below. Before moving to consider the substage 4 problem, we should note the conceptual import of the classical theory we have adumbrated. Piaget suggests that each substage in the development of the object concept reflects a distinctive concept which is peculiar to it. It is natural to ask what the concept might be which is differentiated by the classical theory determined by schema (2). Here it is difficult to be explicit without resorting to metaphor or to the via negativa. Neither seems a satisfactory way to identify the conceptual content of this substage, or any other substage. Quine would seem to point to the only viable approach. Let us suppose that insofar as 'object-sentences' are concerned, the child accepts as true only those belonging to the adumbrated theory. Clearly he is going to have a very strange idea of objects, one that may actually be impossible to characterize other than by the theory itself. But what other characterization do we need? It seems sufficient to identify the range of schemas which the child takes to yield true sentences, in this case schema (2). This fixes, together with the logic, the concept he is entertaining and reveals it to be curious in the extreme. SUBSTAGE

4

The task which characterizes substage 4 is among the most well known of Piaget's experiments, perhaps because the behaviour it engenders is so unexpected. As in the previous substage, the child is presented with an object which is then hidden under a cover (or by some other occluder). Rather than have the child retrieve the object, which he could do, the experimenter does this himself and hides it under another cover which, like the first one, is within reach of the child. At this point the child responds by looking not under the second cover, as one would expect, but under the first one. Piaget speculates that the child is suffering from a conceptual deficit in regard to the relation between changes of location and object identity; identity is not seen to persist through changes of location. Let us try to formulate the substage 4 problem within our theory. We conjecture that the child initially projects two propositions, one atomic and the other molecular.

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T h e object is there,. T h e object is there~ or the object is there2.

T h e subscripted deictics there~ and there2 refer respectively to the first and second places of hiding, and all three occurrences of the object are assumed to be coreferential, This assumption may seem to be at odds with Piaget's understanding of the situation; it would seem to presuppose the very thing that needs to be explained, viz., the growth of the concept of identity. Appearances notwithstanding, we shall indicate below why this is not the case. It may also seem curious that we take the child to entertain a disjunction instead of the corresponding conjunction. Since the child sees an object disappear there~ and also sees an object disappear there2, it is natural to consider the conjunction to be more appropriate. T o appreciate that (4) is in fact the right choice one must understand that the substage 4 problem is conceived by the child as a task which concerns there1, and not there2. He focuses first on there1 where he has seen the object disappear. From previous experience he has good reason to suppose that the object is there; the fact that an object has subsequently emerged does not necessarily imply that the object of interest is not still there. What he has discovered to be a successful strategy in the past is thus a natural strategy in the present situation. As for there2, the child views this to be irrelevant. H e saw the object of primary interest disappear therel and experience justifies him in focusing his attention on this location. What obtains in respect of there2 is not at the m o m e n t of any particular significance. This is not to say that the child holds that no object is there2; presumably he does think there is an object there2 but this is not related to the problem he is now considering. From his current perspective only (3) is relevant; the second disjunct of (4), (5)

T h e object is there2

is irrelevant to the task at hand. Hence the child projects (3) as bivalent and (5) as neither true nor false. Note that (5) is not to be read as saying that some object or other is there2; this the child would certainly hold to be bivalent on the basis of his earlier experience. What is neither true nor false is the hypothesis that the object of his original attention is there2. We can now explain why (4) takes the form of a disjunction. Since (5)

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has the value t3, the conjunction of (3) and (5) also has the value 0 (by the lattice cited above). Thus the conjunction will 'inherit' the irrelevance of its conjunct (5). This is not the case, however, with respect to the disjunction (4). Since the disjunct (3) is bivalent, (4) is not necessarily irrelevant: if (3) is true, so is (4). Given that experience warrants the child's projecting (3) as true, we assume that this is just what he does. He not merely envisages (3) to be relevant but to be so because it is true. In the circumstance he also envisages (4) to be true, even though (5) is held to be neither true nor false. The status of (5) is particularly significant since it allows the substage 4 problem to be formulated. T h e fact that the child takes (5) to be irrelevant is an appropriate characterization of his not appreciating the connection between place and identity. If he made the right connections, the child would project (5) as relevant and would realize that (3) and (5) are genuine alternatives. It is clear from his behaviour, however, that the child does not see things as they are. Taking (3) to be true, he goes to look there1 and discovers that nothing is under the cover. If he had a reasonable concept of identity, in effect if he held (5) to be relevant, he might naturally be expected to look there2. But he does not; he seems to be bewildered, much as he was at the corresponding point in substage 3. His behaviour is random and unpredictable; he seems more bemused than distressed. Having looked there1, the child now knows that (3) is false, but he still believes that (5) is neither true nor false. Given these assignments to (3) and (5), it follows (by the meaning of disjunction in first-degree entailment) that (4) must be neither true nor false and thus can only be held to be irrelevant. In the circumstance it is understandable that the child is at a loss as to what to do. At a certain point during this substage, however, the child shows signs of progress. When presented with the puzzle, he responds not with bewilderment but with genuine frustration. We interpret this as indicating an emerging awareness that (5) is not irrelevant, as he had thought, but may well be true. Of course he still has grounds for supposing that (5) is not true, viz., the very grounds which led him to entertain it as irrelevant. It is not yet clear to him that the object which was hidden there1 has been r e m o v e d to there2, where it can be found. In an attempt to square the facts as he now sees them, the child re-projects (5) as both true and false, which is reflected in his behaviour. He does not look there2 to see if the object is under this cover but remains frustrated with

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puzzlement. And so he should be: it is no use looking if he can neither verify nor falsify what he entertains. No matter what the state of affairs turns out to be, it will coincide with the fact that (5) is both true and false. As in the case of substage 3, there is only one way out of this dilemma, viz., by re-projecting the paradoxical proposition as either true or false. Once it is taken to be bivalent, the child can reason by disjunctive syllogism to the solution. Recall that he knows that (3)

T h e object is theret

is false. Given that he now assumes (5)

T h e object is there2

is bivalent, i.e., either true or false, he can conclude that the disjunction (4)

T h e object is there;

or

the object is there2

must also be bivalent. From this state of affairs it follows that he has a clear reason for going to look there2. Appropriately motivated, the child now looks under the second cover and discovers that the object is indeed there. T h e strategy which brings about this result is exactly the same as in the earlier substage. T h e child pursues the general strategy of attempting to make relevant what he first entertains as irrelevant. Applying this strategy to the substage 4 problem, he falls initially into paradox: his attempt to see (5) as true conflicts with other things he sees and hence, he envisages (5) as both true and false. His next move is more efficacious: he re-projects (5) as either true or false, i.e., as bivalent, and this is sufficient to allow him to deduce the solution. T h e direction of this strategy towards relevance is determined by the nature of the process of reflective abstraction. At each step the child is abstracting a 'sublattice' which forces a reassignment of truth value to the entertained proposition. First he abstracts the lattice consisting of t, f and (t, f), thus forcing a reassignment to (5) which was initially held to be 0. T h e n he abstracts the two valued lattice which requires (5) to be seen as either t or f. At this point (5) is held to be relevant, and the child's behaviour is accordingly both predictable and explicable. He has sufficient reason to act as he does. His ability to solve the substage 4 problem indicates, according to Piaget, that he has progressed to substage 5. T h e r e is reason to think,

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however, that the progress may not be as undiluted as Piaget supposes. Below we shall consider a variant form of the substage 4 task where the results would warrant a certain caution. Nevertheless the child's immediate success is evidence for some conceptual advance and this must be characterized. Here we need only extend in an appropriate way the classical theory constructed in the transition to substage 4. T o this end we add sentence (5) and the following atomic schema. (6)

[ The object is there2]4.

T o reflect the fact that (3) is false we also add its negation and the corresponding molecular schema. (7)

not the object is therel

(8)

[not the object is there1]4.

T h e substage 5 concept of an object, at least its initial version, is determined by the classical theory in which all the admissible instances of schemas (2), (6), and (8) are held as true by the child. Since he is assumed to have no opinions about other 'object-sentences', it is plain that he continues to have a truncated view of objects. How truncated his view must be emerges from the variant of the substage 4 problem which we alluded to above, T h e determinants of this task are basically similar to those of its antecedent: there are two locations in one of which an object is hidden, except in this case the occluders are inverted cups. T h e r e are, moreover, two significant differences. First the object is not transferred from one cup to the other; its location is changed by switching the cups. Second the child does not see the object again after it has been hidden at the first location. Bower et al. have observed that children who can solve the standard substage 4 problem typically fail to solve this problem. T h e y will usually look in the first location there1 rather than in the correct location there2. If the ability to solve the standard problem really marks a major conceptual shift, one wonders why there should be a difficulty with the related task which is structurally similar. Some have speculated that the standard task is not a sure test of genuine development. Others have suggested that the discrepancy may indicate that development ought not to be measured conceptually, at least not in all cases. T h e difference may not always be a matter of concepts but sometimes of search strategies. From a behavioural point of view children divide into two types in respect of their responses to the nonstandard task. Some children, once

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they discover that the object is not there j, will 'correct' their mistake and go to look there2. Others do not do this; they seem to be as bewildered as they were at the corresponding point in the standard task. How are we to explain these different types of response as well as the general difference between the standard and nonstandard tasks? Let us consider the 'correcting' child first. It is reasonable to think that from the beginning he interprets the nonstandard task as a two-place problem; that is, he realizes that both there~ and there2 are relevant. What he does not yet grasp is how the location of the object relates to the switching of the cups. T o reflect his state of mind we suppose that he projects (3)

T h e object is there1

as true and thus also projects the disjunction (4)

T h e object is there1

or

the object is therez

as true. T o this extent the child's response is identical to his earlier response to the standard task. T h e difference is that he now takes the second disjunct of (4) (5)

T h e object is there2

as bivalent. More precisely, he envisages (5) to be false since (3) and (5) cannot be true together. Once he discovers that (3) is false, however, he reasons by disjunctive syllogism that he must have been wrong about (5). If (3) is false and (4) is true, (5) must be true. As a result, he is motivated to look there2 to see if his assumptions are correct. For the 'correcting' child the margin of difference between the standard and nonstandard tasks is small. When he can solve the standard problem, he appreciates the structure of two-place problems and hence has advanced conceptually. It is this which explains why he 'corrects' on the nonstandard task, and it also accounts for how he eventually solves the problem directly. When he realizes that switching the cups implies the falsity of (3), he concludes by straightforward reasoning that (5) is true and thus goes without detour to look there2. T h e 'noncorrecting' child is equally easy to explain but yet more difficult to understand. He too projects (3) and (4) as true but seems to have reverted to a state characteristic of the standard task. He projects (5) as irrelevant, i.e., as neither true nor false. For him the nongtandard task seems entirely new, as if he had never solved the standard task.

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Why he should fail to make the structural connection between the two types of task is something we do not understand. Nevertheless we do understand how he proceeds to solve the problem, viz., by the very same strategy which led him to the solution to the standard problem. With the c o m p e t e n c e to solve the nonstandard task comes a further extension of the classical theory which the child is in the process of constructing. More instances of the schemas (6) (8)

object is there2]4 [not the object is there,]4

[The

become admissible as true. In effect the ability to solve the nonstandard task extends the range of sentences which the child will accept as true. His concept of an object will of course remain truncated, but rather less so than before. SUBSTAGE

5

By this stage Piaget takes the child to have developed a reasonable idea of what objects are. Objects persist when hidden from sight and thus are capable of being retrieved. T h e y preserve their identity when moved from one place of hiding to another and hence are to be sought at the place of removal. What remains to be discovered, according to Piaget, is that the identity of objects can persist through several layers of hiding. T h e substage 5 task suggests that the child has yet to understand this. T h e task is again very simple. An object which the child is attending to is first hidden under a cover and then unseen by the child is placed under another cover which is under the first. The child will typically search for the object by removing the first cover but not the second. Piaget speculates that his behaviour indicates a certain conceptual deficit; the child does not understand that objects can persist through several layers of occluders. Whether this interpretation is correct may be a moot point. What is not in doubt, however, is the curious nature of the child's behaviour. Given that he has successfully handled all the other tasks, one cannot but be struck by the fact that he does not persevere and look under the second cover. It is this curiosity which inspires the conjecture that the puzzle must be conceptual in nature. What else but a conceptual problem could explain the child's failure to pursue such an obvious strategy, especially in view of his experience?

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T h e puzzle may seem even more curious when it is formulated in our theory. It looks not unlike the problem at substage 3. When the child peels off the first cover, we conjecture that he projects the proposition, (1)

T h e object is there.

On this occasion the deictic there refers to the location defined by the second cover. As in the substage 3 task, the child does not seem to think that (1) is relevant to his situation; his response on removing the first c o v e r is one of bewilderment. He does not surmise, as most of us would, that the object must be under the next cover and accordingly proceed to r e m o v e it. Holding (1) to be irrelevant, i.e., neither true nor false, he has no reason to persevere in his search, either in the obvious direction or in any other. T h e strategy by which the child solves his problem is exactly the same as it was at substage 3. He attempts to make (1) relevant and as before, the first step leads him into paradox. He envisages that (1) might be true but finds it difficult to square this with the facts. So, he projects (1) as both true and false. By the process of reflective abstraction, viz., from the three valued lattice to the two valued lattice, he comes to re-project (1) as either true or false, at which point he is motivated to search in the obvious place, T h e proposition which he entertains is now seen to be verifiable or falsifiable, and this provides sufficient reason for him to find out which it is. Once the child learns to solve problems of the substage 5 type Piaget assumes that he has m o v e d to substage 6. What he has constructed in the process can be characterized by adding the following atomic schema to the already constructed theory. (9)

[The object is there]5.

Although similar to schema (2), this schema admits a new set of atomic sentences, viz., those which are held to be true on the basis of their being solutions to substage 5 tasks. H e n c e (9) extends the theory in a significant way. A natural question to ask now is whether the adumbrated theory with the atomic schemas (2), (6), (8), and (9) provides an adequate characterization of the concept of an object. One thing is certainly plain: the theory does not provide an analysis of the concept. Piaget and others have attempted to offer a direct analysis but this approach seems to lead straight to Fodor's dilemma. Such analyses must be given in the form of

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biconditionals, which Fodor shows to be inappropriate for a theory of development. Learning a concept cannot involve the projection of biconditionals for confirmation unless one already understands the analysans of the biconditionals. But since one's understanding of the analysans is precisely what is in need of explanation, Fodor concludes that this direct approach cannot be invoked in a theory of development. It should be noted, therefore, that none of the supposed projections in our theory take the form of biconditionals. The question is whether the schemas arising from these projections can jointly specify (as part of the classical theory to which they belong) the concept under consideration. If one thinks of these schemas as axioms, the resulting theory will be satisfiable in a variety of structures and hence will not uniquely specify any particular concept. If one thinks of the schemas as we intend, however, the situation is rather different. They give rise to atomic sentences which are held to be true; and they are held to be true because they are taken to be solutions to certain problems. These problems constitute the determinants of the analysis; that is, they set the issues to be resolved and thereby fix the parameters necessary to specify the concept. The solutions provide an interpretation for these parameters which is given in the form of propositions which are held to be true. What then is the object concept? It is just the set of solutions to the characteristic problems. Surely this is sufficiently precise. SOME

GENERAL

ISSUES

Whether our theory can be construed as broadly faithful to Piaget's view of stages is a matter open to some dispute. For one thing it is not entirely clear that we have managed to reflect the semantic relation between adjacent stages, viz., that the interpretation of one must be radically different than the interpretation of the other. The propositions projected, for example, at substage 3 are also projected at substage 4; in effect schema (2) is involved in both substages and its instantiations seem to be univocai in the character of their content. Indeed the univocal status of their content seems to be essential for our theory to work. On the other hand, the different constructions which correspond to the two substages seem to be genuinely distinct. Anyone who admits only instances of schema (2) as true atomic object-sentences must surely have a strange idea of objects, and his idea must be radically different (in some ineffable sense) from the concept of someone who

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admits in addition instances of (6) and (8). From this point of view the two substages can only be seen as conceptually distinct. T h e problem is that the distinction cannot be specified other than by pointing to the associated theories; the difference seems otherwise inexpressible. Here we seem to face our own form of paradox. On the one hand, we have a semantic way of specifying the 'increments' of stage development. These are given in the form of schemas which identify solutions to characteristic tasks. Their import remains constant across stages. On the other hand, the admissible substitution instances of these schemas, when they are taken collectively, seem to allow stages to express different general concepts. Different concepts will emerge from different schema sets. Does this imply that the meaning of the substitution instances will therefore differ from one substage to the next? If it does, then a schema cannot have the same import with respect to different substages. T h e solution to this paradox, if it is one, is to invoke a difference of perspective. When viewed by the cognitive theorist, each substage has an identifiable content which is preserved in its successor. This is fixed by the interpretation of the task, which in turn fixes the content of the associated schemas. From the point of view of the child, however, the interpretation of these schemas, i.e., of their instances, will change radically from one substage to another. This can be identified only by pointing to the corresponding theories. T h e difficulty is not merely that one cannot get inside the head of the child but that it would be useless even if one could. One would then have no perspective from which to articulate the difference between the theories. This can only be characterized in an extra-cognitive way, What one shares with the child is the sequence of classical theories, not their incommensurable interpretations. T h e r e are two final points we should like to mention, both of which concern the predictive nature of our theory. T h e first relates to the progression of assignments to the projected propositions. In each substage or stage the progression is supposed to be the same, from 0 to (t, f) to bivalent. As we have indicated, this is given to reflect the progression of how the child views the propositions, as first irrelevant then paradoxical and finally relevant. H e r e the theory is open to empirical confirmation since the child should display behaviour appropriate to his changing state. He should first display behaviour appropriate to thinking the propositions are irrelevant, then he should

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manifest behaviour appropriate to holding them as paradoxical (as 'half-relevant'), and finally he should show behaviour indicating their emergence as relevant. Prima facie analyses of certain experimental data reveal just such a pattern and thus a more detailed investigation would seem a natural idea. The second point concerns the order of substages. From the point of view of our theory there is no apparent reason why substages 4 and 5 should not be reversed. Any child who can deal with the one should be able to deal with the other; that is, after substage 3 a child might be expected to have the ability to solve the substage 5 problem first. Again certain work has been done on this and there is evidence that the problems can be switched. It is significant, therefore, that our theory allows for this. It also suggests that the order of substages may generally be more variable than Piaget envisages. This too is a matter for further empirical investigation. 7 NOTES Here the relevant inference relations extend beyond those that are logically determined. Sometimes sentences may be inferred from other sentences that do not logically imply them. What the conditions are for such 'material' implication may vary from theory to theory. 2 The third stage is seen to be pre-logical in a radical sense. It is not merely that this stage fails to contain an explicit logic; it also fails to have the resources for defining any logic at all. 3 It should be noted that this relation does not generally hold. That is, in the range of many valued logics it is not typically the case that reducing the number of truth values increases the number of entailments. 4 For further details on the three logics see Makinson. A more elaborate treatment of the logic of first-degree entailment can be found in Anderson and Belnap. The concept of a proposition is somewhat tricky here. We take the proposition expressed by (1) to be identified by the referents of its terms, viz., the object and there, and the meanings of the expressions involved, in particular the meanings of is and there. Such a proposition, i.e., one so characterized, is distinguished in a logic-independent manner. From the perspective of formal semantics, however, the proposition expressed by (1) may be characterized in a logic-dependent way. If one extends classical logic and the logic of first-degree entailment into intensional logics, the proposition expressed will be distinguished by different functions in the two cases. In effect there wilt be distinct propositions involved, the 'classical' one and the 'first-degree' one. This is a subtle matter which we shall not pursue further just now, except to say that here we have a manifestation of stage holism. ~' Here consistency means first-degree consistency, i.e., consistency as defined in firstdegree entailment.

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7 This paper grew out of a lecture course on Piaget's constructivism which was given in 1981 in the Cognitive Science Program at the University of Massachusetts (Amherst). I am grateful to my co-lecturer Michael Arbib who first stimulated an interest in Piaget's work. This has been nurtured in the Edinburgh Workshop 'Natural Logic and Development' whose members have been constant in their encouragement, patience and criticism of my attempts to rework Piaget's theory. I should particularly like to thank Tom Bower, Tom Pitcairn, David Wallace, and Jennifer Wishart. I am also grateful to Henri Wermus and Barbel Inhelder for their generous hospitality when I delivered an antecedent of the present paper in Geneva. Finally I should like to thank Kit Fine, David McCarty, and Mark Steedman for initiating a number of clarifications. REFERENCES Anderson, A. and N. Belnap: 1972, Entailment, Princeton University Press, Princeton. Beth, E. W. and J. Piaget: 1966, Mathematical Epistemology and Psychology, D. Reidel, Dordrecht. Bower, T. G. R.: !974, Development in Infancy, W. H. Freeman, San Francisco. Brainerd, C. J.: 1976, Piaget's Theory of Intelligence, Prentice-Hall, Englewood Cliffs, NJ. Fodor, J. A.: 1976, The Language of Thought, Harvester Press, Sussex. Furth, H. G.: 1981, Piaget and Knowledge, 2nd edition, University of Chicago Press, Chicago. Johnson-Laird, P. N.: 1983, Mental Models, Cambridge University Press, Cambridge. Makinson, D. C.: 1973, Topics in Modern Logic, Methuen, London. Quine, W. V. O.: 1960, Word and Object, MIT Press, Cambridge. Piaget, J.: 1954, The Construction of Reality in the Child, Basic Books, New York. Piaget, J.: 1967, Six Psychological Studies, Random House, New York. Piaget, J.: 1970, Genetic Epistemology, Columbia University Press, New York. Piaget, J.: 1971, Structumlism, Routledge & Kegan Paul, London. Piaget, J.: 1971, Biology and Knowledge, University of Chicago Press, Chicago. Piaget, J. and B. Inhelder: 1969, The Psychology of the Child, Routledge & Kegan Paul, London. Piaget, J. and B. Inhelder: 1973, Memory and Intelligence, Routledge & Kegan Paul, London. Wishart, J. G.: 1979, The Development of the Object Concept in Infancy (unpublished doctoral thesis, University of Edinburgh). Centre for Cognitive Science University of Edinburgh 2 Buccleuch Place Edinburgh, EH8 9LW Scotland