Note from chapter author, Dr. Mark A. Winstanley: Broadly, my interests lie in epistemology and my approach to the questions that have vexed Occidental philosophers for over two millennia is naturalistic on the one hand and in accord with a shift in style in the philosophy of science in the second half of the 20th century away from foundational concerns to the investigation of the actual production of scientific knowledge on the other. Like most post-Kuhnian philosophers of science, I acknowledge the value of the history of science for understanding the way scientific knowledge is generated; however, I also recognize that tacit assumptions concerning human cognition are made when interpreting historical record; I, therefore, believe that cognitive science should inform the history of science. I am particularly interested in genetic epistemology since it was conceived by Jean Piaget as a scientific epistemology in which both the historiogenesis and psychogenesis of knowledge are methodological pillars. Moreover, genetic epistemology aims to provide science with a scientific rather than a philosophical foundation; measured in terms of consensus among its practitioners, it thus remains true to the historical origins of science’s success – emancipation from philosophy.
In remarking that ‘Piaget’s views on logic are idiosyncratic’ (Johnson-Laird et al. 1992, p. 418), Johnson-Laird reveals his adherence to a popular assessment of Piaget’s theory of reasoning among Anglophone cognitive psychologists (Bond 1978, 2005). However, Johnson-Laird does concede that ‘[i]t is not always easy to understand Piaget’s theory’ (Johnson-Laird 2006, p. 249), thereby corroborating Piaget’s own impression that his work was not well understood(Smith et al. 2009, pp. 1–10). Unfortunately, difficulties in understanding Piaget’s theory are exacerbated by the inaccessibility of his original works in a predominately Anglophone research environment. He wrote in French, and translations into English are selective and not rarely dubious in quality (Smith et al. 2009, pp. 28–44). ‘[R]easoning is nothing more than the propositional calculus itself’ (Inhelder and Piaget 1958, p. 305; in: Johnson-Laird et al. 1992, p. 418), for example, is the citation Johnson-Laird uses to support his interpretation of Piaget’s theory as a mental logic theory of reasoning. However, Lesley Smith considers ‘reasoning is nothing more than the calculus embodied in propositional operations’ (Smith 1987, p. 344) to be a more faithful rendition. The difference in the translations may appear insignificant, but it makes all the difference between correct and incorrect interpretations of Piaget’s operatory theory of reasoning.
Operatory Theory of Reasoning
Operations and their Structures
Piaget characterises operations as follows:
PSYCHOLOGICALLY, operations are actions which are internalizable, reversible, and coordinated into systems characterized by laws which apply to the system as a whole. They are actions, since they are carried out on objects before being performed on symbols. They are internalizable, since they can also be carried out in thought without losing their original character of actions. They are reversible as against simple actions which are irreversible. In this way, the operation of combining can be inverted immediately into the operation of dissociating, whereas the act of writing from left to right cannot be inverted to one of writing from right to left without a new habit being acquired differing from the first. Finally, since operations do not exist in isolation they are connected in the form of structured wholes (Piaget 1957, p. 8, see also 1971, pp. 21–2, 2001, Chapter 2; Piaget and Beth 1966, p. 172; Piaget and Grize 1972, p. 55).
Consider the affirmations and negations of two propositions, [latex]p[/latex], [latex]p ̅[/latex]; and [latex]q[/latex], [latex]q ̅[/latex]. Combining them conjunctively gives rise to four different conjunctions:
[latex]pq[/latex]
[latex]pq̅[/latex]
[latex]p̅q[/latex]
[latex]p̅[/latex][latex]q̅[/latex]
Logically, the conjunctions cannot be true simultaneously since they are incompatible. From an operatory point of view, however, all four conjunctions play a role in determining the relation between the propositions [latex]p[/latex] and [latex]q[/latex]. By systematically considering the various combinations of true and false conjunctions, 16 distinct propositions about the possible combinations of truth and falsity of these conjunctions can be distinguished, and these propositions can be abbreviated by logical operators. (see Table 1).
Table 1: (Click on the table to open a PDF that can be read by a screenreader.) The 16 Logical Operators of Propositional Logic. The columns of this table are comprised of four conjunctions, but for the sake of brevity only those that are true are listed. Moreover, the columns are set out in pairs so that each pair contains the full complement of conjunctions. By connecting the conjunctions listed in each column disjunctively, the disjunctive normal form of the binary operators in the bottom row are generated. With the exception of *, o, w, p[q], and q[p] the binary operators are familiar. * represents the complete affirmation; o, the complete negation; w, an exclusive disjunction, and p[q] as well as q[p] are affirmations of p and q conjointly with either [latex]\overline{q}[/latex] or [latex]\overline{p}[/latex], respectively (After Piaget and Grize 1972, fig. 100).
The logical operators can be combined disjunctively with each other; for example, [latex](p≡q)∨p ̅q=p⊃q[/latex] ; [latex](p\ w\ q)\vee\ \bar{p}\ \bar{q}=p|q[/latex]; etc. In fact, starting from the complete negation, all 16 operators can be generated by disjunctively combining suitable conjunctions from the four possible conjunctions available; for example, [latex]o\vee pq=pq[/latex]; [latex]pq\vee p\ \bar q=p[q][/latex]; [latex]\ pq\vee\bar{p}q\vee\bar{p}\bar{q}=p\supset q[/latex]; etc. For Piaget (Piaget and Grize 1972, p. 335), the disjunctive composition of combinations of these conjunctions constitutes the direct operation on the logical operators. However the outcome of disjunctively combining any combination of conjunctions to the complete affirmation is again the complete affirmation. As the complete affirmation plainly illustrates, all operators cannot therefore be transformed into each other by means of the direct operations alone. Nevertheless, each operation has a dual expression; for example, [latex]\ p\supset q=pq\vee \overline{p}q\vee \overline{p}\overline{q}=\overline{p\overline{q}}[/latex]; [latex]p\vee q=pq\vee\bar{p}q\vee p\bar{q}=\bar{\bar{p}\bar{q}}[/latex]; etc. Furthermore, [latex]pq\vee \overline{p}q\vee \overline{p}\overline{q}=(p\ast q)\bullet \overline{p\overline{q}}[/latex] and [latex]pq\vee\bar{p}q\vee p\bar{q}=(p\ast q)\bullet\bar{\bar{p}\bar{q}}[/latex]; in contrast to the successive accumulation of conjunctions through the direct operation, the dual expression combined conjunctively with the complete affirmation [latex]p*q[/latex] reduces the conjunctions in the complete affirmation. Based on the dual expression, a second operation, the inverse operation, can now be defined as the conjunction of negations of combination of the four conjunctions (Piaget and Grize 1972, p. 335). By means of the direct and inverse operations in combination, all 16 binary operators can be transformed into each other. Moreover, by judicious implementation of these operations any operator can be reduced to the complete negation; for example, [latex]pq\bullet\bar{pq}=o[/latex]; [latex](p\supset q)\bullet p\overline{q}=\overline{p\overline{q}}\bullet p\overline{q}=o[/latex], etc. In fact, it is clear from Table 1, in which the columns are organised in complementary pairs with respect to the full complement of conjunctions, that there is an operation that transforms each of the 16 binary operators into the complete negation. Piaget (Piaget and Grize 1972, p. 335) defines the general identity operation, [latex]\vee o[/latex], as the transformation that is composed of the direct and inverse operation, on the one hand, and that leaves any operator it is composed with unaltered, on the other hand.
The direct inverse and identity operations are reversible operations, which the system of operations has in common with a mathematical group. However, the system also has special identities, such as [latex]pq\vee pq=pq[/latex]; [latex]pq\vee[pq\vee p\bar{q}]=[pq\vee p\bar{q}][/latex]; [latex]pq\bullet[p\ast q]=pq[/latex]; etc. (Piaget and Grize 1972, p. 335). These are lattice-like operations, and they are incommensurate with the operations of a group. In particular, the operations of a group are associative. Whereas associativity works for disjunctions of the conjunctions [latex]\left[pq\vee p\bar{q}\right]\vee\bar{p}q=pq\vee[p\bar{q}\vee\bar{p}q][/latex], for example, it does not always hold for disjunctive-conjunctive mixtures [latex]\left[pq\vee p q\right]\bullet\bar{pq}\neq pq\vee[pq\bullet\bar{pq}][/latex] , since [latex]pq\bullet\bar{pq}=o[/latex] whereas [latex]pq\vee o=pq[/latex]. Associativity is, therefore, limited in this system of operations due to these special identities.
In summary, the relations between propositions represented by the binary operators form a system of transformations, and the operational structure effecting these transformations incorporates reversible operations characteristic of a group as well as non-reversible operations typical of lattices.
‘Grouping’ is the term Piaget (Piaget and Grize 1972, Chapters 38–9) used to denoted the structured whole constituted by these operations, and with the assistance of his co-workers he made several attempts at formalising it using the algebraic tools of logic (Piaget and Grize 1972, n. 1 p. 92); ultimately, however, the grouping resisted their efforts (Piaget and Grize 1972, p. XIV–XV).
Despite the logical garb, the operations of the grouping do not necessarily preserve truth; [latex]p\equiv q\vee\bar{p}q=p\supset q[/latex], for example, is an application of the direct operation [latex]\vee\bar{p}q[/latex], and the transformation from the equivalence to the conditional preserves truth; however, the outcome of the inverse operation [latex]\bullet\bar{\bar{p}q}[/latex] applied to the conditional is the equivalence [latex]p\supset q\bullet\bar{p\ \bar q}=p\equiv q[/latex]. If the operational transformation were a rule of inference, a false conclusion would follow from true premises (see Table 2).
[latex]p⊃q[/latex]
[latex]p≡q[/latex]
[latex]p[/latex]
[latex]⊃[/latex]
[latex]q[/latex]
[latex]p[/latex]
≡
[latex]q[/latex]
T
T
T
T
T
T
T
F
F
T
F
F
F
T
T
F
F
T
F
T
F
F
T
F
Table 2: Truth conditions of equivalence and the conditional. Truth values of the conditional are set out in the left column, and those of equivalence, in the right. Whereas [latex]p≡q[/latex]⊨ [latex]p⊃q[/latex], the shaded cells highlight the truth values that make [latex]p⊃q[/latex] ⊨ [latex]p≡q[/latex] invalid.
The fact that the operatory model of reasoning does not allow deduction is a common criticism of Piaget’s theory; however, it is, as Grize(2013, pp. 153–4) points out, one of the peculiarities of Piaget’s theory that a calculus of propositions from the point of view of validity does not exist, let alone rules of inference. The next section characterises the actual relationship between logic and experimental psychology Piaget had in mind.
Psycho-logic
Logic is concerned with what conclusions follow from what premises, and it develops techniques for determining the validity of inferences. Piaget’s operatory theory, on the other hand, is not directly concerned with logical consequence, and it does not provide techniques for assessing the validity of arguments. Piaget understood his theory in analogy to mathematical physics. Physics investigates the physical world experimentally, and its criterium for truth is agreement with empirical facts; mathematics, on the other hand, is neither based on experiment nor does its truth depend on agreement with empirical facts. It is a formal science, whose truth depends solely on the formal consistency of the deductive systems constructed. With the aim of explaining the physical world, mathematical physics draws on both deductive and empirical sources and applies mathematics to physics to construct a deductive theory based on the experimental findings of physics. Like mathematical physics, Piaget (see also Bond 1978, 2005; 1957, p. 25) also envisages ‘psycho-logic’ or ‘logico-psychology’ as a tertium quid. On the one hand, psychology investigates mental life empirically, and its criterion for truth is agreement with experimental findings; on the other hand, logic, like mathematics, is a deductive science, which is not concerned with correspondence with facts, but with formal rigour, and, again like mathematics, it has developed algebraic techniques. Psycho-logic is an application of these formal algebraic tools to the findings of experimental psychology, and it aims to construct a deductive theory based on the experimental facts of psychology. In other words, psycho-logic uses of the formal algebraic tools of logic to model the structured wholes systems of operations form, namely groupings.
The Grouping as a Cognitive Tool
Psycho-logic aims to model the intellectual operations that form the foundation of reasoning. On the logical operators of propositional logic, for instance, the interpropositional grouping models the operations transforming the 16 operators into one another. However, the operations do not correspond to rules of inference and do not necessarily preserve truth. They simply describe the intellectual activity transforming one operator into another without consideration of logical consequence. Whatever its true relationship to reasoning psycho-logic is therefore not synonymous with logic.
Piaget describes how adolescents reason using the operations of the grouping when attempting to grasp the connection between phenomena as follows:
Let us take as an example the implication [latex]p⊃q[/latex], and let us imagine an experimental situation in which a child between twelve and fifteen tries to understand the connections between phenomena which are not familiar to him but which he analyses by means of the new propositional operations rather than by trial and error. Let us suppose then that he observes a moving object that keeps starting and stopping and he notices that the stops seem to be accompanied by lighting of an electric bulb. The first hypothesis he will make is that the light is the cause (or an indication of the cause) of the stops, or [latex]p⊃q[/latex] (light implies stop). There is only one way to confirm the hypothesis, and that is to find out whether the bulb ever lights up without the object stopping, or [latex]pq̅[/latex] ([latex]pq̅[/latex] is the inverse of or negation of [latex]p⊃q[/latex]). But he may also wonder whether the light, instead of causing the stop, is caused by it, or [latex]p⊃q[/latex] (now the reciprocal and not the inverse of [latex]p⊃q[/latex]). To confirm [latex]p⊃q[/latex] (stop implies light), he looks for the opposite case which would disconfirm it; that is, does the object ever stop without the light going on? This case, [latex]p̅q[/latex], is the inverse of [latex]p⊃q[/latex]. The object stopping every time the light goes on is quite compatible with its sometimes stopping for some other reason. Similarly, [latex]pq̅[/latex], which is the inverse of [latex]p⊃q[/latex], is also the correlative of [latex]q⊃p[/latex]. If every time there is a stop the bulb lights up ([latex]q⊃p[/latex]), there can be lights without stops. Similarly, if [latex]q⊃p[/latex] is the reciprocal of [latex]p⊃q[/latex], then [latex]p̅q[/latex] is also the reciprocal of [latex]pq̅[/latex] (Inhelder and Piaget 1969, p. 139).
More generally, given any two observable phenomena represented by propositions [latex]p[/latex] and [latex]q[/latex], it is not immediately obvious how they are related. The relation between [latex]p[/latex] and [latex]q[/latex] can be determined by means of the four possible coincidences of these phenomena, which are represented by the conjunctions [latex]pq[/latex], [latex]p̅q[/latex], [latex]pq̅[/latex] and [latex]p̅q̅[/latex]. However, individually each observation does not allow the relationship between the phenomena to be determined unequivocally. Observation of [latex]p[/latex] and [latex]q[/latex] always occurring together, [latex]pq[/latex], for example, could mean that [latex]p[/latex]and [latex]q[/latex]are related in any of the 8 ways represented by the columns in Table 1 in which [latex]pq[/latex]occurs. Through observation of all four possible coincidences of the phenomena, on the other hand, the exact relationship between [latex]p[/latex]and [latex]q[/latex]can be determined unequivocally. Observation of [latex]pq[/latex] and [latex]p̅q̅[/latex] occurring without either [latex]p̅q[/latex] or [latex]pq̅[/latex] occurring, for example, means that the phenomena represented by [latex]p[/latex] and [latex]q[/latex] are equivalent; whereas observation of [latex]pq[/latex], [latex]p̅q[/latex], and [latex]pq̅[/latex] but no cases of [latex]p̅q̅[/latex] means that [latex]p∨q[/latex] (see Table 2). Conversely, if [latex]p⊃q[/latex] is postulated, [latex]pq[/latex], [latex]p̅q[/latex], and [latex]p̅q̅[/latex] are observations that would support the hypothesis, whereas [latex]pq̅[/latex] would falsify it. In short, the interpropositional grouping generates a framework of possible relations between phenomena in which the connection between actual phenomena can be determined rationally (Smith 1987, sec. Piaget’s Logic: A Constructivist Interpretation). Although observations are necessary, the rationale for judging the relationship between phenomena is derived from the framework of possible observations, which the interpropositional grouping generates. Thus, ‘reasoning is … the calculus embodied in propositional operations’ (Smith 1987, p. 344).
Logic and Reasoning
‘Logic is an essential tool for all sciences, but it is not a psychological theory of reasoning’ (Johnson-Laird 2006, p. 17) is the conclusion Johnson-Laird comes to after enumerating the many differences between logic and reasoning. After reviewing and assessing the achievements of the German psychological school of thought known as Denkpsychologie, Piaget concludes:
“Thought Psychology” finished by making thought the mirror of logic, and in this lies the root of the difficulties it has found insurmountable. The question is then to ascertain whether it would not be better simply to reverse the terms and make logic the mirror of thought” (Piaget 2001, p. 27).
Without explication, the mirror metaphor is perhaps obscure, but Piaget sheds light on it in subsequent elaborations:
“logic and the psychology of thought began by being confused and not differentiated at all; Aristotle no doubt thought he was writing a natural history of the mind (as well as of physical reality itself) by stating the laws of the syllogism. When psychology was set up as an independent science, psychologists came to understand (taking a considerable time over it) that the reflections contained in text-books of logic on the concept, judgment and reasoning did not exempt them from seeking to sort out the causal mechanism of intelligence. But as a residual effect of their original failure to draw a distinction, they still continued to think of logic as a science of reality, placed, in spite of its normative character, on the same plane as psychology, but concerned exclusively with “true thought” is [sic] opposed to thought in general, freed from all norms. Hence the deluded outlook of Denkpsychologie, according to which thought, a psychological fact, constitutes a reflection of logical laws (Piaget 2001, pp. 28–9).
In this quotation, Piaget clearly distinguishes between causal and normative sciences. The former treats thought as a psychological fact and is a science of reality concerned with the causal mechanisms of intelligence; the latter reflects on thought, its concepts, judgments and reasoning but only from the point of view of validity. Moreover, his criticism is levelled at the paucity of differentiation between causal and normative laws that misled thought psychologists to found a science of reality on a normative science. In short, logic is no more a psychological theory of reasoning for Piaget than it is for Johnson-Laird. In fact ‘logic is the mirror of thought, and not vice versa’ (Piaget 2001, p. 27) according to Piaget.
Additional Resources
Piaget, J. (1957). Logic and Psychology. (W. Mays & F. Whitehead, Trans.). New York: Basic Books Inc.
Piaget, J. (1977). The Stages of Intellectual Development in Childhood and Adolescence. In H. E. Guber & J. J. Vonèche (Eds.), H. E. Gruber (Trans.), The Essential Piaget (pp. 814–819). New York: Basic Books, Inc.
Piaget, J. (2001). The Psychology of Intelligence. (M. Piercy & D. E. Berlyne, Trans.). London; New York: Routledge.
Piaget, J., & Evans, R. I. (1973). Jean Piaget, the Man and his Ideas. (E. Duckworth, Trans.). New York: E P Dutton.
References
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Bond, T. G. (2005). Piaget and Measurement II: Empirical Validation of the Piagetian Model. In L. Smith (Ed.), Critical Readings on Piaget (ebook., pp. 178–208). Routledge.
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