Piaget's logical formalism for formal operations: An interpretation in context

Two interpretations of the equations used by B. Inhelder and J. Piaget in The growth of logical thinking from childhood to adolescence (London: Routledge & Kegan Paul, 1958) are discussed, with implication as the central example. The expression (p ? q) (p implies q) is said to be equal to $\text{(p · q)}\text{V}\text{(}\text{p}\text{· q)}\text{V}\text{(}\text{p}\text{·}\text{q}\text{)}$ (i.e.,p and q, or not p and q, or neither p nor q). According to one, (p ? q) asserts the existence of each case mentioned. According to the other, (p ? q) only asserts that current knowledge allows the possibility of each of the cases. Neither interpretation makes sense of all the relevant passages. Both can be combined in a consistent interpretation when attention is paid to the functional context of the subject's use of logical operations in this book: the “operations” describe the knowledge states which the subject can differentiate and relate, on his way to solving Inhelder's tasks. The logical notation used to represent these states is not a representational format attributed to the subject and manipulated by his cognitive processes, but part of a structuralist account of the subject's reasoning capacities.