In contrast, just like in the case of addition or subtraction transformations, they would make no specific prediction as to whether this number word or another number word applies, if one or more individual members of the set are replaced by other individuals – unless the pragmatics of the task leads them to the correct answer. This mTOR phosphorylation explanation in terms
of set identity predicts children’s failure at the one-to-one comparison task, which was left unexplained in Brooks et al.’s (2012) account. Indeed, in both the one-to-one comparison task and the single-set transformation task, children must choose between a previously-heard label and a new label, thus in terms of pragmatics the two tasks are equivalent. In terms of quantities involved, the two tasks are equivalent too. Therefore, if children reason in terms of quantity, they should succeed in
the comparison task when the two sets are equal in number, just as they succeed in the single-set task when no transformation is applied. If however children reason in terms of set identity, then in the one-to-one comparison task there is no reason why information about one set should help them RNA Synthesis inhibitor solve a question about another set. To get a better understanding of this interpretation, think of first names, which are defined in terms of identity. If a set is called “five” and is put in exact one-to-one correspondence with another set, we predict that children are undecided as to whether this second set should be called “five” like the other set. Nevertheless, children should know that if the members of a set called “five” remain in the set, and no new item is added, then the set is still called “five”.5 Interpreting children’s usage of the number words in terms of set identity makes an important prediction. In the published versions of the single-set transformation task
(Brooks et al., 2012 and Sarnecka and Gelman, 2004), the transformation leaving numerosity constant left the identity of the set PIK3C2G unchanged as well. Under our interpretation, subset-knowers should not choose to conserve the initial number word for an identity-changing substitution transformation, even though the cardinal value of the set remains constant in this condition. At 5 years of age, children have clearly overcome the limitations of their understanding of numerical equality, since they know how set transformations impact number words, even for number words that fall beyond their counting range, and even for substitution transformations that keep number constant while altering the identity of a set’s members (Lipton & Spelke, 2006).
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