Inductive generalization is ubiquitous in human cognition; however, the factors that support this ability early in development remain controversial.
What is Inductive Reasoning?
Inductive reasoning involves making generalizations from instances. It is a powerful and effective tool to generate new knowledge. Consider this example: When told a novel fact about alligators (eg, "crocodile embryos lack sex chromosomes"), most adults correctly conclude that crocodile embryos also lack sex chromosomes. Therefore, making an inductive inference on the basis of what is known creates new knowledge. This mode of inference is not guaranteed to generate correct knowledge (a fact about crocodile embryos could be incorrectly overgeneralized to all oviparous animals).
How is induction born in the human mind?
Induction in children
In particular, it is not clear whether young children use object-type information in the course of induction and what role linguistic labels play in this process. In recent years, the debate on the development of category-based induction has been dominated by two alternative perspectives: an explanation of naive theory (Gelman and Markman, 1986; Markman, 1990) and an explanation based on similarity (Sloutsky and Fisher, 2004, p. 2012). Two fundamental differences between these explanations are:
(1) if category-based induction arises gradually in the course of development or if children are initially predisposed to depend on object-type information in the course of induction and
(2) whether linguistic labels contribute to inference induction by providing information about the object type or by increasing the overlap of features between the presented entities.
How Adults Conduct Induction
In the absence of perceptual information to guide inferences, it has been hypothesized that adults make inferences based on their knowledge about the type of object: objects that belong to the same or related categories are likely to have many properties in common (Rips, 1975; Osherson et al., 1990; Kemp and Tenenbaum, 2008; Hayes et al., 2010; Murphy and Ross, 2010). In other words, adults are said to use category-based induction to generalize from the known to the unknown. Despite general agreement that category-based induction is a ubiquitous component of mature cognition (although see Sloman, 1993), there is little agreement on the origins of the development of this ability.
The Naive Theory
According to the naive theory approach, from very early in development, people first identify the category membership of the items under consideration and then generalize a known property to items of the same type: “At 2½ years, children expect categories to promote rich inductive inferences ... and may overlook contradictory perceptual appearances in doing so ”(Gelman & Coley, 1990, p. 802).
Furthermore, it has been suggested that the ability to make category-based inferences is not a product of development and learning. Instead, children are "initially biased" to recognize that labels denote categories and make inferences on the basis of shared labels, and therefore membership in shared categories (Gelman & Markman, 1986, p. 207), a idea that has had a great influence on literature. (for example, Keil, 1989; Gelman and Coley, 1990; Booth and Waxman, 2002; Kalish, 2006; Jaswal and Markman, 2007).
Thus, as explained by the naive theory, from a very young age children are expected to perform category-based inductions even if the perceptual information conflicts with the category membership information. As a result, one would expect even young children to perform relatively well on simple initiation tasks when category membership information is available. Furthermore, any observed improvement in performance with age is believed to be due to a reduction in statistical noise rather than changes in induction mechanisms.
In contrast to the naive theory approach, Sloutsky and Fisher (2004) proposed a similarity-based explanation called SINC (for Similarity, Induction, Naming, and Categorization). According to SINC, children make inferences based on the general similarity of the presented entities calculated on all the characteristics of the perceived object. Within this approach, tags are viewed as characteristics of the object (rather than category markers) that contribute to overall perceived similarity.
Therefore, according to SINC, an inference can be based on labels without necessarily being based on categories. Several findings suggest that children rely primarily on the general perceptual similarity of objects (but not category membership information) to make inferences well beyond the preschool years, possibly up to 7-9 years of age ( Fisher and Sloutsky, 2005; Sloutsky et al., 2007; Badger and Shapiro, 2012; Sloutsky and Fisher, 2012). In other words, in contrast to the notion of initial competence advocated by proponents of the naive theory explanation, proponents of SINC suggest that category-based induction is not a developmental defect, but rather that category-based induction follows a long developmental course.
Gelman and Markman's Seminal Study
Perhaps the strongest evidence for the naive theory account comes from the seminal study of Gelman and Markman (1986). In this study, researchers asked preschool-age children and college students to make inferences about natural-type objects when perceptual information was ambiguous or in conflict with category membership (cf. Sloutsky and Fisher, 2004).
Tags were used to communicate information about categories. For example, participants were asked whether a target item (eg, a brown squirrel) shared a non-obvious property with the test item that was designed to resemble the target (eg, a brown rabbit) or with the test item that was designed to look different from the target but belonged to the same category (for example, a gray squirrel; but see Sloutsky and Fisher, 2004 for divergent arguments and data on calibrating perceptual similarity in this study) .
The overall rate of category matching options was above chance for both preschool-age children and college students. These findings were taken as evidence that even young children have the belief (or a naive theory) that natural-type objects share a number of unobservable properties if they belong to the same category and make inductive inferences based on this belief. Later studies reported similar findings in younger children and even infants (eg, Gelman and Coley, 1990; Graham et al., 2004).
Basic Principle of Inductive Generalization
The basic principle of inductive generalization is that what you get from known instances can be generalized to all. Its best-known form is the venerable simple enumeration induction or, more briefly, enumerative induction. We know that some A's are B's; from it we infer that all A's are B. In many treatises on logic, even in the last century, "induction" simply meant induction by simple enumeration.
One difficulty with this form of inductive inference is its limited scope. If the evidence is not presented as the sentence "A is B", we cannot continue; if the result to be supported is not of the form "All A are B". we cannot continue. There have been many attempts to expand the scope of this form of inductive inference. These attempts have generated a family of forms of inductive inference that I call "inductive generalization." The principle that governs this family is the notion that what is obtained from known cases can be generalized to all.
Therefore, we are licensed to infer from instances to their generalizations. What generates the family is the use of more expansive ways of characterizing and describing instances and their generalizations. Mill's methods allow us to talk about causes. If we have many cases of A followed by B, we are licensed to infer not only that A is always followed by B, but A causes B
Hempel's satisfaction criterion is based on the much greater expressive power of first-order predicate logic. We can consider phrases like "If someone is a god or descends from a god, then that someone is immortal." Hempel offered an accurate description of what it is like to be an example of such a universal prayer. In one respect, however, Mill's methods outperformed Hempel's. Because Mill's methods allow an instance like "These A are followed by B", to confirm a generalization that A causes B The crucial novelty is the new term "cause" that did not appear in the instance sentence.
Glymour's bootstrap approach to commit is an attempt to repair this flaw. His account allows us to use sentences from the theory under investigation as an interpretive device to introduce new theoretical vocabulary into sentences connected by confirming relations. Demonstrative induction makes this interpretive use of theoretical sentences the full content of induction. Its inductions become "demonstrative", that is, deductive.
In all these forms of inductive inference, a single problem remains unsolved. Sometimes an instance provides strong evidentiary support for a hypothesis. A single instance may be enough to establish it. At other times, many instances may provide only limited support. While we may have strong intuitions about which case is which, we somehow just know - that knowledge does not come from explanations of inductive generalization. Two inductive generalizations that seem essentially similar in form can provide very different supporting strengths; nothing in the stories that will follow gives us a systematic way of distinguishing.