The Psychology of Induction

Concept Learning and Concept Formation

Example 1: I point to things saying either ‘oogle’ or ‘aagle’ each time. What does ‘oogle’ and ‘aagle’ mean? This is an example in which you must learn a concept.

You assume that these terms refer to the attributes of the objects to which I am pointing, like ‘metal’, ‘wooden’, ‘big’, ‘small’, ‘brown’, ‘rectangular’, and so on. In fact, there is no such connection because I say ‘oogle’ if my finger points up, and ‘oogle’ if it points down. This is an example in which not all associations are equally learnable.

Example 2: Mayer (1992) p. 83. You will be given a series of stimuli, individually, with each item varying in shape (circle or square), size (large or small), color (red or green), and number (one or two). Out of all possible stimuli (there are 16 of them), some are in a target group and some are not. You will be shown an item, and you have to predict whether it is in the group (initially it will just be a guess). Then I will give you the correct answer before going on to the next item. Here are 8 training examples:

1

2

3

4

5

6

7

8

1 red large square

1 green large square

2 red small squares

2 red large circles

1 green large circle

1 red small circle

1 green small circle

1 red small square

NO

NO

YES

NO

NO

YES

YES

YES

Conjecture: The group consists of all and only small objects (either 1 or 2, and either red or green).

Here are the 8 test examples:

9

10

11

12

13

14

15

16

2 green large squares

1 red large circle

2 green small circles

2 red small circles

2 green large circles

2 green small squares

2 red large squares

1 green small circle

NO

NO

YES

YES

NO

YES

NO

YES

Question: Have we proved that our conjecture is right? Yes, assuming that the group membership does not change over time.

Note: Example 2 is harder than simple enumerative induction because the associations between ‘yes’ and ‘small’ are mixed in with accidental associations between ‘yes’ and other attributes. Those accidental associations are commonly referred to as noise.

Example 3: Here is a variation on the previous example. 8 training cases:

1

2

3

4

5

6

7

8

1 red large square

1 green large square

2 red small squares

2 red large circles

1 green large circle

1 red small circle

1 green small circle

1 red small square

YES

NO

YES

YES

NO

NO

NO

NO

Conjecture: The Much-Red hypothesis: 2 red objects of any size or 1 large red object.

8 test examples:

9

10

11

12

13

14

15

16

2 green large squares

1 red large circle

2 green small circles

2 red small circles

2 green large circles

2 green small squares

2 red large squares

1 green small circle

YES

YES

NO

YES

YES

NO

YES

NO

Discussion: The conjecture is proved false in example 3. The group actually consists of much-red objects plus very-much-green objects (2 large green things). Note that shape proved to be irrelevant to group membership. It’s as if the total ‘intensity’ of ‘redness’ has to be higher than a certain threshold in order to gain group membership, and while green objects have a certain degree of ‘redness’, they don’t have enough unless there are two large green things present.

Scientific Analogy:

Concept Learning versus Concept Formation

Continuity and Noncontinuity Theories of Learning

Parallel to the distinction between concept learning and concept formation, we have different theories about how inductive learning takes place.

Simple Bayesianism

Note: In examples 2 and 3, there are 216 or 6,5536 possible hypotheses concerning membership in the target group!

Simple Updating:

Question: Does simple Bayesianism involve continuous or noncontinuous learning?

Answer:

  1. Either the list of hypotheses with nonzero probabilities includes the true hypothesis, or it does not.
  2. If it does, then the procedure will converge to the true hypothesis without jumps. But if it does not, then every hypothesis will be refuted eventually, which will force a Bayesian learner to jump to a new set of hypotheses.
  3. While this move is noncontinuous, it is not Bayesian.
  4. Therefore Bayesian learning is continuous.

Remark: In all the recently discussed learning tasks, the observations have been fed to the learner. But in science, the learner has a great deal of control over what observations are made (though not over what the outcomes are). This adds another element in the psychological problem of induction—what strategies do inductive learners use in making such choices and what effect to they have effects do they have on learning rates? (See Mayer 1994, around p. 93).

What Makes Learning Hard?

  1. Learning by simple enumerative induction is very easy because only the observational premises relevant to the conclusion are listed as premises. In all the examples here, there is added noise, and the learner has to learn to distinguish between the two (e.g., is shape relevant to target group membership, is color relevant, and so on).
  2. Concept formation is harder then concept learning which does not involve the invention of new concepts. Simple enumerative induction is the easiest of all because there is no need to identify any concept at all!

A look ahead: The kind of inductive reasoning typically discussed by psychologists, which involves the invention of a new concept, is quite different from simple enumerative induction, which does not involve the learning of new concepts. The great influence of Hume and Mill has tended to steer philosophers away from this phenomenon, although William Whewell is one notable exception here.

What’s next?

We have looked at inductive inference according to philosophers, and we have looked at inductive inference according to logicians, and we have looked at inductive inference according to psychologists. Next, we look at inductive inference in science according to historians of science.