1. Models as the representations of Boolean concepts Geoff Goodwin Princeton University

2. Boolean concepts Boolean combination (and, or, not) Examples: –tall and handsome –a person over the age of 18 and not a party to the case –variable that is random and normally distributed –DSM diagnostic criteria Many concepts go beyond Boolean combination (Rosch, Mervis, Medin): –chair, game, tall, table –table: worktop and something supporting it

3. Major Themes Structure, complexity What accounts for the difficulty of acquiring and communicating Boolean concepts? –Mental models –Task factors

4. Basic Task Instances Non-instances Concept: Triangle

5. Basic Task Instances Non-instances Concept: Red and large, or yellow and small

6. Difficulty of Boolean concept learning Shepard, Hovland & Jenkins (1961)

7. Simple concept

13. Difficulty of Boolean concept learning Shepard, Hovland & Jenkins (1961) I < II < III, IV, V < VI (Replicated by Nosofksy et al., 1994)

14. Feldman (2000), Nature Minimal Boolean complexity –Length of the shortest logical expression that captures the concept I < II < III, IV, V, < VI 1 4 6 6 6 10 II: (a and b) or (not a and not b) VI: (a and ((not b and c) or (b and not c))) or (not a and ((not b and not c) or (b and c))) Extended to larger battery

15. Problems with minimal complexity Ad hoc choice of connectives (and, or, not) –Include or else: I <II <IIIIVV <VI 124533 Wrong computations, not minimal! (Mathy & Bradmetz, 2004; Vigo, 2006) –Corrected: I <II <IIIIVV <VI 1445610 No psychological mechanism for forming minimal descriptions

16. Model theory of concepts Models: Possibilities consistent with assertion (Johnson-Laird, 1983, 2001, 2006) –Complete models vs. mental models Positive instances  mental models Eliminate irrelevant variables

17. Model theory of concepts Concept I: abcabc a b ¬c a ¬b c a ¬b ¬c

18. Model theory of concepts Concept I: abc a b ¬c a ¬b c a ¬b ¬c

19. Model theory of concepts Concept I: a

20. Model theory of concepts Concept I: a[+ b][+ c]

21. Model theory of concepts Concept I: Large yellow triangle Large red triangle Small yellow triangle Small red triangle

22. Model theory of concepts Concept I: Triangle

23. Basic principles: Recap Positive instances  models Eliminate irrelevant variables Number of models predicts difficulty

24. Initial observation SHJ concepts: I <II <IIIIVV <VI 122334 Not perfect, but better than minimal complexity

25. More formal observations Feldman’s battery (76 concepts) Correlations –Minimal complexity, r (74) = -.56 –Mental models, r (74) = -.75 p <.01 Regressions –Minimal complexity + parity, R 2 =.53 –Models dominant in regression, R 2 =.56 –Models explains unique variance Models more powerful and more parsimonious

26. Other factors in concept learning Communicability of concepts Categorization vs. description Description taps explicit overall representation of the concept

27. The switch task Acquire concept and then describe it Task (computerized): –3 switches control a light, according to Boolean function –Series of tests: adjust switches and “submit” –Goal: discover the rule which turns the light on

28. S1 S2 S3

31. Example: Switch 1 or else Switch 3 S1 S2 S3

32. Switch 1 or else Switch 3 S1 S2 S3

33. Switch 1 or else Switch 3 S1 S2 S3

34. Switch 1 or else Switch 3 S1 S2 S3

35. Switch 1 or else Switch 3 S1 S2 S3

36. Switch 1 or else Switch 3 S1 S2 S3

37. Switch 1 or else Switch 3 S1 S2 S3

38. Switch 1 or else Switch 3 S1 S2 S3

39. Switch 1 or else Switch 3 S1 S2 S3

40. Switch 1 or else Switch 3 S1 S2 S3

41. Switch 1 or else Switch 3 S1 S2 S3

42. Description tasks Explicit overall representation What predicts accuracy of descriptions? Models? Other factors? How are descriptions formulated? –Boolean formulae, models?

43. Experiment 1 28 participants 2 switches 1 connective 3 switches 2 connectives 3 switches 3 connectives A and not B (1 model) A and (not B or else C) (2 models) (A or else not B) and (A or else C) (2 models) A or else not B (2 models) A or else (not B and C) (3 models) (A or B) and (not B or C) (2 models)

44. Accuracy Models 1 model96% 2 models78% 3 models39% p <.0001, η 2 =.57 Minimal complexity 1 connective89% 2 connectives59% 3 connectives75% p <.06, η 2 =.13 Number of switches (relational complexity) 2 switches89% 3 switches67% p <.01, η 2 =.29

45. Descriptions Minimal logical descriptions, 4% –“S1 and not S2” Models, 72% –Complete lists of instances –More succinct models Relations and quantifiers, 19% –“The light will go on when switches one and two are in the same position” –“Switch three must be up by itself, or all switches must be up for the light to come on” Conditional expressions, 4%

46. Summary: Experiment 1 Models predict accuracy of descriptions Relational complexity How people formulate descriptions: –Models not minimal Boolean descriptions –Relations and quantifiers

47. Experiment 2 Replication and extension to more problems More complete test of what causes difficulty

48. Experiment 2: Problem Set 28 participants 2 switches 1 connective 3 switches 2 connectives 3 switches 3 connectives A and B (1 model) A and (B or else C) (2 models) (A or else B) and (A or else C) (2 models) A or else B (2 models) A or else (B and C) (3 models) (A or not B) and (B or C) (2 models) A or B (2 models) A or (B or else C) (3 models) (A or else B) or (A or else C) (4 models)

49. Preliminary observation 2 switches 1 connective 3 switches 2 connectives 3 switches 3 connectives A and B (1 model) A and (B or else C) (2 models) (A or else B) and (A or else C) (2 models) A or else B (2 models) A or else (B and C) (3 models) (A or not B) and (B or C) (2 models) A or B (2 models) A or (B or else C) (3 models) (A or else B) or (A or else C) (4 models) 100%

50. Accuracy Models 1 model100% 2 models82% 3 models65% p <.001, η 2 =.45 Minimal complexity 1 connective93% 2 connectives75% 3 connectives67% p <.01, η 2 =.32 Number of switches (relational complexity) 2 switches93% 3 switches71% p <.001, η 2 =.39

51. Descriptions Minimal logical descriptions, 19% –“S1 and S2” Models, 39% –Complete lists of instances –Succinct models Relations and quantifiers, 30% –“The light bulb will be on if both switches 1 and 2 are the same and are different from 3” –“When only two switches exactly are on the light comes on”

52. Summary: Experiment 2 Models predominant: accuracy and descriptions Broad use of relations and quantifiers (30%): affects accuracy –Upset predictions of all theories!

53. General Conclusions Difficulty of Boolean concept learning (and, or, not) not resolved, despite recent theories (Feldman, 2000) Minimal complexity theory has major errors: –ad hoc logical language –wrong computations! –no plausible psychological mechanisms

54. General Conclusions Mental model theory can explain Boolean concept acquisition Models minimized by eliminating irrelevant variables Best predictive success

55. General Conclusions Model theory explains descriptions reasonably well Also relational complexity Relations and quantifiers –beyond Boolean algebra

56. General Conclusions Relations and quantifiers affect accuracy: –aberrant problem –Other experiments: contrast with standard SHJ result Broadly compatible with model theory (Bucciarelli & Johnson-Laird, 1999; Goodwin & Johnson- Laird, 2005). Integration

57. Thanks! Phil Johnson-Laird Adam Alter, Greg Detre, Sam Glucksberg, Adele Goldberg, Dena Gromet, Sunny Khemlani, Louis Lee