1. Conditionals and Non- constructive reasoning David Over Department of Psychology University of Durham david.over@durham.ac.uk

2. A great influence F. P. Ramsey (1903-1930)

3. The THOG problem Four figures: a black square, a black circle, a white square, a white circle. Peter Wason was thinking of one colour and one shape. A THOG is a figure with the colour or the shape but not both. The black square is a THOG. Thus it is so called because it is white or a square (but not both) or black or a circle (but not both). Which is it?

4. Non-constructive disjunctive reasoning Modified example from Levesque (1986) and Toplak & Stanovich (2002): Jack is looking at Ann but Ann is looking at George. Jack is a cheater but George is not. Is a cheater looking at non-cheater? A) Yes B) No C) Cannot tell

5. The non-constructive aspect Jack is looking at Ann but Ann is looking at George. Jack is a cheater but George is not. Is a cheater looking at non-cheater? Ann is either a cheater or not. If she is, then a cheater (Ann) is looking at a non- cheater (George). If she is not, then a cheater (Jack) is looking at a non-cheater (Ann). Therefore, the answer is “Yes”.

6. A constructive approach Jack is looking at Ann but Ann is looking at George. Jack is a cheater but George is not. Is a cheater looking at non-cheater? We get hold of Ann and try to cooperate with her in reciprocal altruism. She does not cooperate. Our “cheater detection mechanism” fires, and we conclude she is a cheater. Therefore, the answer is “Yes”.

7. The distinction and dual process theory Non-constructive inference is the purest example of an analytic process in System 2. It is an inference from “above”, using logic. Constructive inference is from “below”: it is grounded in heuristic processes of System 1, such as those of perception.

8. Another example of non-constructive inference Given a “sample” of 100 people, we infer that everyone in it has some disease d or does not have d. We assume that a number x have d and so that 100 - x do not have d, and the same with a symptom s. We can then infer the proportion of people with d out of those with s. All this thought is “from above” and has no relation to anything properly called “natural sampling”. It cannot, therefore, give support to the claims of some evolutionary psychologists.

9. Non-constructive inference is not always difficult It is difficult to answer the question about Ann, but not to follow the step by step process of inferring the answer when it is given to us. It is difficult to derive Bayes’s theorem for oneself using non- constructive inference, but not to follow its step by step application given in so- called “natural sampling”.

10. Yet another example of non- constructive inference Johnson-Laird & Byrne (2002, p. 650) hold that the following inference is valid: In a hand of cards, there is an ace or a king or both. So if there isn’t an ace in the hand, then there is a king. Now if the above were valid, then the ordinary conditional would be the material conditional, which Johnson-Laird & Byrne deny, but this inference is acceptable only if the disjunction is inferred “from above”.

11. The form of the inference referred to by Johnson-Laird & Byrne (2002) Inferring “if not-p then q” from “p or q”. From “not-p or q”, we get “if not-not-p then q” by the form, from which we infer by double negation “if p then q”. So one could equally well study inferring “if p then q” from “not-p or q”.

12. More logical points For all conditionals we must have that “if p then q” logically implies “not-p or q” But only for the material conditional, can the converse hold, as the material conditional just means “not-p or q”.

13. If Johnson-Laird & Byrne (2002) are right Then “if p then q” is logically equivalent to the truth function material conditional, “not-p or q”. And those of us who deny the equivalence are wrong. But wait! Johnson-Laird & Byrne (2002) themselves deny the equivalence. So how do we make their theory consistent?

14. The inference “from below” We see that there is an ace in the hand. So in the hand, there is an ace or a king or both. And so if there isn’t an ace in the hand, there is a king. This is one of the paradoxes of the material conditional and is obviously invalid for the ordinary conditional in natural language, even a “basic” one. If it were not, the ordinary conditional would be the material conditional.

15. The Ramsey test Ramsey (1931) suggested that people could judge “if p then q” by “...adding p hypothetically to their stock of knowledge …” They would thus fix '...their degrees of belief in q given p…”, which is their subjective conditional probability of q given p, P(q/p).

16. What the Ramsey test implies ( Over, Hadjichristidis, Evans, Handley, & Sloman, in press) The probability of an indicative conditional, P(p > q), is the subjective conditional probability, P(q/p).

17. P(q/p) high implies high P(not-p or q) Suppose we find that P(q/p) is high. Then we will find that P(not-p or q), the material conditional, is high. P(not-p or q) = P(not-p) + P(q) - P(not-p & q) = P(not-p) + P(q/p) - P(not-p)P(q/p)

18. P(not-p or q) high does not imply high P(q/p) Suppose we find that P(not-p) is high. Thus P(not-p or q) is high, but recall: P(not-p or q) = P(not-p) + P(q/p) - P(not-p)P(q/p) And that means that P(q/p) can be low when P(not-p or q) is high.

19. Validity and strength Inferring “if not-p then q” from “p or q”is not logically valid, as P(p or q) can be higher than P(q/not-p). However, the inference can be a strong probabilistic inference.

20. Constructive example We think that we see Ann going into the library. We infer with high confidence that Ann is in the library or the computer lab. But we could not infer from this that, if she is not in the library, then she is in the computer lab.

21. Constructive details P(library & lab) = 0 P(library & not-lab) =.9 P(not-library & lab) =.01 P(not-library & not-lab) =.09 P(library or lab) =.91 P(lab/not-library) =.01/.1 =.1

22. Non-constructive example We infer from reading the module guide that everyone in the class is in the library or the lab. Ann is in the class. So Ann is in the library or the lab. And so, if Ann is not in the library, then she is in the lab.

23. Non-constructive details 1 P(library & lab) = 0 P(library & not-lab) =.5 P(not-library & lab) =.5 P(not-library & not-lab) = 0 P(library or lab) = 1 P(lab/not-library) =.5/.5 = 1

24. Non-constructive details 1: Stalnaker conditional P(library & lab) = 0 P(library & not-lab) =.5 P(not-library & lab) =.5 P(not-library & not-lab) = 0 P(library or lab) = 1 P(if not-library S-then lab) = 1

25. Non-constructive details 2 P(library & lab) = 0 P(library & not-lab) =.45 P(not-library & lab) =.45 P(not-library & not-lab) =.1 P(library or lab) =.9 P(lab/not-library) =.45/.55 =.81 P(if not-library S-then lab) =.9

26. Non-constructive details 3 P(library & lab) = 0 P(library & not-lab) =.7 P(not-library & lab) =.2 P(not-library & not-lab) =.1 P(library or lab) =.9 P(lab/not-library) =.2/.3 =.66

27. Non-constructive details 4 P(library & lab) = 0 P(library & not-lab) =.8 P(not-library & lab) =.1 P(not-library & not-lab) =.1 P(library or lab) =.9 P(lab/not-library) =.1/.2 =.5

28. Example from Evans & Over, 2004, Ch. 7 Ann has anaemia or is pregnant. If Ann does not have anaemia, then she is pregnant.

29. Counterfactual version Ann had anaemia or she was pregnant. If Ann had not had anaemia, then she would have been pregnant.

30. What the last two examples seem to imply P(if Ann does not have anaemia, then she will be pregnant) was high. P(if Ann had not had anaemia, then she would have been pregnant) is low.

31. Over, Hadjichristidis, Evans, Handley, & Sloman (in press) appear to show: The subjective probability of a past tense counterfactual, P(p >> q), is the subjective probability of an indicative conditional, P(p > q), at an earlier time, which is P(q/p) at that time.

32. The experiment in more detail Compare: If Turkey improves its human rights practices in 2000, then it will be awarded membership of the EU. If Turkey had improved its human rights practices in 2000, then it would have been awarded membership of the EU.

33. The probability of counterfactuals task 26 participants at Plymouth University were asked the following question. What is the probability of the truth of: “If Turkey had improved its human rights practices in 2000, then Turkey would have been awarded membership of the EU.”

34. The truth table task What was the probability of the following from a perspective of 5 years ago? Turkey will improve its human rights practices and will be admitted to the EU. Turkey will improve its human rights practices and will not be admitted to the EU. Turkey will not improve its human rights practices and will be admitted to the EU.. Turkey will not improve its human rights practices and will not be admitted to the EU.

35. Analyzing P(p > q) and P(p >> q) We performed multiple regression analyses on P(p > q) and P(p >> q) using P(p), P(q/p), and P(q/not-p) as predictors.

36. The results for items Regressions across conditionals. Cells = beta weights EXP1 (indicatives) EXP2 (indicatives) EXP3 (counterf.) TrueFalse P(p).05.00.14*.06 P(q/p).90*-.93*.93*.87*

37. The results for participants Regressions for individual participants. Cells = beta weights EXP1 (indicatives) EXP2 (indicatives) EXP3 (counterf.) TrueFalse P(p).02.16*-.04 P(q/p).42*-.38*.51*.42*

38. Summary of results P(q/p) was by far the strongest predictor. There was a smaller negative effect of P(q/not-p). This could suggest a relation to the delta-p rule, which takes P(q/p) - P(q/not-p) to measure the degree of covariation between p and q. Does this mean that a counterfactual states a causal relation between p and q?

39. The apparent conclusion The causal conditional becomes the past tense counterfactual when time passes and we learn that Turkey did not improve its human rights practices and was not admitted into the EU.

40. More precisely The subjective probability of a past tense counterfactual, P(p >> q), is the subjective probability of an indicative conditional, P(p > q), at an earlier time, which is P(q/p) at that time.

41. The extended Ramsey test When you disbelieve p, judge the probability of “if p then q” by adding p hypothetically to what you believe, making minimal changes to preserve consistency. Then determine your degree of belief in q given p.

42. The minimal changes Our results imply that the minimal changes for judging a past tense counterfactual P(p >> q) should start by going back in time to an earlier mental state and then judging the subjective probability of the relevant indicative conditional, P(p > q), at that time.

43. Recall the problematic case P(if Ann does not have anaemia, then she will be pregnant) was high. P(if Ann had not had anaemia, then she would have been pregnant) is low.

44. For the doctor in the past: TT P(a & p) = 0 TF P(a & not-p) =.45 FT P(not-a & p) =.45 FF P(not-a & not-p) =.1 P(a or p) =.9 P(p/not-a) =.45/.55 =.81

45. For us looking back: TF P(a & not-p) =.45 FT P(not-a & p) =.45 FF P(not-a & not-p) =.1 P(a or p) =.9 P(not-a > p) = high P(not-a >> p) = low P(not-a >> not-p) = high

46. What we know with hindsight: TF P(a & not-p) =.45 FT P(not-a & p) =.45 FF P(not-a & not-p) =.1 P(a or p) =.9 P(not-a > p) = high P(not-a >> p) = low P(not-a >> not-p) = high

47. TF is the actual state of affairs but which not-a state is closer? TF (a & not-p) Is it FT? (not-a & p) Or FF ? (not-a & not-p)

48. Surely FF is closer to TF TF (a & not-p) FF (not-a & not-p) FT (not-a & p)

49. Suppose (not-d >> p) is the Stalnaker conditional: TF (a & not-p) FF (not-a & not-p) So (if not-a S-then not-p) holds

50. Surely the probabilities should change as well TF P(a & not-p) =.45 FF P(not-a & not-p) =.1? FT P(not-a & p) =.45?

51. Surely we have: TF P(a & not-p) =.45 FF P(not-a & not-p) =.45 P(if not-a S-then not-p) =.9

52. Counterfactuals and constructive inference We get the constructive knowledge over time that Ann has anaemia, and this knowledge of the actual facts affects our evaluation of the counterfactual. Hindsight “bias” and counterfactuals?

53. Constructive knowledge and closeness We get the constructive knowledge over time that Ann has anaemia, and this knowledge of the actual facts affects our evaluation of the closeness of possibilities.

54. Constructive knowledge and intervention Intervening to cure the anaemia - this intervention would be constructive. It leads to the “closest” state, but would not preserve the truth of “anaemia or pregnant”.

55. Intervention and probability Need to understand the relation between intervention, closeness, and probability. Relevant work: Pearl (2000) on the “do” operator and Teigen (2005) on the proximity heuristic.

56. Controllable actions After Kahneman & Miller (1986), there has been much research in JDM and social psychology on closeness. This finds that the results of controllable actions are judged to be “close”.

57. Closeness distinctions Mr C is 5 minutes late for his flight and Mr D is 30 minutes late for his flight. If Mr C had been 5 minutes earlier, he would have made his flight. If Mr D had been 30 minutes earlier, he would have made his flight.

58. What our results imply about closeness: Mr C is 5 minutes behind time and Mr D is 30 minutes behind time. If Mr C saves 5 minutes, he will catch his flight. If Mr D saves 30 minutes, he will catch his flight.

59. The deontic case: Over, Manktelow, & Hadjicristidis (2004) Should Ann be treated for anaemia? If Ann is not-pregnant, then she has anaemia. P(anaemia/not-pregnant) is high. Not treating anaemia is a serious health cost. Therefore, If Ann is not pregnant, then she should be treated for anaemia.

60. The deontic account in a non- constructive context Ann is pregnant or has anaemia. If she is not-pregnant, then she has anaemia, so P(anaemia/not-pregnant) is high. But if she were not pregnant, then she would have not have anaemia. True If she is not pregnant, then she should not be treated for anaemia.

61. Conclusion: Some of the ideas I have supported are not quite right in the non-constructive case, but I hope the people I work with will correct them for me.