1. Henrik Singmann Karl Christoph Klauer David Over
Testing the Empirical Adequacy of Coherence as a Norm for Conditional Inferences
Henrik Singmann
Karl Christoph Klauer
David Over

2. New Paradigm Psychology of Reasoning
Normative System:
Subjective Probability Theory (de Finetti 1936, 1937/1964; Ramsey, 1931/1990)
Logic of Probability (Adams, 1998; Gilio, 2002; Gilio & Over, 2012)
Bayesianism (Oaksford & Chater, 2007)
Everyday Conditionals:
Draw on background knowledge
(At least weak) causal connection between antecedent and consequent
If oil prices continue to rise then German petrol prices will rise.
The Equation: P(q|p) = P(if p then q)
removes "Paradoxes" (e.g., Pfeifer, 2013)

3. 4 Conditional Inferences
Modus Ponens (MP):
If p then q. p
Conclusion: q
Affirmation of the consequent (AC):
If p then q. q
Conclusion: p
Modus Tollens (MT):
If p then q.
Not q
Conclusion: Not p
Denial of the antecedent (DA):
If p then q.
Not p
Conclusion: Not q

4. Normative Standards for Conditonal Inferences
p-validity (Adams, 1998)
p-valid inferences (MP and MT) confidence preserving: conclusion cannot be more uncertain than premises
Uncertainty: U(p) = 1 – P(p)
No restriction for AC and DA
e.g. MP: U(q) < U(if p then q) + U(p)
coherence (Mental Probability Logic; Pfeifer & Kleiter, 2005, 2010)
Inferences probabilistically coherent (i.e., drawing inference does not expose to dutch book)
If not all probabilities are specified, mental probability logic predicts coherence intervals (assuming unspecified probabilites in [0, 1])

5. Normative Standards for Conditonal Inferences
p-validity (Adams, 1998)
p-valid inferences (MP and MT) confidence preserving: conclusion cannot be more uncertain than premises
Uncertainty: U(p) = 1 – P(p)
No restriction for AC and DA
e.g. MP: U(q) < U(if p then q) + U(p)
coherence (Mental Probability Logic; Pfeifer & Kleiter, 2005, 2010)
Inferences probabilistically coherent (i.e., drawing inference does not expose to dutch book)
If not all probabilities are specified, mental probability logic predicts coherence intervals (assuming unspecified probabilites in [0, 1])

6. Mental Probability Logic: MP
if p then q P(q|p) p P(p) q P(q) ?
Law of total probability: P(q) = P(q|p)P(p) + P(q|¬p)(1 − P(p))
Setting P(q|¬p) to 0 and 1: P(q) = [ P(q|p)P(p) , P(q|p)P(p) + (1 − P(p)) ]

7. Coherence Intervals
Intervals for all inferences (Pfeifer & Kleiter, 2005):

8. Overview Goal: Assess empirical adequacy of coherence.
Fully probabilized task (i.e., all premises uncertain): probabilized conditional reasoning task
Only highly believable conditionals (Evans et al., 2010).
Participants provide all required estimates directly and independently.

9. Exp 1
N = 30
16 highly believable conditionals (13 from Evans et al., 2010):
If car ownership increases then traffic congestion will get worse.
If jungle deforestation continues then Gorillas will become extinct.
If the cost of fruit and vegetables is subsidised then people will eat more healthily.
Participants work on 4 randomly selected conditionals.
For each conditional participant work on 1 inference (MP, MT, AC, or DA).
Singmann, Klauer, Over (2014). New Normative Standards of Conditional Reasoning and the Dual-Source Model. Frontiers in Psychology.

10. Procedure I
If car ownership increases then traffic congestion will get worse. In your opinion, how probable is the above statement/assertion?
Car ownership increases. In your opinion, how probable is it that the above event occurs? X X

11. Procedure II
If car ownership increases then traffic congestion will get worse. (Probability 80%) Car ownership increases. (Probability 95%) Under these premises, how probable is that traffic congestion will get worse? X

12. Procedure II
If car ownership increases then traffic congestion will get worse. (Probability 80%) Car ownership increases. (Probability 95%) Under these premises, how probable is that traffic congestion will get worse? X

13. Procedure III in random order:
P(q|p) : How probable is that traffic congestion will get worse should car ownership increase?
P(p ∧ q): Car ownership increases and traffic congestion will get worse. In your opinion, how probable is it that the above event occurs?
P(¬p ∨ q): Car ownership does NOT increase or traffic congestion will get worse. In your opinion, how probable is it that the above event occurs?
P(q|¬p): How probable is that traffic congestion will get worse should car ownership NOT increase?
P(q): Traffic congestion will get worse. In your opinion, how probable is it that the above event occurs?

14. Procedure Exp I P(if p then q)? P(minor premise)?
P(MP / MT / AC / DA)?
P(…)?
show previous response

15. Procedure Exp I ×4 P(if p then q)? P(minor premise)?
P(MP / MT / AC / DA)?
P(…)? ×4
show previous response

16. Coherence

17. Coherence: Chance CorrectionMP MT AC DA
87% / 45%
63% / 65%
60% / 58%
60% / 46%
Percentage of coherent responses / coherent responses predicted by chance
Only for MP and DA evidence for above chance coherence.
LMM on difference between coherence (0/1) and interval size:
significant intercept: F(1, 16.07) = 7.37, p = .02
effect of inference: F(3, 9.26) = 2.88, p = .09
Post-hoc (Bonferroni-Holm): only MP (.40) and to a lesser degree DA (.14) above 0. MT = -.02; DA = .02

18. Exp II
n = 29
same 16 highly believable conditionals (13 from Evans et al., 2010):
If car ownership increases then traffic congestion will get worse.
If jungle deforestation continues then Gorillas will become extinct.
If the cost of fruit and vegetables is subsidised then people will eat more healthily.
Participants work on 4 randomly selected conditionals.
For each conditional participant work on 2 inference (either MP & DA or MT & AC).

19. Procedure II a X X In random order:
If car ownership increases then traffic congestion will get worse. In your opinion, how probable is the above statement/assertion?
If car ownership NOT increases then traffic congestion will get worse. In your opinion, how probable is the above statement/assertion? X X

20. Procedure II b X X In random order:
Car ownership increases. In your opinion, how probable is it that the above event occurs?
Car ownership does NOT increase. In your opinion, how probable is it that the above event occurs? X X

21. Procedure II c (random order)
If car ownership increases then traffic congestion will get worse. (Probability 80%) Car ownership increases. (Probability 75%) Under these premises, how probable is that traffic congestion will get worse? X

22. Procedure II c (random order)
If car ownership increases then traffic congestion will get worse. (Probability 80%) Car ownership does NOT increases. (Probability 28%) Under these premises, how probable is that traffic congestion will NOT get worse? X

23. Procedure II c (random order)
If car ownership increases then traffic congestion will get worse. (Probability 80%) Car ownership does NOT increases. (Probability 28%) Under these premises, how probable is that traffic congestion will NOT get worse? X

24. Procedure Exp II P(if p then q)? P(if ¬p then q)? P(p / q)?
P(MP / AC)?
P(DA / MT)?

25. Procedure Exp II P(if p then q)? P(if ¬p then q)? P(p / q)?
show previous response
P(MP / AC)?
P(DA / MT)?
show previous response

26. Procedure Exp II ×4 P(if p then q)? P(if ¬p then q)? P(p / q)?
show previous response
P(MP / AC)?
P(DA / MT)?
show previous response

27. Coherence

28. Coherence: Chance CorrectionMP MT AC DA
66% / 39%
45% / 52%
50% / 56%
55% / 39%
Percentage of coherent responses / coherent responses predicted by chance
Replication:
Only for MP and DA evidence for above chance coherence.
LMM on difference between coherence (0/1) and interval size:
intercept not signficant: F(1, 19.48) = 1.79, p = .20
effect of inference: F(3, 11.11) = 3.23, p = .06
Post-hoc: only MP (.26) and to lesser degree DA (.14) above 0. MT = -.07; AC = -.07

29. Conclusions
Coherence is not cognitive mechanism underlying probabilistic conditional inferences.
Temporal and procedural focus increases coherence somewhat (see Evans, Thompson, & Over, 2015; Cruz et al., 2015).
Normative ideas are bad building blocks for descriptively adequate psychological theories.

30. Possible Limitations
Intervals for all inferences (Pfeifer & Kleiter, 2005):

31. Potential Limitations
If car ownership increases then traffic congestion will get worse. (Probability 80%) Car ownership does NOT increases. (Probability 28%) Under these premises, how probable is that traffic congestion will NOT get worse? Π X μ

32. product:
mean:

33. 100%
26%
86%
50%
product:
mean:

34. Thank you for your attention

36. Coherence

37. MP MT AC DA 87% / 45% 63% / 65% 60% / 58% 60% / 46% MP MT AC DA
87% / 45%
63% / 65%
60% / 58%
60% / 46% MP MT AC DA
66% / 39%
45% / 52%
50% / 56%
55% / 39%

38. Exp I
Exp II

39. Summary Exp I Methods Participants work on 4 conditionals
One inference (MP, MT, AC, & DA) per conditional
8 estimates per conditional:
P(conditional)
P(minor premise)
P(conclusion)
P(q|p)
P(p ∧ q)
P(¬p ∨ q)
P(q|¬p)
[ P(conclusion without premises) ]

40. Exp I: Procedure I & II Always in this sequence:
P(conditional) (estimate = .80)
P(minor premise) (estimate = .95)
P(conclusion) (estimate = .70) [while previous responses are displayed]
Allows us to assess p-validity and mental probability logic predictions:
Is U(conclusion) < U(premises)? = < ( ) + ( ) = .25
Is .70 in the coherence interval? [ .80 × .95, .80 × ( )] = [.76, .81]

41. Exp 1: p-validity

42. Summary Exp II Methods Participants work on 4 conditionals
For each conditional participant work on 2 inference
4 estimates per conditional (each block in random order):
P(conditional)
P(q|¬p)
P(minor premise) [e.g., p]
P(other minor premise) [e.g., not-p]
2 inferences per conditional (in random order)
MP/MT
DA/AC

43. Only for MP there are above chance p-valid responses.
p-validity
Replication:
Only for MP there are above chance p-valid responses.
LMM on difference (violation 0/1 - summed uncertainty):
Intercept significantly > 0: F(1, 10.87) = 9.95, p = .02
Effect of Inference significantish: F(1, 17.51) = 3.83, p = .07
Post-Hoc (Bonferroni-Holm): Only MP > 0 (.21), not MT (-0.002)