1. Henrik Singmann Karl Christoph Klauer David Over
Assessing the Empirical Adequacy of New Normative Standards of Conditional Reasoning
Henrik Singmann
Karl Christoph Klauer
David Over

2. Traditional View of Conditionals
Conditional "if p then q" understood as truth-functional material implication, equivalent to "not-p or q" (Johnson-Laird & Byrne, 1991).
"Paradoxes" - infer "if p then q" from "not-p"
If we spin a fair coin 100 times then we will get 100 heads.
Probability of conditional increases with increasing probability of not flipping the coin.
Deduction paradigm
Participants asked to assume truth of premises.
Only draw logically necessary conclusions.

3. Traditional View of Conditionals
Unrealistic
Conditional "if p then q" understood as truth-functional material implication, equivalent to "not-p or q" (Johnson-Laird & Byrne, 1991).
"Paradoxes" - infer "if p then q" from "not-p"
If we spin a fair coin 100 times then we will get 100 heads.
Probability of conditional increases with increasing probability of not flipping the coin.
Deduction paradigm
Participants asked to assume truth of premises.
Only draw logically necessary conclusions.
absurd
Most human reasoning from uncertain premises or more or less confidently held beliefs

4. 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)

5. 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

6. 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
valid in standard logic (i.e., truth of premises entails truth of conclusion)
NOT valid in standard logic (i.e., truth of premises does NOT entail truth of conclusion)

7. 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)
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 interval (assuming unspecified probabilites in [0, 1])

8. 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))]

9. Mental Probability Logic
Intervals for all inferences (Pfeifer & Kleiter, 2005):

10. Overview
Goal: Assess empirical adequacy of The Equation, p-validity, and mental probability logic as computational-level account.
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.

11. Experiment 1 (N = 30):
Singmann, Klauer, Over (in revision). New Normative Standards of Conditional Reasoning and the Dual-Source Model. Frontiers in Psychology.

12. Materials
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).

13. 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

14. 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

15. 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]

16. 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?

17. Summary 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) ]

18. Meaning of the conditional
Bold: Significant correlations (p < .05). *: Significant after Bonferroni-Holm correction.
Problem: Independence assumption of correlation violated (more than one data point per participant)

19. Meaning of Conditional: LMM
Only P(q|p) adds unique variance to the to the prediction of P(if p then q).
Linear Mixed Model (LMM) analysis (Baayen, Davidson, & Bates, 2008; Barr, Levy, Scheepers, & Tily, 2013)
Crossed random effects for participants and items.

20. p-validity

21. p-validity: Chance Correction
Only for MP there are above chance p-valid responses.
Larger summed uncertainty of premises → larger probability that response is p-valid.
If P(premises) < .5, all responses are p-valid.
Assumption: Chance responses uniformly distributed
Compare whether or not a response is p-valid (0/1) with sum of U(premises).
Difference > 0: Above chance performance (analysis suggested by Jonathan Evans)
LMM on the difference:
Intercept significantly > 0: F(1, 10.48) = 8.39, p = .02
Effect of Inference significant: F(1, 28.98) = 8.41, p = .007
Post-Hoc (Bonferroni-Holm): Only MP > 0 (.26), not MT (-0.004)

22. Coherence

23. Coherence: Chance Correction
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

24. Summary Experiment 1 The Equation can be replicated.
Only P(q|p) adds unique variance in predicting P(if p then q).
Neither P(p ∧ q) nor P(q|¬p) (Delta-p) is predictive as well.
Data appears to be mostly p-valid, only for MP p-validity above chance (not for MT).
Responses are above chance coherent for MP and (less) DA.

25. No assessment of The Equation
Experiment 2 (N = 29)
new data
No assessment of The Equation

26. Materials
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).

27. Procedure Ia 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

28. Procedure Ib 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

29. Procedure IIa (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

30. Procedure IIb (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

31. Procedure IIb (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

32. Summary 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

33. 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)

34. Coherence

35. Coherence: Chance Correction
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 a lesser degree DA (.14) above 0. MT = -.07; DA = -.06

36. Summary Experiment 2
Albeit changing order of items, we replicate Experiment 1
Data appears to be mostly p-valid, only for MP p-validity above chance (not for MT).
Responses are above chance coherent for MP and (less) DA.

37. General Discussion Individuals understand conditional as P(q|p).
p-validity or mental probability logic no empirically adequate computational level accounts of reasoning.
Only for MP, performance was according to normative standards: Bayesian updating

38. Possible Lmitations
Intervals for all inferences (Pfeifer & Kleiter, 2005):

39. 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? X X

40. Coherence Heuristic: Coverage