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

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) 16.01.2019

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 16.01.2019

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]) 16.01.2019

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]) 16.01.2019

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)) ] 16.01.2019

Coherence Intervals Intervals for all inferences (Pfeifer & Kleiter, 2005): 16.01.2019

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. 16.01.2019

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.

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 16.01.2019

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 16.01.2019

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 16.01.2019

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? 16.01.2019

Procedure Exp I P(if p then q)? P(minor premise)? P(MP / MT / AC / DA)? P(…)? show previous response 16.01.2019

Procedure Exp I ×4 P(if p then q)? P(minor premise)? P(MP / MT / AC / DA)? P(…)? ×4 show previous response 16.01.2019

Coherence 16.01.2019

Coherence: Chance Correction MP 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 16.01.2019

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). 16.01.2019

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 16.01.2019

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 16.01.2019

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 16.01.2019

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 16.01.2019

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 16.01.2019

Procedure Exp II P(if p then q)? P(if ¬p then q)? P(p / q)? P(MP / AC)? P(DA / MT)? 16.01.2019

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 16.01.2019

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 16.01.2019

Coherence 16.01.2019

Coherence: Chance Correction MP 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 16.01.2019

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. 16.01.2019

Possible Limitations Intervals for all inferences (Pfeifer & Kleiter, 2005): 16.01.2019

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 16.01.2019

product: mean:

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

Thank you for your attention 16.01.2019

16.01.2019

Coherence 16.01.2019

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% 16.01.2019

Exp I Exp II 16.01.2019

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) ] 16.01.2019

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)? 1 - .70 = .30 < (1 - .80) + (1 - .95) = .25 Is .70 in the coherence interval? [ .80 × .95, .80 × .95 + (1 - .95)] = [.76, .81] 16.01.2019

Exp 1: p-validity 16.01.2019

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 16.01.2019

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) 16.01.2019