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

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. 21.11.2018

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 21.11.2018

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) 21.11.2018

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 21.11.2018

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) 21.11.2018

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]) 21.11.2018

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))] 21.11.2018

Mental Probability Logic Intervals for all inferences (Pfeifer & Kleiter, 2005): 21.11.2018

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. 21.11.2018

Experiment 1 (N = 30): Singmann, Klauer, Over (in revision). New Normative Standards of Conditional Reasoning and the Dual-Source Model. Frontiers in Psychology. 21.11.2018

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). 21.11.2018

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 21.11.2018

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 21.11.2018

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] 21.11.2018

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? 21.11.2018

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) ] 21.11.2018

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) 21.11.2018

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. 21.11.2018

p-validity 21.11.2018

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) 21.11.2018

Coherence 21.11.2018

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 21.11.2018

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. 21.11.2018

No assessment of The Equation Experiment 2 (N = 29) new data No assessment of The Equation 21.11.2018

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). 21.11.2018

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 21.11.2018

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 21.11.2018

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 21.11.2018

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 21.11.2018

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 21.11.2018

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 21.11.2018

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) 21.11.2018

Coherence 21.11.2018

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 21.11.2018

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. 21.11.2018

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 21.11.2018

Possible Lmitations Intervals for all inferences (Pfeifer & Kleiter, 2005): 21.11.2018

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 21.11.2018

Coherence Heuristic: Coverage 21.11.2018