Henrik Singmann Christoph Klauer Sieghard Beller Beyond updating: Disentangling form and content with the dual-source model of probabilistic conditional reasoning. Henrik Singmann Christoph Klauer Sieghard Beller

A girl had sexual intercourse. How likely is it that the girl is pregnant? A girl is NOT pregnant. How likely is it that the girl had NOT had sexual intercourse? A girl is pregnant. How likely is it that the girl had sexual intercourse? A girl had NOT had sexual intercourse. How likely is it that the girl is NOT pregnant?

Joint probability distribution A girl had sexual intercourse. How likely is it that the girl is pregnant? A girl is NOT pregnant. How likely is it that the girl had NOT had sexual intercourse? Inference "MP" "MT" "AC" "DA" p  q ¬q  ¬p q  p ¬p  ¬q Response reflects P(q|p) P(¬p|¬q) P(p|q) P(¬q|¬p) A girl is pregnant. How likely is it that the girl had sexual intercourse? A girl had NOT had sexual intercourse. How likely is it that the girl is NOT pregnant? 3 free parameters Provides conditional probabilities/predictions: P(MP) = P(q|p) = P(p  q) / P(p) P(MT) = P(¬p|¬q) = P(¬p  ¬q) / P(¬q) P(AC) = P(p|q) = P(p  q) / P(q) P(DA) = P(¬q|¬p) = P(¬p  ¬q) / P(¬p) Joint probability distribution q ¬q p P(p  q) P(p  ¬q) ¬p P(¬p  q) P(¬p  ¬q) Oaksford, Chater, & Larkin (2000) Oaksford & Chater (2007)

Experimental Paradigm Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Klauer, Beller, & Hütter (2010) Singmann, Klauer, & Beller (2016)

Experimental Paradigm Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Inference "MP" "MT" "AC" "DA" p  q ¬q  ¬p q  p ¬p  ¬q Response reflects P(q|p) P(¬p|¬q) P(p|q) P(¬q|¬p) Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2+) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Inference MP MT AC DA p → q p  q p → q ¬q  ¬p p → q q  p p → q ¬p  ¬q Response reflects P(q|p) P(¬p|¬q) P(p|q) P(¬q|¬p) Klauer, Beller, & Hütter (2010) Singmann, Klauer, & Beller (2016)

? Bayesian Updating Role of conditional in Bayesian models: Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Full Inferences (Week 2) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Role of conditional in Bayesian models: PROB: increases probability of conditional, P(q|p) (Oaksford et al., 2000) EX-PROB: increases probability of conditional PMP(q|p) > Pother(q|p) (Oaksford & Chater, 2007) KL: increases P(q|p) & Kullback-Leibler distance between g and g' is minimal (Hartmann & Rafiee Rad, 2012) Consequence of updating: Effect is content specific. Joint probability distribution: g q ¬q p P(p  q) P(p  ¬q) ¬p P(¬p  q) P(¬p  ¬q) Updated joint probability distribution: g' q' ¬q' p' P(p'  q') P(p'  ¬q') ¬p' P(¬p'  q') P(¬p'  ¬q') ?

Effect of Conditional Reduced Inferences (Week 1) Full Inferences (Week 2) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Klauer, Beller, & Hütter (2010, Exp. 1)

Dual-Source Model (DSM) Klauer, Beller, & Hütter (2010) Singmann, Klauer, & Beller (2016) Response + × (1 – λ) ξ(C,x) × λ τ(x) + (1 – τ(x)) × ξ(C,x) knowledge-based form-based C = content (one for each p and q) x = inference (MP, MT, AC, & DA)

Meta-ANalysis 7 data sets (Klauer et al., 2010; Singmann et al., 2016) total N = 179 reduced and full conditional inferences only no additional manipulations each model fitted to data of each individual participant. mean free parameters: 17.3 17.7 22.1 17.7 Singmann, Klauer, & Beller (2016)

Dual-Source Model (DSM) Exp. 2: validate Exp. 1: validate Response + × (1 – λ) ξ(C,x) × λ τ(x) + (1 – τ(x)) × ξ(C,x) knowledge-based form-based Singmann, Klauer, & Beller (2016) C = content (one for each p and q) x = inference (MP, MT, AC, & DA)

Exp. 1: Manipulating Form Conditional Inferences (Week 2/3) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Exp. 1: Manipulating Form Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Biconditional Inferences (Week 2/3) If a girl had sexual intercourse, then and only then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? 4 different conditionals 4 inferences (MP, MT, AC, DA) per conditional N = 31

Exp. 1: Manipulating Form Conditional Inferences (Week 2/3) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Exp. 1: Manipulating Form Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Biconditional Inferences (Week 2/3) If a girl had sexual intercourse, then and only then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? 4 different conditionals 4 inferences (MP, MT, AC, DA) per conditional N = 31

Exp. 1: Manipulating Form Conditional Inferences (Week 2/3) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Exp. 1: Manipulating Form Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? Biconditional Inferences (Week 2/3) If a girl had sexual intercourse, then and only then she is pregnant. A girl had sexual intercourse. How likely is it that the girl is pregnant? ns. *** 4 different conditionals 4 inferences (MP, MT, AC, DA) per conditional N = 31

Exp. 2: Manipulating Expertise Full Inferences, Expert (Week 2) A nutrition scientist says: If Anne eats a lot of parsley then the level of iron in her blood will increase. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? Exp. 2: Manipulating Expertise Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? or Full Inferences, Non-Expert (Week 2) A drugstore clerk says: If Anne eats a lot of parsley then the level of iron in her blood will increase. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? 6 different conditionals 3 expert 3 non-exprt 4 inferences (MP, MT, AC, DA) per conditional N = 47

Exp. 2: Manipulating Expertise Full Inferences, Expert (Week 2) A nutrition scientist says: If Anne eats a lot of parsley then the level of iron in her blood will increase. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? Exp. 2: Manipulating Expertise Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? or Full Inferences, Non-Expert (Week 2) A drugstore clerk says: If Anne eats a lot of parsley then the level of iron in her blood will increase. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? 6 different conditionals 3 expert 3 non-exprt 4 inferences (MP, MT, AC, DA) per conditional N = 47

Exp. 2: Manipulating Expertise Full Inferences, Expert (Week 2) A nutrition scientist says: If Anne eats a lot of parsley then the level of iron in her blood will increase. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? Exp. 2: Manipulating Expertise Reduced Inferences (Week 1) If a girl had sexual intercourse, then she is pregnant. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? or Full Inferences, Non-Expert (Week 2) A drugstore clerk says: If Anne eats a lot of parsley then the level of iron in her blood will increase. Anne eats a lot of parsley. How likely is it that the level of iron in her blood will increase? ns. * 6 different conditionals 3 expert 3 non-exprt 4 inferences (MP, MT, AC, DA) per conditional N = 47

Dual-Source Model (DSM) Exp. 2: validate Exp. 1: validate Oaksford & Chater, … Response + × (1 – λ) ξ(C,x) × λ τ(x) + (1 – τ(x)) × ξ(C,x) knowledge-based form-based Singmann, Klauer, & Beller (2016) C = content (one for each p and q) x = inference (MP, MT, AC, & DA)

Summary Bayesian updating does not seem to explain effect of conditional. Probability theory cannot function as wholesale replacement for logic as computational-level theory of what inferences people should draw (cf. Chater & Oaksford, 2001). DSM adequately describes probabilistic conditional reasoning: When formal structure absent, reasoning purely Bayesian (i.e., based on background knowledge only). Formal structure provides reasoners with additional information about quality of inference (i.e., degree to which inference is seen as logically warranted). Responses to full inferences reflect weighted mixture of Bayesian knowledge-based component and form-based component. DSM useful and parsimonious measurement model.

That was all

Suppression Effects: MP Disablers Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is inflated to begin with then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Suppression Effects: MP Byrne (1989) Baseline Condition If a balloon is pricked with a needle then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Alternatives Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is pricked with a knife then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Additional disablers reduce endorsement to MP and MT. Additional alternatives do not affect endorsement to MP and MT.

Suppression Effects: AC Disablers Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is inflated to begin with then it will quickly lose air. A balloon quickly looses air. How likely is it that the balloon was pricked with a needle? Suppression Effects: AC Byrne (1989) Baseline Condition If a balloon is pricked with a needle then it will quickly lose air. A balloon quickly looses air. How likely is it that the balloon was pricked with a needle? Alternatives Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is pricked with a knife then it will quickly lose air. A balloon quickly looses air. How likely is it that the balloon was pricked with a needle? Additional disablers do not affect endorsement to AC and DA. Additional alternatives reduce endorsement to AC and DA.

Exp. 3: Procedure Full Baseline Condition Reduced Baseline Condition If a balloon is pricked with a needle then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Exp. 3: Procedure Reduced Baseline Condition If a balloon is pricked with a needle then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Full Disablers Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is inflated to begin with then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Reduced Disablers Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is inflated to begin with then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Reduced Alternatives Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is pricked with a knife then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air? Full Alternatives Condition If a balloon is pricked with a needle then it will quickly lose air. If a balloon is pricked with a knife then it will quickly lose air. A balloon is pricked with a needle. How likely is it that the balloon quickly looses air?

Exp. 3: Disabling Condition Total N: 167 Exp. 3: Disabling Condition Reduced Inferences (Week 1) Full Inferences (Week 2) If a person drinks a lot of coke then the person will gain weight. A person drinks a lot of coke. How likely is it that the person will gain weight? Please note: A person only gains weight if the metabolism of the person permits it, the person does not exercise as a compensation, the person does not only drink diet coke. If a person drinks a lot of coke then the person will gain weight. A person drinks a lot of coke. How likely is it that the person will gain weight? Please note: A person only gains weight if the metabolism of the person permits it, the person does not exercise as a compensation, the person does not only drink diet coke.

Exp. 3: Disabling Condition Total N: 167 Exp. 3: Disabling Condition Reduced Inferences (Week 1) Full Inferences (Week 2) If a person drinks a lot of coke then the person will gain weight. A person drinks a lot of coke. How likely is it that the person will gain weight? Please note: A person only gains weight if the metabolism of the person permits it, the person does not exercise as a compensation, the person does not only drink diet coke. If a person drinks a lot of coke then the person will gain weight. A person drinks a lot of coke. How likely is it that the person will gain weight? Please note: A person only gains weight if the metabolism of the person permits it, the person does not exercise as a compensation, the person does not only drink diet coke.

Exp. 3: Disabling Condition Total N: 167 Exp. 3: Disabling Condition Reduced Inferences (Week 1) Full Inferences (Week 2) If a person drinks a lot of coke then the person will gain weight. A person drinks a lot of coke. How likely is it that the person will gain weight? Please note: A person only gains weight if the metabolism of the person permits it, the person does not exercise as a compensation, the person does not only drink diet coke. If a person drinks a lot of coke then the person will gain weight. A person drinks a lot of coke. How likely is it that the person will gain weight? Please note: A person only gains weight if the metabolism of the person permits it, the person does not exercise as a compensation, the person does not only drink diet coke. *** *** ** * *** ***

Suppression Effects in Reasoning In line with formal accounts: Disablers and alternatives suppress form-based evidence for „attacked“ inferences. In line with probabilistic accounts: Alternatives (and to lesser degree disablers) decreased the knowledge-based support of the attacked inferences. Difference suggests that disablers are automatically considered, but not alternatives: neglect of alternatives in causal Bayesian reasoning (e.g., Fernbach & Erb, 2013). Only disablers discredit conditional (in line with pragmatic accounts, e.g., Bonnefon & Politzer, 2010)

That was all