Disentangling form and content with the Dual-Source Model of probabilistic conditional reasoning

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

A girl had NOT had sexual intercourse. How likely is it that the girl is NOT 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 sexual intercourse. How likely is it that the girl is 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) A girl had NOT had sexual intercourse. How likely is it that the girl is NOT 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 sexual intercourse. How likely is it that the girl is pregnant? Oaksford, Chater, & Larkin (2000) Oaksford & Chater (2007)

Experimental Paradigm Full Inferences (Week 2+) Full Inferences (Week 2+) Full Inferences (Week 2+) Full Inferences (Week 2+) Reduced Inferences (Week 1) Reduced Inferences (Week 1) Reduced Inferences (Week 1) 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? 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? 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? 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? 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)

Bayesian Updating Full Inferences (Week 2) 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? 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. ?

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) knowledge-based form-based C = content (oneforeachpandq) x = inference (MP, MT, AC, & DA)

Meta-ANalysis • 7 datasets (Klauer et al., 2010; Singmann et al., 2016) • total N = 179 • reducedandfullconditionalinferencesonly • no additional manipulations • eachmodelfittedtodataofeach individual participant. meanfreeparameters: 17.3 17.7 22.1 17.7 Singmann, Klauer, & Beller (2016)

Dual-Source Model (DSM) Exp. 2: validate Exp. 1: validate knowledge-based Singmann, Klauer, & Beller (2016) form-based C = content (oneforeachpandq) x = inference (MP, MT, AC, & DA)

Exp. 1: Manipulating Form Conditional Inferences (Week 2/3) Biconditional Inferences (Week 2/3) 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? 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 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) Biconditional Inferences (Week 2/3) 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? 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 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) Full Inferences, Non-Expert (Week 2) Reduced Inferences (Week 1) 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? 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? 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 • 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) Full Inferences, Non-Expert (Week 2) Reduced Inferences (Week 1) 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? 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? 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 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 knowledge-based Oaksford & Chater, … Singmann, Klauer, & Beller (2016) form-based C = content (oneforeachpandq) 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.

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Suppression Effects: MP Byrne (1989) Baseline Condition Disablers Condition 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? 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? 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? Additional disablers reduce endorsement to MP and MT. Additional alternatives do not affect endorsement to MP and MT.

Suppression Effects: AC Byrne (1989) Baseline Condition Disablers Condition 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? 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? 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? 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 Full Disablers Condition Reduced Disablers Condition Full Alternatives Condition 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? 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? 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? 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? 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? 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?

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.

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)

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Disentangling form and content with the Dual-Source Model of probabilistic conditional reasoning