Henrik Singmann Karl Christoph Klauer Sieghard Beller Validating the Parameters of the Dual-Source Model of Probabilistic Conditional Reasoning Henrik Singmann Karl Christoph Klauer Sieghard Beller

Conditional Reasoning Conditional is a rule with form: If p then q. Conditional inferences usually consist of conditional rule (major premise), minor premise (e.g., p), and conclusion (e.g., q). e.g.: If a person drinks a lot of coke then the person will gain weight. A person drinks a lot of coke. Conclusion: The person will gain weight. 26.11.2018 Dual-Source Model

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 26.11.2018 Dual-Source Model

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) 26.11.2018 Dual-Source Model

2-Phase Probabilistic Reasoning Task knowledge phase (week 1) rule phase (week 2) Observation: A balloon is pricked with a needle. How likely is it that it will pop? Rule: If a balloon is pricked with a needle, then it will pop. Observation: A balloon is NOT pricked with a needle. How likely is it that it will NOT pop? Observation: A balloon is NOT pricked with a needle. How likely is it that it will NOT pop? Rule: If a balloon is pricked with a needle, then it will pop. Observation: A balloon is pricked with a needle. How likely is it that it will pop? 26.11.2018 Dual-Source Model

How do we explain the effect of the conditional? Dual-source model for probabilistic conditional inferences (Klauer, Beller, & Hütter, 2010): Conditional absent (knowledge phase) Conditional probability of conclusion given minor premise (from background knowledge). E.g., Given p, how likely is q? P(q|p) Conditional present (rule phase) Conditional adds form-based evidence: Subjective probability to which inference is seen as warranted by (logical) form of inference. E.g., How likely is the conclusion given that the inference is MP? 26.11.2018 Dual-Source Model

How do we explain the effect of the conditional? Dual-source model for probabilistic conditional inferences (Klauer, Beller, & Hütter, 2010): Conditional absent (knowledge phase) Conditional probability of conclusion given minor premise (from background knowledge). E.g., Given p, how likely is q? P(q|p) Conditional present (rule phase) Conditional adds form-based evidence: Subjective probability to which inference is seen as warranted by (logical) form of inference. E.g., How likely is the conclusion given that the inference is MP? 26.11.2018 Dual-Source Model

How do we explain the effect of the conditional? Dual-source model for probabilistic conditional inferences (Klauer, Beller, & Hütter, 2010): Conditional absent (knowledge phase) Conditional probability of conclusion given minor premise (from background knowledge). E.g., Given p, how likely is q? P(q|p) Conditional present (rule phase) Conditional adds form-based evidence: Subjective probability to which inference is seen as warranted by (logical) form of inference. E.g., How likely is the conclusion given that the inference is MP? Our hypotheses: Conditional provides form-based information which is integrated with background knowledge. 26.11.2018 Dual-Source Model

Formalizing the Dual-Source Model Response + × (1 – λ) ξ(C,x) × λ τ(x) + (1 – τ(x)) × ξ(C,x) knowledge-based form-based C = content (1 – 4) x = inference (MP, MT, AC, & DA) 26.11.2018 Dual-Source Model

Are dual-source model parameters valid? Validating form parameter τ: Does type of conditional solely affect form based component? Manipulation of form: "If … then …" versus "If … then and only then …" Validating mixture weights λ: Effect of speaker expertise? Manipulation of who utters conditional: expert versus non-expert 26.11.2018 Dual-Source Model

Experiment 1 Does type of conditional solely affect form based component? Probabilistic conditional/biconditional reasoning tasks (n = 31): Session (knowledge phase): Problems without conditional or biconditional. Session (rule phase I): Problems with conditional (2 ×) or biconditional (2 ×). Session (rule phase II): Problems with conditional or biconditional (mapping conditional-biconditional flipped). Example Item: If a person drinks a lot of coke, then the person will gain weight. If a person drinks a lot of coke, then and only then the person will gain weight. 26.11.2018 Dual-Source Model

Estimating the dual-source model If a person drinks a lot of coke, then the person will gain weight. If a person drinks a lot of coke, then and only then the person will gain weight. 48 data points and 20 parameters per participant. Fitted by minimizing squared deviations. Mean R² = .92 (.73 - .99) if – then and only then if - then knowledge phase ξ predictions 26.11.2018 Dual-Source Model

Selective Influence Mixture weights λ Form parameters τ ** ns. 26.11.2018 Dual-Source Model

Experiment 2 What is the effect of speaker expertise? (Stevenson & Over, 2001) Probabilistic conditional reasoning tasks (n = 47): Session (knowledge phase): 6 problems without conditional. Session (rule phase): 6 problems with conditional, either uttered by expert (3 ×) or by non-expert (3 ×). Example item: If Anne eats a lot of parsley then the level of iron in her blood will increase. Nutrition scientist versus drugstore clerk 26.11.2018 Dual-Source Model

Selective Influence * Mixture weights λ Form parameters τ mean R² = .85 (.27 - 1.00) Mixture weights λ Form parameters τ ns. * 26.11.2018 Dual-Source Model

General Discussion Parameters of dual-source model show selective influence: Manipulation of form solely affects form component. Manipulation of speaker expertise solely affects mixture weights. (Conditional) Reasoning based on different integrated information: No single-process theory adequate (Singmann & Klauer, 2011). „Pure“ Bayesianism adequate only for knowledge phase (i.e., without conditional) Dual-source model is not tied to normative model (e.g., Elqayam & Evans, 2011). 26.11.2018 Dual-Source Model

Thanks for your attention Thanks for your attention. Thanks to Karl Christoph Klauer Sieghard Beller Slides are available at my website: http://www.psychologie.uni-freiburg.de/Members/singmann/presentations 26.11.2018 Dual-Source Model

Formalizing the Dual-Source Model Knowledge phase Rule phase Joint probability distribution: Provides conditional probabilities: P("MP") = P(q|p) = P(p  q) / P(q) P("MT") = P(¬p|¬q) P("AC") = P(p|q) P("DA") = P(¬q|¬p) Three independent parameters (e.g., P(p), P(q), and P(¬q|p), Oaksford, Chater, & Larkin, 2000). Knowledge phase parameters ξ Mixture weights λ one τ per inference: τ(MP) τ(MT) τ(AC) τ(DA) τ(x) represent subjective probability to which inference x is seen as (logically) warranted. q ¬q p P(p  q) P(p  ¬q) ¬p P(¬p  q) P(¬p  ¬q) 26.11.2018 Dual-Source Model

Experiment 1 Does the conditional only affect the form based component? Probabilistic conditional/biconditional reasoning tasks (n = 31): Session (knowledge phase): Problems without conditional or biconditional rule. Session (rule phase I): Problems with conditional rule (2 ×) or biconditional rule (2 ×). Session (rule phase II): Problems with conditional rule or biconditional rule (mapping conditional-biconditional flipped). Two additional control conditions: Participants work on the problems without rule 3 times (n = 25). Participants work on the problems with rules 2 times (n = 29). 26.11.2018 Dual-Source Model

Materials and Results Four contents orthogonally cross two types of defeaters (e.g., Cummins, 1995): disablers and alternatives If a predator is hungry, then it will search for prey. If a predator is hungry, then and only then it will search for prey. (few disablers, few alternatives) If a balloon is pricked with a needle, then it will quickly lose air. If a balloon is pricked with a needle, then and only then it will quickly lose air. (few disablers, many alternatives) If a girl has sexual intercourse, then she will be pregnant. If a girl has sexual intercourse, then and only then she will be pregnant. (many disablers, few alternatives) If a person drinks a lot of coke, then the person will gain weight. If a person drinks a lot of coke, then and only then the person will gain weight. (many disablers, many alternatives) 26.11.2018 Dual-Source Model

Estimating the dual-source model 48 data points per participants 20 parameters: 4 * 3 parameters underlying knowledge (ξ, xsi) 2 * 4 parameters for rule weight (λ, lambda) and form (τ, tau) Fitted by minimizing squared deviations. Mean R² = .92 (.73 - .99) 26.11.2018 Dual-Source Model

Exp 2: Materials Stimuli selected from list of 20 (web prestudy): If Anne eats a lot of parsley then the level of iron in her blood will increase. Nutrition scientist versus drugstore clerk If Katherina grows 10 cm then her ballet partner will fail to perform the lifting routine with her. ballet dancer versus ballet audience member If the shampoo „Fresh and Soft“ is removed from the assortment then the yearly volume of sales will decrease. Manager versus Cashier If the Airbus A380 flies through turbulences then its voltage level will fluctuate. Pilot versus airline passenger If individuals in industrialized nations continue to emit similar amounts of CO2 then the Gulf stream will stop. Enviromental Scientist versus newspaper reader. If Lisa invests in hedge funds then she will loose all the inversted money. Asset consultant versus bank teller. If Miroslav Klose’s lactat level is within the norm then he plays all games for the German national football team. Der ärztliche Betreuer der Nationalmannschaft versus ein Journalist 26.11.2018 Dual-Source Model

If Anne eats a lot of parsley then the level of iron in her blood will increase. Nutrition scientist versus drugstore clerk 26.11.2018 Dual-Source Model

dual-source 48 data points per participants 26 parameters: 6 * 3 parameters underlying knowledge (ξ, xsi) 2 * 4 parameters for rule weight (λ, lambda) and form (τ, tau) Fitted by minimizing squared deviations. Mean R² = .85 (.27 - 1.00) 26.11.2018 Dual-Source Model

How else explain effect of conditional? Alternative (Bayesian) views: New paradigm psychology of reasoning (e.g., Oaksford & Chater, 2007; Pfeifer & Kleiter, 2010) Causal Bayes nets (e.g., Sloman, 2005; Fernbach & Erb, in press) Reasoning basically probability estimation from coherent probability distribution of background knowledge. Presence of conditional changes knowledge base: Form affects each item idiosyncratically. 26.11.2018 Dual-Source Model

Model differences Dual-Source Model Alternative Views Rule Phase: Knowledge Phase + Form-based evidence per inference (+ 4 parameters) Knowledge Phase: Probability distribution over P(p), P(¬p), P(q), P(¬q) per content. Rule Phase: Probability distribution per content/conditional updates: P'(¬q|p) < P(¬q|p) (Oaksford, Chater & Larkin, 2000) P'(q|p) > P(q|p), then minimizing KL-distance to prior distribution (Hartmann & Rafiee Rad, 2012) 26.11.2018 Dual-Source Model

Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15) error bars: difference adjusted Cousineau-Morey-Baguley intervals different lines: different conditionals (i.e., different contents/items) conditional absent conditional present 26.11.2018 Dual-Source Model

↑ conditional present ↓ Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15) error bars: difference adjusted Cousineau-Morey-Baguley intervals different lines: different conditionals (i.e., different contents/items) ↑ conditional absent ↓ ↑ conditional present ↓ new data (n = 29) 26.11.2018 Dual-Source Model

↑ conditional present ↓ Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15) error bars: difference adjusted Cousineau-Morey-Baguley intervals different lines represent different contents of the conditional Presence of conditional increases participants' estimates of probability of conclusion. Especially for formally valid inferences MP and MT. ↑ conditional absent ↓ ↑ conditional present ↓ new data (n = 29) 26.11.2018 Dual-Source Model

↑ conditional present ↓ Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15) error bars: difference adjusted Cousineau-Morey-Baguley intervals different lines represent different contents of the conditional How can we explain effect of presence of conditional? Our data challenge pure Bayesian or probabilistic approaches that solely rely on background knowledge. ↑ conditional absent ↓ ↑ conditional present ↓ new data (n = 29) 26.11.2018 Dual-Source Model