Henrik Singmann Christoph Klauer Sieghard Beller

Henrik Singmann Christoph Klauer Sieghard Beller
Beyond Bayesian Updating: A Descriptively Adequate Model of Conditional Reasoning Henrik Singmann Christoph Klauer Sieghard Beller

Overview Singmann, H., & Klauer, K. C. (2011). Deductive and inductive conditional inferences: Two modes of reasoning. Thinking & Reasoning, 17(3), 247– Singmann, H., Klauer, K. C. & Beller, S. (under review). Probabilistic Conditional Reasoning: Disentangling Form and Content with the Dual-Source Model. Revised manuscript submitted for publication. Please interrupt me with questions.

What is Reasoning Reasoning is a "transition in thought, where some beliefs (or thoughts) provide the ground or reason for coming to another" (Adler, 2008). Deductive Reasoning: The current prince will be the next king. Prince Charles is the current prince. Therefore, Prince Charles will be the next king. Inductive Reasoning: The beer I have tasted in the UK so far was rather bland. Therefore, all British beer is bland. Irrational Reasoning: Too many immigrants coming to the UK. Most of these immigrants are coming from outside the EU. Therefore, the UK should leave the EU.

Reasoning and Logic Syllogisms (Aristotle)
Set Interpretation of Syllogisms All men are mortal. Socrates is a man. Therefore, Socrates is mortal. All men are mortal. Socrates is mortal. Therefore, Socrates is a man. mortal beings men Socrates

Reasoning and Logic Syllogisms (Aristotle) Conditional Inferences
All men are mortal. Socrates is a man. Therefore, Socrates is mortal. All men are mortal. Socrates is mortal. Therefore, Socrates is a man. If someone is human then she is mortal. Socrates is human. Therfore, Socrates is mortal. If someone is human then she is mortal. Socrates is mortal. Therfore, Socrates is human.

4 Conditional Inferences
Modus Ponens (MP): If p then q p Therefore, q Affirmation of the consequent (AC): If p then q q Therefore, p Inferences that start with M are valid, other are not. Modus Tollens (MT): If p then q Not q Therefore, not p Denial of the antecedent (DA): If p then q Not p Therefore, not q

Modus Ponens (MP): If p then q p Therefore, not q Affirmation of the consequent (AC): If p then q q Therefore, not p Logic tells us how people should reason. But how do they reason? Modus Tollens (MT): If p then q Not q Therefore, not p Denial of the antecedent (DA): If p then q Not p Therefore, not q

Theories of Human Reasoning
Mental Rules/Logic (Inhelder & Piaget, 1958; Rips, 1994; Stenning & van Lambalgen, 2008) Mental Model Theory (Johnson-Laird, 1983; Johnson-Laird & Byrne, 1991) Bayesian/Probabilistic Approach (Oaksford & Chater, 2007) Suppositional Theory (Evans & Over, 2004)

Effect of Inference MP (valid): AC (invalid):
If a person fell into a swimming pool, then the person is wet. A person fell into a swimming pool. How valid is the following conclusion from a logical perspective? The person is wet. If a person fell into a swimming pool, then the person is wet. A person is wet. How valid is the following conclusion from a logical perspective? The person fell into a swimming pool. Explain axis: y-axis response scale (endorsement) X-axis inference Singmann & Klauer (2011, Exp. 2)

Effect of Content MP (valid) AC (invalid) prological counterlogical
If a person fell into a swimming pool, then the person is wet. A person fell into a swimming pool. Therefore, the person is wet. If a girl had sexual intercourse, then she is pregnant. A girl had sexual intercourse. Therefore, the girl is pregnant. If a person fell into a swimming pool, then the person is wet. A person is wet. Therefore, the person fell into a swimming pool. If a girl had sexual intercourse, then she is pregnant. A girl is pregnant. Therefore, the girl had sexual intercourse. counterlogical

Effect of Instruction MP (valid) AC (invalid) deductive probabilistic
If a person fell into a swimming pool, then the person is wet. A person fell into a swimming pool. How valid is it that the person is wet? How likely is it that the person is wet? If a person fell into a swimming pool, then the person is wet. A person is wet. How valid is it that the person fell into a swimming pool? How likely is it that the person fell into a swimming pool? probabilistic

Theories of Human Reasoning
Deductive Reasoning Probabilistic/Everyday Reasoning Klauer & Singmann (2011): At least two processes contribute to reasoning. Single process theories (e.g., Mental Models; Bayesian approaches) cannot explain both, deductive and probabilistic reasoning. Mental Rules/Logic (Inhelder & Piaget, 1958; Rips, 1994 ; Stenning & van Lambalgen, 2008) Mental Model Theory (Johnson-Laird, 1983; Johnson-Laird & Byrne, 1991) Bayesian/Probabilistic Approach (Oaksford & Chater, 2007) Suppositional Theory (Evans & Over, 2004)

Modus Ponens (MP): If p then q p Therefore, q Affirmation of the consequent (AC): If p then q q Therefore, p Modus Tollens (MT): If p then q Not q Therefore, not p Denial of the antecedent (DA): If p then q Not p Therefore, not q

Joint probability distribution
Probabilistic Model 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) Joint probability distribution q ¬q p P(p  q) P(p  ¬q) ¬p P(¬p  q) P(¬p  ¬q) 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) Reasoning amounts to assessing probabilities of conclusions based on one’s background knowledge. For example for MP, “Given ‘If p then q’ and p, how likely is q?”, individuals consult their background knowledge regarding p and q and assess the conditional probability of the conclusion q given minor premise p. Oaksford, Chater, & Larkin (2000) Oaksford & Chater (2007)

Joint probability distribution
Probabilistic Model 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) Joint probability distribution q ¬q p P(p  q) P(p  ¬q) ¬p P(¬p  q) P(¬p  ¬q) 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) Oaksford, Chater, & Larkin (2000) Oaksford & Chater (2007)

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)

Effect of Conditional Data: Bayesian Model (Oaksford & Chater, 2007):
Conditional increases endorsement. Validity effect: Stronger increase for valid (MP & MT) than invalid (AC & DA) inferences. Bayesian Model (Oaksford & Chater, 2007): Conditional changes background knowledge. Probability distribution updates given conditional. Dual-Source Model (Klauer et al., 2010): Background knowledge determines responses for reduced inferences: Bayesian Reasoning Conditional provides form-based information. Responses to full inferences reflect mixture of knowledge and form information.

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 (under review) 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? 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? ns. * 6 different conditionals 3 expert 3 non-exprt 4 inferences (MP, MT, AC, DA) per conditional N = 47

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 (under review) C = content (one for each p and q) x = inference (MP, MT, AC, & DA)

? 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) 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') ?

Meta-ANalysis 7 data sets (Klauer et al., 2010; Singmann et al., under review) 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 (2010, Exp. 2) MP (valid) AC (invalid) deductive If a person fell into a swimming pool, then the person is wet. A person fell into a swimming pool. How valid is it that the person is wet? How likely is it that the person is wet? If a person fell into a swimming pool, then the person is wet. A person is wet. How valid is it that the person fell into a swimming pool? How likely is it that the person fell into a swimming pool? probabilistic

C = content (one for each p and q) x = inference (MP, MT, AC, & DA) Response + × (1 – λp) ξ(C,x) × λp τ(x) + (1 – τ(x)) × ξ(C,x) knowledge-based form-based probabilistic Response + × (1 – λd) ξ(C,x) × λd τ(x) knowledge-based form-based deductive

24 data points 15 free parameters (9 ξ, 4 τ, 2 λ) τ(MP) = 1.00, τ(AC) = .46 λp = .45, λd = .64 Effect of Instruction MP (valid) AC (invalid) deductive If a person fell into a swimming pool, then the person is wet. A person fell into a swimming pool. How valid is it that the person is wet? How likely is it that the person is wet? If a person fell into a swimming pool, then the person is wet. A person is wet. How valid is it that the person fell into a swimming pool? How likely is it that the person fell into a swimming pool? probabilistic

Summary Single process theories not able to account for full pattern of conditional inferences. 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. *** *** ** * *** ***

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

Formal Account of Uncertain Reasoning
Pfeifer and Kleiter’s (2005; 2010) mental probability logic Inferences should be probabilistically coherent: estimated probabilities agree with known/fixed probabilities according to elementary probability theory. Missing/unkown probabilities in [0, 1] → responses should lie in predicted interval E.g., MP: P(q) = [ P(q|p)P(p) , P(q|p)P(p) + (1 - P(p)) ] Example: If car ownership increases then traffic congestion will get worse. (P = 0.8) Car ownership increases. (P = 0.95) Under these premises, how probable is that traffic congestion will get worse? [ .80 × .95 , .80 × ( ) ] = [ .76 , .81 ]

Formal Account of Probabilistic Reasoning
Probabilized conditional reasoning task: all premises uncertain Only highly believable conditionals (Evans et al., 2010), e.g., If car ownership increases then traffic congestion will get worse. If jungle deforestation continues then Gorillas will become extinct. Two phase experiment: Participants provide estimates of premises directly and independently. Participants estimate probably of conclusion, while estimates (1.) are presented.

Example Item 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

Individuals do not reason according to probabilistic norms.
Replicating main finding from research on deductive reasoning: Individuals do not reason according to probabilistic norms.

Henrik Singmann Christoph Klauer Sieghard Beller

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