1. Henrik Singmann Karl Christoph Klauer Sieghard Beller
New results on the dual-source model of probabilistic conditional reasoning
Henrik Singmann
Karl Christoph Klauer
Sieghard Beller

2. Conditional Reasoning
A conditional is a rule with the form: If p then q.
Conditional inferences usually consist of the conditional rule (major premise), a minor premise (e.g., p) and the 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.
Dual-Source Model

3. 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
Dual-Source Model

4. 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)
Dual-Source Model

5. Dual-Source Model
Formal model for probabilistic conditional reasoning (Klauer, Beller, & Hütter, 2010) .
Reasoners integrate two types of information:
Background knowledge: Conditional probability of the conclusion given minor premise. E.g., Given p, how likely is q? P(q|p)
Form-based evidence: Subjective probability to which an inference is seen as logically warranted by the form of the inference. E.g., How likely is the conclusion given that the inference is MP?
Dual-Source Model

6. Example Item: Knowledge Phase
Observation: A balloon is pricked with a needle.
How likely is it that it will pop?
Dual-Source Model

7. Example Item: Knowledge Phase
Observation: A balloon is pricked with a needle.
How likely is it that it will pop? X
Dual-Source Model

8. Example Item: Rule Phase
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?
Dual-Source Model

9. Example Item: Rule Phase
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? X
Dual-Source Model

10. ↑ conditional present ↓
Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15)
different lines represent different conditionals (i.e., different contents/items)
↑ conditional absent ↓
↑ conditional present ↓
new data (n = 29)
Dual-Source Model

11. ↑ conditional present ↓
Klauer, Beller, & Hütter (2010): Experiment 1 (n = 15)
different lines represent different contents of the conditional
The presence of the rule increases participants' estimates of the probability of the conclusion.
Especially so for the formally valid inferences MP and MT.
↑ conditional absent ↓
↑ conditional present ↓
new data (n = 29)
Dual-Source Model

12. The Knowledge Phase
Participants estimate a conclusion given minor premise only, e.g., Given p, how likely is q?
Response should reflect conditional probability of conclusion given minor premise, e.g., P(q|p)
"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)
Dual-Source Model

13. Formalizing the Knowledge Phase
Joint probability distribution over P(p), P(q), and their negations per content:
Provide the conditional probabilities, e.g.: P(MP) = P(q|p) = P(p  q) / P(q)
Three independent parameters describe the joint probability distribution (e.g., P(p), P(q), and P(¬q|p), Oaksford, Chater, & Larkin, 2000). q ¬q p
P(p  q)
P(p  ¬q) ¬p
P(¬p  q)
P(¬p  ¬q)
Dual-Source Model

14. 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)
Dual-Source Model

15. Research Questions
Does the conditional only affect the form based component?
Manipulation of the form: "If … then …" versus "If … then and only then …"
What is the effect of speaker expertise?
Manipulation of who utters the conditional: expert versus non-expert
Dual-Source Model

16. Experiment 1
Does the conditional only affect the form based component?
Participants work on 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).
Dual-Source Model

17. What is the Effect of the rule?
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)
Dual-Source Model

18.
Dual-Source Model

19. 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 ( )
Dual-Source Model

20. Rule weight parameters
| Effect | df | MSE | F | η² | p | | | | lambda | 1, 30 | 0.03 | 1.31 | .04 | .26 |
Dual-Source Model

21. | Effect | df | MSE | F | η² | p | | | | rule_type | 1, 30 | 0.1 | 8.46 ** | .04 | .007 | | inference | 2.13, | 0.14 | 1.19 | .02 | .31 | | rule_type:inference | 2.56, | 0.13 | *** | .26 | <.001 |
Dual-Source Model

22. Experiment 1: Summary
Observed behavior shows expected differences between two types of rules.
Dual-source model adequately captures those differences:
Only form-based (τ) component differs across the rules
Weighting of rule (λ) is not affected
Knowledge component adequately captures knowledge phase
Dual-Source Model

23. Experiment 2
What is the effect of speaker expertise? (Stevenson & Over, 2001)
Participants work on 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 ×).
Two additional control conditions:
Participants work on the problems without rule 2 times (n = 46).
Participants work on the problems with rule 1 time (n = 45).
Dual-Source Model

24. Exp 2: Materials Stimuli selected from a list of 20 (web prestudy):
Wenn Anne viel Petersilie isst, dann verbessert sich ihr Eisenwert im Blut.
Ernährungswissenschaftler versus Drogerieangestellte
Wenn Katharina 10 cm wächst, dann scheitert ihr Ballettpartner an der Hebefigur mit ihr.
Balletttänzer versus Ballettbesucher
Wenn das Haarshampoo "Fresh and Soft" aus dem Sortiment der Drogerie genommen wird, dann sinken die Jahresumsatzzahlen.
Filialleiter versus Kassierer
Wenn der Airbus A380 durch Turbulenzen fliegt, dann schwankt die Spannung im Stromnetz der Maschine.
Pilot versus Fluggast
Wenn die Menschen in den Industrienationen weiterhin so viel CO2 ausstoßen, dann wird der Golfstrom zum Erliegen kommen.
Umweltwissenschaftler versus Zeitungsleser
Wenn Lisa in Hedgefonds investiert hat, dann verliert sie ihr investiertes Geld.
Vermögensberater versus Banklehrling
Wenn Miroslav Kloses Lactatwert im Normbereich ist, dann spielt er jedes Spiel der Fußballnationalmannschaft.
Der ärztliche Betreuer der Nationalmannschaft versus ein Journalist
Dual-Source Model

25.
Dual-Source Model

26.
Dual-Source Model

27. 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 ( )
Dual-Source Model

28. Rule weight parameters
| Effect | df | MSE | F | η² | p | | | | lambda | 1, 46 | 0.03 | 5.17 * | .1 | .03 |
Dual-Source Model

29. | Effect | df | MSE | F | η² | p | | | | role | 1, 46 | 0.12 | 0.28 | <.001 | .60 | | inference | 2.62, | 0.14 | 4.58 ** | .04 | .006 | | role:inference | 2.86, | 0.12 | 1.72 | .02 | .17 |
Dual-Source Model

30. Experiment 1: Summary
Observed behavior shows expected differences between different levels of speaker expertise.
Dual-source model adequately captures those differences:
Only weighting of rule (λ) differs in expected direction: Rule weighted stronger for experts
Form-based (τ) component differs across the rules
Knowledge component captures knowledge phase
Speaker expertise does not affect how individuals perceive logical form, but only weighting of sources.
Dual-Source Model

31. Discussion
Dual-source model is validated and adequate model for disentangling cognitive processes in probabilistic conditional reasoning.
Dual-source model can even handle unfamiliar inferences.
Probabilistic conditional reasoning doesn't seem to be psychologically "special".
Not all factors surpressing inferences (speaker expertise) do so via changing the perceived logical correctness.
Dual-Source Model