1. Henrik Singmann Karl Christoph Klauer Sieghard Beller
Probabilistic Conditional Reasoning: The Dual-Source Model Provides Evidence for an Influence of Form
Henrik Singmann
Karl Christoph Klauer
Sieghard Beller

2. 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.
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. Research Question:
Effect of presence of conditional in everyday probabilistic conditional reasoning?
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
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)
Dual-Source Model

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

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

14. The Knowledge Phase
Participants estimate 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

15. 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 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

16. 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?
Dual-Source Model

17. 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 intuition:
Conditional provides form-based information which is integrated with background knowledge.
Dual-Source Model

18. 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

19. Are dual-source model parameters valid?
Empirical Part I:
Are dual-source model parameters valid?
Dual-Source Model

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

21. Experiment 1 Does conditional solely affect form-based component?
Participants work on conditional/biconditional reasoning task (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 problems without rule 3 times (n = 25).
Participants work on problems with rules 2 times (n = 29).
Dual-Source Model

22. 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

23.
Dual-Source Model

24. 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

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

26. | 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

27. Experiment 1: Summary
Observed behavior shows 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

28. Experiment 2 Effect of speaker expertise? (Stevenson & Over, 2001)
Participants work on conditional reasoning task (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

29. Exp 2: Materials Stimuli selected from 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

30. 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 shw 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
Dual-Source Model

31.
Dual-Source Model

32.
Dual-Source Model

33. 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

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

35. | 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

36. Experiment 2: 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 inferences only
Knowledge component captures knowledge phase
Speaker expertise does not affect how individuals perceive logical form, but only weighting of sources.
Dual-Source Model

37. Empirical Part I: Discussion
Dual-source model parameters valid measures for separating cognitive processes in probabilistic conditional reasoning.
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

38. 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?
Dual-Source Model

39. How else explain effect of conditional?
Alternative 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 background knowledge.
Presence of conditional changes knowledge base: Form affects each item idiosyncratically.
Dual-Source Model

40. 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 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)
Dual-Source Model

41. Which model provides the best account to the data?
Empirical Part II:
Which model provides the best account to the data?
Dual-Source Model

42. The Data Sets
Data sets (or parts thereof) that implement default paradigm:
Knowledge phase without conditional.
Rule phase with conditional.
No additional manipulations.
6 datasets (N = 148):
Klauer et al. (2010, Exps. 1 & 3, n = 15 & 18) and 2 new data sets (n = 26 & 29) exactly implemented default paradigm. (For Klauer et al.'s experiments third sessions were omitted)
For Klauer et al. (2010, Exp 4, n = 13) we omitted trials which did not follow default paradigm.
New data set (n = 47) in which speaker expertise was manipulated (Experiment 2, Part I)
Dual-Source Model

43. General Methodology 4 different conditionals (i.e., contents)
Participants responded to all four inferences per content.
Participants gave two estimates per inference: original and converse inference (pooled for analysis).
Two measurements (> one week in between):
without conditional
with conditional
→ 32 data points per participant.
Exceptions:
Klauer et al. (2010, Exp. 3): 5 conditionals = 40 data points per participant
New data on speaker expertise: 6 conditionals (3 experts, 3 novice) = 48 data points
Dual-Source Model

44. Candidate Models
Fitted all five models to individual data by minimizing sum of squared deviance from data and predictions.
Knowledge Phase (for all models):
3 parameters per content/conditional = 12 parameters for 4 conditionals.
Rule Phase:
Dual-source model (DS): + 4 form-based parameters = 16 parameters (independent of the number of conditionals)
Oaksford et al. (2000, OCL): + 1 parameter per content = 16 parameters.
Extended Oaksford & Chater (2007, eOC & eOC*): + 2 parameters per content = 20 parameters. (eOC* does not restrict posterior probability distribution)
Minimizing the Kullback-Leibler divergence (KL): + 1 parameter per content = 16 parameters
Dual-Source Model

45. Results I Klauer et al. (2010), Exp. 4
5 measurement points - we only use data that is comparable to other conditions (spread across 5 measurement points)
Dashed line shows mean fit value of DS.
Numbers behind model names = numbers of parameters per model.
Ordering: DS = eOC* > OCL > KL
Dual-Source Model

46.
Dual-Source Model

47. DS > KL > eOC* > OCL DS > eOC* > OCL = KL
DS > eOC* = KL > OCL
DS > eOC* = KL = OCL
Dual-Source Model

48. Results Speaker Expertise (Experiment 2)
48 data points.
In addition to DS, we fitted DS* which allowed for separate form parameters for experts and novices, respectively.
Ordering: eOC* = DS* ≥ OCL = DS > KL
Dual-Source Model

49. Summary Empirical Part II
Dual-source model provides overall best account for 6 datasets.
For two cases where it shares the first place (with the extended model by Oaksford & Chater, 2007), it has less parameters
Provides best account for 83 of 148 data sets (56%).
Extended Oaksford & Chater model (48 best accounts) performs better than Kullback-Leibler model (17 best account), but has more parameters.
Strong evidence for our interpretation of effect of the conditional: Provide formal information that cannot easily be accounted for by changes in participants' knowledge base only.
Dual-Source Model

50. General Discussion
Dual-source model parameters adequate & provides best account among candidate models.
(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

51. General Discussion
Dual-source model is not tied to normative model (e.g., Elqayam & Evans, 2011).
Parameters of dual-source model offer insights into cognitive processes:
Dissociation of suppression effects: disablers decrease weighting parameters, whereas alternatives decrease background-knowledge based evidence for AC and DA (and both affect the form-based component)
Speaker expertise affects weighting parameter only
Dual-Source Model

52. Thanks for your attention
Thanks for your attention. Thanks to Karl Christoph Klauer Sieghard Beller
Dual-Source Model