1. Suppression Effects in the Dual-Source Model of Conditional Reasoning
Henrik Singmann
Sieghard Beller
Karl Christoph Klauer

2. Research Goals How do humans reason with probabilistic relations?
Which cognitive processes contribute to reasoning from everyday conditionals?
How can we disentangle them and measure their contributions?
How do different linguistic forms (of potentially the same logical relation) influence the reasoning process?
Dual-Source Model

3. 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

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
Dual-Source Model

5. 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

6. Content Effects
Content can influence reasoning from conditionals via two types of counterexamples: disabling conditions and alternative antecedents.
If a balloon is pricked with a needle, then it will quickly lose air.
Few disablers for the consequent and many alternatives for the antecedent lead to strong endorsement of valid inferences (MP & MT) and weak endorsement of the invalid inferences (AC & DA)
If a girl has sexual intercourse, then she will be pregnant.
Many disablers for the consequent and few alternatives for the antecedent lead to weak endorsement of valid inferences (MP & MT) and strong endorsement of the invalid inferences (AC & DA)
To tap everyday reasoning studies often use probabilistic instructions: "How likely is it that the conclusion holds?"
Dual-Source Model

7. Explaining Content Effects
Probabilistic theory (Oaksford et al., 2000)
Reasoning is estimation of subjective probabilities based on background knowledge of the problem.
Problem Type MP MT AC DA
p → q 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

8. Explaining Content Effects
Probabilistic theory (Oaksford et al., 2000)
Reasoning is estimation of subjective probabilities based on background knowledge of the problem.
What is the effect of the rule? (e.g. Liu, 2003)
Typically endorsement of MP and MT is heightened
Problem Type 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

9. Explaining Content Effects
Probabilistic theory (Oaksford et al., 2000)
Reasoning is estimation of subjective probabilities based on background knowledge of the problem.
What is the effect of the rule? (e.g. Liu, 2003)
Typically endorsement of MP and MT is heightened
Problem Type 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

10. What is the Effect of the Rule?
Klauer, Beller, & Hütter (2010, Exp. 1):
Participants work on conditional reasoning tasks with different contents on two sessions:
Session (knowledge phase): Problems without conditional rule.
Session (rule phase): Problems with conditional rule.
Four contents cross disablers and alternatives:
If a predator is hungry, then it will search for prey. (few disablers, few alternatives)
If a balloon is pricked with a needle, then it will quickly lose air. (few disablers, many alternatives)
If a girl has sexual intercourse, then she will be pregnant. (many disablers, few alternatives)
If a person drinks a lot of coke, then the person will gain weight. (many disablers, many alternatives)
Dual-Source Model

11. What is the Effect of the Rule?
In both sessions participants work on all 4 contents and all four conditional inferences (MP, MT, AC, DA)
To obtain better estimates of the perceived probability with which a conclusion holds, participants work on two versions of each inference, the original conclusion and the converse conclusion:
MP (Original): p → q; p : q
MP' (Converse): p → q; p : ¬q
Dependent Variable: How likely does the conclusion hold given minor premise (Session 1) or given conditional rule and minor premise (Session 2)? Response on scale from 0% to 100%
Dual-Source Model

12. Klauer et al. (2010): Experiment 1
predator: few disablers, few alternatives
balloon: few disablers, many alternatives
girl: many disablers, few alternatives
coke: many disabler, many alternatives
Dual-Source Model

13. The Dual-Source Model
The finding that the presence of the rule affects reasoning from conditional problems is difficult for theories assuming that reasoning is a process of subjective probability estimation based on background knowledge.
According to the dual-source model of conditional reasoning (Klauer, Beller, & Hütter, 2010) the effect of the rule is to provide evidence in addition to the background knowledge: form-based evidence (i.e., the subjective probability to which an inference is seen as logically warranted)
Dual-Source Model

14. The Dual-Source Model
The dual-source model is a mathematical measurement model that describes probabilistic conditional reasoning with three parameters:
ξ (xsi): quantifies the knowledge-based component (formalized as in Oaksford et al.'s, 2000, model)
τ (tau): form-based evidence
λ (lambda): weighting factor
The model is estimated by fitting individual data from both sessions to one set of parameters.
Dual-Source Model

15. The Dual-Source Model
Observable response on one inference: With rule = λ{τ(x) × 1 + (1 – τ(x)) × ξ(C,x)} + (1 – λ)ξ(C,x) Without rule = ξ(C,x)
τ(x) = form-based evidence, subjective probability for accepting inference x (i.e., MP, MT, AC, DA) based on the logical form.
ξ(C,x) = knowledge-based evidence, subjective probability for accepting inference x for content C based on the background knowledge.
Dual-Source Model

16. Estimating the Dual-Source Model
Participants work on conditional reasoning tasks with four different contents on two sessions:
Session (knowledge phase): Problems without conditional rule.
Session (rule phase): Problems with conditional rule.
Four contents cross disablers and alternatives.
In both sessions participants work on all 4 contents and all four conditional inferences (MP, MT, AC, DA)
The model is fitted by minimizing the sum of the squared deviance from data and predictions.
Dual-Source Model

17. Klauer et al. (2010): Experiment 1
predator: few disablers, few alternatives
balloon: few disablers, many alternatives
girl: many disablers, few alternatives
coke: many disabler, many alternatives
Dual-Source Model

18. Klauer et al. (2010): Experiment 1
ξ (xsi) & τ (tau) Model parameters
τ (tau)
predator: few disablers, few alternatives
balloon: few disablers, many alternatives
girl: many disablers, few alternatives
coke: many disabler, many alternatives
Dual-Source Model

19. Parameter Validation
The knowledge parameter (ξ) generally maps on the responses from the knowledge phase (i.e., session without rule).
The form parameter τ follows a conditional pattern: MP > MT > AC ≥ DA
The form parameter τ was further validated in Experiment 3 (Klauer et al., 2010):
An additional sentence with little background knowledge: “If the level of oxytocin in the blood is increased, lactation is elevated.”
Two rules: "If p, then q" and "p only if q"
Both rules were randomly and counterbalanced presented for all contents in two rule phases (seesions two and three).
Dual-Source Model

20. Klauer et al. (2010), Experiment 3
predator: few disablers, few alternatives
balloon: few disablers, many alternatives
girl: many disablers, few alternatives
coke: many disabler, many alternatives
oxytocin: unkown content
Dual-Source Model

21. Klauer et al. (2010), Experiment 3
The model was fitted to 60 data points per individual with 25 parameters:
8 τ parameters (4 per rule)
2 λ parameters (1 per rule)
15 ξ paramaters (3 per content)
The additional rule only affected the form-parameters, the weighting parameters were identical (.88 versus .88).
Knowledge parameters followed the knowledge phase (session without rule)
Model fit was acceptable: .89
only if
if then
Dual-Source Model

22. Summary Dual-Source Model
The dual-source model tries to explain the empirical differences between responses to conditional problems without rule and with rule.
It describes probabilistic conditional reasoning as a weighted integration of two types of information, knowledge-based and form-based information.
The three model parameters map onto those processes:
ξ (xsi): quantifies the knowledge-based component (formalized as in Oaksford et al.'s, 2000, model)
τ (tau): form-based evidence
λ (lambda): weighting factor
Dual-Source Model

23. Suppression Effects (Byrne, 1989)
Counterexamples (disablers and alternatives) implicitly influence endorsement to inferences.
Byrne (1989) showed that additional premises with explicit counterexamples defeat otherwise valid inferences (i.e., human reasoning is non-monotonic)
Dual-Source Model

24. Suppression Effects Disabler Alternative
If a predator is hungry, then it will search for prey. A predator is hungry. How likely is it that it will search for prey? Please note: A predator will only search for prey, if it is physically fit.
If a predator is hungry, then it will search for prey. A predator is hungry. How likely is it that it will search for prey? Please note: A predator will also search for prey, if it needs to feed its offspring.
Disabler
Alternative
Dual-Source Model

25. Research question:
Which type of information (background knowledge or perceived validity) is affected by suppression effects?
Dual-Source Model

26. Suppression Effects – Experiment
We asked 41 participants to generate either alternatives or disablers for the four contents and used the three most frequently generated counterexamples for the next study.
Participants work on original and converse conditional inferences (same 4 contents) in two phases:
knowledge phase: without rule
rule phase: with rule
Three experimental conditions:
baseline (no explicit counterexamples), n = 29
disablers (three additional disablers), n = 31
alternatives (three additional alternatives), n = 31
Six control conditions:
3 groups (baseline, …) with 2 knowledge phases (n = 92)
3 groups with only a rule phase (no prior knowledge phase) (n = 90)
Dual-Source Model

27. Example Item (Disablers Condition)
Observation: A predator is hungry.
How likely is it that it will search for prey?
Please note:
A predator will only search for prey, if
it is physically fit,
it is living in the wild,
it has no prey left.
Dual-Source Model

28. Example Item (Disablers Condition)
Rule: If a predator is hungry, then it will search for prey.
Observation: A predator is hungry.
How likely is it that it will search for prey?
Please note:
A predator will only search for prey, if
it is physically fit,
it is living in the wild,
it has no prey left.
Dual-Source Model

29. predator
without rule
with rule
coke
without rule
with rule

30. girl
without rule
with rule
balloon
without rule
with rule

31. Dual-Source Model
We find suppression effects for both disablers and alternatives. The effect of the disablers seems to be weaker in the knowledge phase without rule.
We now fit the dual-source model with 16 parameters to the individual 32 datapoints:
4 τ parameters (1 per inference)
12 ξ paramaters (3 per content)
1 λ parameters (note the λ is calculated from the τ parameters)
Dual-Source Model

32. Fits, Weighting (λ), & Form (τ)
Model Fit was good (R² = .92), no differences between the conditions.
weight λ (lambda)
form τ (tau) **
Dual-Source Model

33. Knowledge Paramaters (Xsi)
3-way interaction of condition × content × inference is significant (p < .001), as well as the 2-way interaction of condition × inference (p < .001):
knowledge ξ (xsi)
Dual-Source Model

34. predator
xsi
girl
xsi
coke
balloon

35. Summary Suppression Effects
The dual-source model shows a dissociation for different suppression effects (results replicate from a prestudy):
Explicit disabling conditions
undermine the credibility of the rule: lower weighting λ
undermine the perceived form-based evidence for the valid conditional inferences: lower τ(MP) and τ(MT)
slight effects on knowledge parameter for MP: lower ξ(predator, MP) and ξ(balloon, MP)
Explicit alternative conditions
undermine the perceived form-based evidence for the invalid conditional problems: lower τ(AC) and τ(DA)
undermine the perceived knowledge-based evidence for all contents on the invalid inferences: lower ξ(C, AC) and ξ(C, DA)
Dual-Source Model

36. Summary
Modeling suppression effects with the dual-source model shows that both disabling conditions and alternative conditions suppress the subjective probability with which the relevant inferences seem warranted by the logical form of the inference (i.e., lower τ parameters).
Disabling conditions suppress the credibility of the conditional rule (i.e., lower λ parameters).
Alternative antecedents suppress the subjective certainty that an inference seems warranted by background knowledge (i.e., lower ξ parameters).
Dual-Source Model

37. Outlook: Next Study
It seems desirable to disentangle the contribution of the causal direction from the contribution of the conditional form of the presented inferences.
In the so far used materials the conditionals were all of the form "If cause, then effect". How will the role of disablers and alternatives to the conditional rule be, for conditional of the form "If effect, then cause" (diagnostic conditionals)?
To run such a study we need to develop new materials in which disablers and alternatives can be kept constant across the direction of the conditional rule.
Dual-Source Model

38. Research Goals How do humans reason with probabilistic relations?
Usage and integration of two different types of information: background knowledge and logical form.
Different linguistic forms alter the subjective probability to which an inference is seen as logically warranted.
The dual-source model allows us to estimate the contribution of those processes.
Dual-Source Model

39. Thank you.
Dual-Source Model

40. Consistency weighting
The model is originally fitted with a linear weighting function truncated at 40 and 160 (Klauer et al. 2010). Original R² = .92
We fitted the model with a number of different weighting functions (without truncation) :
linear (R² = .89)
logarithmic (R² = .92)
quadratic (R² = .89)
The results hold between the different weighting functions
Dual-Source Model

41. Control conditions Exp. 2#
condition
Phase 1
Phase 2 1
baseline
without rule
with rule 2
disabler 3
alternative 4 5 6 7
nothing 8 9
Dual-Source Model

42. Suppression Effects: Prestudy
Participants work on original and converse conditional inferences (4 contents) in two phases:
knowledge phase: without rule
rule phase: with rule
Three conditions:
baseline (no explicit counterexamples), n = 26
disablers (three additional disablers), n = 27
alternatives (three additional alternatives), n = 24
Dual-Source Model

45. Fits, Weighting (λ), & Form (τ)
Model fit is good (R² = .91), no difference between conditions
***
Dual-Source Model

46. Knowledge Paramaters (Xsi)
3-way interaction of condition × content × inference is not significant (p = .07), only the 2-way interaction of condition × inference (p < .001):
Dual-Source Model

48. Suppression Effects - prestudy
The dual-source model shows a dissociation for different suppression effects:
Explicit disabling conditions
undermine the credibility of the rule: lower λ
undermine the perceived form-based evidence for the valid conditional inferences: lower τ(MP) and τ(MT)
Explicit alternative conditions
undermine the perceived form-based evidence for the invalid conditional inferences: lower τ(AC) and τ(DA)
undermine the perceived knowledge-based evidence for all contents on the invalid inferences: lower ξ(C, AC) and ξ(C, DA)
Dual-Source Model