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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.
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4 Conditional Inferences
Modus Ponens (MP):
Affirmation of the consequent (AC):
If p then q.
p
Conclusion: q
If p then q.
q
Conclusion: p
Modus Tollens (MT):
Denial of the antecedent (DA):
If p then q.
Not q
Conclusion: Not p
If p then q.
Not p
Conclusion: Not q
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4 Conditional Inferences
Modus Ponens (MP):
Affirmation of the consequent (AC):
If p then q.
p
Conclusion: q
If p then q.
q
Conclusion: p
Modus Tollens (MT):
Denial of the antecedent (DA):
If p then q.
Not q
Conclusion: Not p
If p then q.
Not p
Conclusion: Not q
NOT valid in standard logic (i.e., truth
of premises does NOT entail truth of
conclusion)
valid in standard logic (i.e., truth of
premises entails truth of conclusion)
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2-Phase Probabilistic Reasoning Task
knowledge phase (week 1)
rule phase (week 2)
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?
Observation: A balloon is NOT
pricked with a needle.
Observation: A balloon is NOT
pricked
with
needle.is pricked with a
Rule:
If aaballoon
needle, then it will pop.
How likely is it that it will NOT
pop?
Observation: A balloon is pricked
with a needle.
How likely is it that it will NOT
pop?
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How likely is it that it will pop?
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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?
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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?
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Dual-Source Model
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How do we explain the effect of the conditional?
Dual-source model for probabilistic conditional inferences
(Klauer, Beller, & Hütter, 2010):
hypotheses:
Our
Conditional
absent (knowledge phase)
Conditional probability
of conclusion
given minor premise
Conditional
provides
form-based
(from background knowledge). E.g.,
information
which
is integrated with
Given p, how likely
is q? P(q|p)
background knowledge.
 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?
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Formalizing the Dual-Source Model
knowledge-based
form-based
C = content (1 – 4)
x = inference (MP, MT, AC, & DA)
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Are dual-source model parameters valid?
1. Validating form parameter τ:
Does type of conditional solely affect form based
component?
-
Manipulation of form:
"If … then …" versus "If … then and only then …"
2. Validating mixture weights λ:
Effect of speaker expertise?
-
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Manipulation of who utters conditional:
expert versus non-expert
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Experiment 1
Does type of conditional solely affect form based component?
 Probabilistic conditional/biconditional reasoning tasks (n = 31):
1.
2.
3.

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:
-
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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.
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Estimating the dual-source model


If a person drinks a lot of
 48 data points and 20
coke, then the person will
parameters per participant.
gain weight.
 Fitted by minimizing squared
If a person drinks a lot of
deviations.
coke, then and only then the  Mean R² = .92
person will gain weight.
(.73 - .99)
if – then and only then
if - then
ξ predictions
knowledge phase
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Selective Influence
Mixture weights λ
Form parameters τ
**
ns.
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Experiment 2
What is the effect of speaker expertise? (Stevenson &
Over, 2001)
 Probabilistic conditional reasoning tasks (n = 47):
1. Session (knowledge phase): 6 problems without conditional.
2. 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
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Selective Influence
Mixture weights λ
mean R² = .85 (.27 - 1.00)
Form parameters τ
ns.
*
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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).
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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
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Formalizing the Dual-Source Model
Knowledge phase
Rule phase
 Joint probability distribution:
 Knowledge phase parameters ξ
q
¬q
p
P(p  q)
P(p  ¬q)
¬p
P(¬p  q)
P(¬p  ¬q)
 Mixture weights λ
 one τ per inference:
 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)
-
τ(MP)
τ(MT)
τ(AC)
τ(DA)
 Three independent parameters
 τ(x) represent subjective
(e.g., P(p), P(q), and P(¬q|p),
probability to which inference x is
Oaksford, Chater, & Larkin, 2000).
seen as (logically) warranted.
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Experiment 1
Does the conditional only affect the form based component?
 Probabilistic conditional/biconditional reasoning tasks (n =
31):
1.
2.
3.

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:
-
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Participants work on the problems without rule 3 times (n = 25).
Participants work on the problems with rules 2 times (n = 29).
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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)
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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)
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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.
-

If Katherina grows 10 cm then her ballet partner will fail to perform the lifting
routine with her.
-

Enviromental Scientist versus newspaper reader.
If Lisa invests in hedge funds then she will loose all the inversted money.
-

Pilot versus airline passenger
If individuals in industrialized nations continue to emit similar amounts of
CO2 then the Gulf stream will stop.
-

Manager versus Cashier
If the Airbus A380 flies through turbulences then its voltage level will
fluctuate.
-

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.
-

Nutrition scientist versus drugstore clerk
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.
-
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Der ärztliche Betreuer der Nationalmannschaft versus ein Journalist
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If Anne eats a lot of parsley then
the level of iron in her blood will
increase.
Nutrition scientist versus drugstore
clerk
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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)
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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.
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Model differences
Dual-Source Model
Alternative Views
Knowledge Phase:
 Probability distribution over
P(p), P(¬p), P(q), P(¬q) per content.
Rule Phase:
 Knowledge Phase +
 Form-based evidence per
inference (+ 4 parameters)
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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)
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Klauer, Beller, &
Hütter (2010):
Experiment 1 (n = 15)
error bars:
difference adjusted
Cousineau-MoreyBaguley intervals
conditional absent
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conditional present
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different lines:
different conditionals
(i.e., different
contents/items)
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Klauer, Beller, &
Hütter (2010):
Experiment 1 (n = 15)
error bars:
difference adjusted
Cousineau-MoreyBaguley intervals
↑ conditional absent ↓
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↑ conditional present ↓
Dual-Source Model
different lines:
different conditionals
(i.e., different
contents/items)
new data
(n = 29)
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Klauer, Beller, &
Hütter (2010):
Experiment 1 (n = 15)
error bars:
difference adjusted
Cousineau-MoreyBaguley intervals
different lines
Presence of conditional increases participants'represent
estimates
↑ conditional of
absent
↓
↑ conditional present ↓
of probability
conclusion.
different contents
the conditional
Especially for formally valid inferences MP andofMT.
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new data
(n = 29)
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Klauer, Beller, &
Hütter (2010):
Experiment 1 (n = 15)
error bars:
difference adjusted
Cousineau-MoreyBaguley intervals
different lines
How can we explain effect of presence of conditional?
represent
↑ conditional
absent ↓pure Bayesian
↑ conditional
↓
Our data
challenge
or present
probabilistic
different contents
of the conditional
approaches that solely rely on background knowledge.
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new data
(n = 29)
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