1. Henrik Singmann Sieghard Beller Karl Christoph Klauer
Conditional Reasoning: The Dual Source Model of Conditional Reasoning versus Minimizing the Kullback-Leibler Divergence
Henrik Singmann
Sieghard Beller
Karl Christoph Klauer

2. Conditional Reasoning
A conditional is a statement with the form: If p, then q.
Conditional inferences usually consist of the conditional (major premise), a minor premise (e.g., p) and the conclusion (e.g., q), e.g.:
If a balloon is pricked with a needle, then it will pop.
A balloon is pricked with a needle.
Therefore, the balloon will pop.
Dual-Source Model

3. 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
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
Dual-Source Model

4. 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
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
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:
What is the effect of the presence of the 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 contents of the conditional
↑ 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. Example Item: Knowledge Phase
Observation: A balloon is pricked with a needle.
How likely is it that it will pop? X
Dual-Source Model

13. The Knowledge Phase
Participants are asked to estimate a conclusion given a minor premise only. E.g., Given p, how likely is q?
The response should reflect the conditional probability of the 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

14. Formalizing the Knowledge Phase
We have the joint probability distribution over P(p), P(q), and their negations in the knowledge phase per content:
From this we can obtain the conditional probabilities, e.g.: P(MP) = P(q|p) = P(p  q) / P(q)
We need at least three independent parameters (e.g., P(p), P(q), and P(¬q|p), Oaksford, Chater, & Larkin, 2000) to describe the joint probability distribution. q ¬q p
P(p  q)
P(p  ¬q) ¬p
P(¬p  q)
P(¬p  ¬q)
Dual-Source Model

15. How do we explain the effect of the conditional?
When the conditional is absent, participants use their background knowledge to estimate the conditional probability of the conclusion given minor premise. E.g., Given p, how likely is q? P(q|p)
The presence of the conditional adds a different type of information: form-based evidence (i.e., the 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?
The dual-source model (Klauer, Beller, & Hütter, 2010) posits that people integrate these two types of information in the conditional inference task.
Dual-Source Model

16. Formalizing the Dual-Source Model
Observable response on one inference With conditional = λ{τ(x) × 1 + (1 – τ(x)) × ξ(C,x)} + (1 – λ)ξ(C,x) Without conditional = 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

17. How else could we explain it?
A prominent alternative to our assumptions is that the presence of the conditional simply changes the knowledge base from which people reason.
So far we have only had one formalization of this approach, Oaksford, Chater, and Larkin's (2000) model of conditional reasoning.
In their model the joint probability distribution is defined by P(p), P(q), and P(¬q|p) (the exceptions parameter), from which all four conditional probabilities can be calculated. (This is also the parametrization of the knowledge in the dual-source model)
The presence of the conditional decreases possible exceptions, P'(¬q|p) < P(¬q|p), so only affecting one of the underlying parameters.
Dual-Source Model

18. An alternative formalization
According to Hartmann and Rafiee Rad (2012, SPP workshop organized by Niki Pfeifer) when reasoning from conditionals and learning the premises, participants update their joint probability distribution by minimizing the Kullback-Leibler divergence with respect to the new information.
We therefore describe the joint probability distribution by a = P(p), α = P(q|p), and β = P(¬q|¬p).
When learning the conditional, participants update the conditional probability α' > α, and the other parameters (a' and β') update by minimizing the Kullback-Leibler distance to the original distribution.
With only one additional free parameter (α') we therefore also update the other parameters.
Dual-Source Model

19. Empirical Question:
Which of the three models provides the best account to the empricial data?
Dual-Source Model

20. Klauer, Beller, & Hütter (2010)
Experiment 1: emphasis on conditional
15 participants; 4 different contents; participants responded to all four inferences per content.
Three measurements (at least one week in between):
without conditional
with conditional (conditional stated "as in everyday context")
with conditional (treat conditional as true without exceptions)
Experiment 3: if-then versus only-if
18 participants; 5 different contents
with conditional: if-then
with conditional: only-if
(actually contents and type of rule, if-then versus only-if, were counterbalanced across measurements 2 and 3)
Dual-Source Model

21. Klauer, Beller, & Hütter (2010)
Experiment 1, 48 data points: 4 × 3 knowledge parameters +
Dual-source (DS) model: 2 × 4 independent form-based parameters = 20 parameters
Kullback-Leibler (KL) model: 2 × 4 updating parameters (updating of previous measurement) = 20 parameters
Oaksford et al.'s (OAK) model: 2 × 4 updating parameters (increasing across measurements) = 20 parameters
Experiment 3, 60 data points: 5 × 3 knowledge parameters +
Dual-source model: 2 × 4 independent form-based parameters = 23 parameters
Kullback-Leibler model: 2 × 5 updating parameters (independent) = 25 parameters
Oaksford et al.'s model: 2 × 5 updating parameters (independent) = 25 parameters
Dual-Source Model

22.
Dual-Source Model

23. all direct comparisons are significant.
only Kullback-Leibler and Oaksford et al.'s model do NOT differ significantly.
Dual-Source Model

24. Klauer, Beller, & Hütter (2010)
Experiment 4: Negations
13 participants; 4 contents
5 measurements:
without conditional
with conditional (if p, then q)
with conditional (if p, then not q)
with conditional (if not p, then q)
with conditional (if not p, then not q)
(as before, contents and negation of rule were actually counterbalanced so that at each measurement each content and each type of negation was presented once)
Dual-Source Model

25. Klauer, Beller, & Hütter (2010) – Exp 4
80 data points: 4 × 3 knowledge parameters +
Dual-source model: 4 form-based parameters = 16 parameters
Kullback-Leibler model: 4 × 4 independent updating parameters = 28 parameters
Oaksford et al.'s model: 4 × 4 independent updating parameters = 28 parameters * *
Dual-Source Model

26. Supression Effects Supression effects (data presented last year):
77 (dataset 1) and 91 (dataset 2) participants; 4 contents
3 between-subjects conditions:
Baseline (same as before)
Disablers (additional premises undermining the sufficiency of p for q)
Alternatives (additional premises undermining the necessity of p for q)
2 measurements:
without conditional (but with additional premises)
with conditional (and with additional premises)
All models have 4 × 3 knowledge parameters plus 4 form-based or updating parameters = 16 parameters for 32 data points
Dual-Source Model

27. all direct comparisons are significant.
Supression Effects
all direct comparisons are significant.
Dual-Source Model

28. Effects of Speaker Expertise
32 participants (still collecting); 7 contents
3 & 3 contents are randomly selected per participant
2 measurements:
without conditionals
with conditionals (3 conditionals uttered by expert, 3 conditionals uttered by non-expert)
Dual-Source Model

29. Effects of Speaker Expertise
48 data points: 6 × 3 knowledge parameters +
DS: 4 form parameters + 1 weighting parameter = 21 parameters
KL: 6 updating parameters = 22 parameters
OAK: 6 updating ps. = 22 parameters * *
Dual-Source Model

30. Summary Results
The dual-source model provides the overall best account for 6 datasets. For the two cases where it shares the first place (with Oaksford et al.'s model), it has less parameters.
The dual-source model provides the best account for 144 of 246 data sets (59%).
The Kullback-Leibler model (4 times second place, 53 best accounts) and Oaksford et al.'s model (2 times first place, one time second, 49 best account) share the second place.
This is strong evidence for our interpretation of the effect of the conditional. It seems to provide formal information that can not easily be accounted for by changes in participants' knowledge base.
Dual-Source Model

31. Beyond Model Fit
The parameters of the dual-source model offer an insight into different underlying cognitive processes, e.g., a dissociation of different suppression effects (disablers decrease the weighting parameters, whereas alternatives decrease the knowledge base for AC and DA).
The dual-source model can be easily extended to other types of connectives such as e.g., or, as it is not strictly tied to conditional reasoning.
Dual-Source Model