=Paper=
{{Paper
|id=Vol-1419/paper0108
|storemode=property
|title=Uncertain Conditionals and Counterfactuals in (Non-)Causal Settings
|pdfUrl=https://ceur-ws.org/Vol-1419/paper0108.pdf
|volume=Vol-1419
|dblpUrl=https://dblp.org/rec/conf/eapcogsci/PfeiferS15
}}
==Uncertain Conditionals and Counterfactuals in (Non-)Causal Settings==
https://ceur-ws.org/Vol-1419/paper0108.pdf
Uncertain Conditionals and Counterfactuals in (Non-)Causal Settings
Niki Pfeifer (niki.pfeifer@lmu.de)
Munich Center for Mathematical Philosophy
Ludwig-Maximilians-Universität München
Geschwister-Scholl-Platz 1, D-80539 München
Richard Stöckle-Schobel (richard.stoeckle-schobel@lmu.de)
Munich Center for Mathematical Philosophy
Ludwig-Maximilians-Universität München
Geschwister-Scholl-Platz 1, D-80539 München
Abstract allow also for representing causal information: If some
Conditionals are basic for human reasoning. In our pa- cause (e.g., taking aspirin) is present, then an effect oc-
per, we present two experiments, which for the first time curs (alleviates headache). Such causal conditionals are
systematically compare how people reason about indica- closely related with counterfactuals. When people think
tive conditionals (Experiment 1) and counterfactual con- about whether taking aspirin and headache are causally
ditionals (Experiment 2) in causal and non-causal task
settings (N = 80). The main result of both experiments
related, they ask whether the corresponding counter-
is that conditional probability is the dominant response factual “If aspirin were taken, headache would be alle-
pattern and thus a key ingredient for modeling causal, viated” holds. Thus, understanding how people reason
indicative, and counterfactual conditionals. In the paper, about causal and counterfactual conditionals is crucial
we will give an overview of the main experimental re- for understanding causal cognition. Compared to the
sults and discuss their relevance for understanding how
vast psychological literature on indicative conditionals
people reason about conditionals.
(for an overview see, e.g., Evans & Over, 2004), stud-
Keywords: Causality; Conditionals; Conditional Proba-
bility; Counterfactuals; Reasoning; Uncertainty
ies on adult reasoning about counterfactuals are sur-
prisingly rare. Within the probabilistic truth table task
Introduction paradigm, counterfactuals were investigated only by
Classical logic used to be the dominating rationality Over et al. (2007). In our paper, we present two exper-
framework for psychological reasoning research in the iments, which for the first time systematically compare
20th century. To deal with the defeasibility and uncer- how adults reason about indicative conditionals (Exper-
tainty of everyday life inference, probabilistic rational- iment 1) and counterfactual conditionals (Experiment 2)
ity norms have gained popularity (e.g., Baratgin, Over, in causal and non-causal probabilistic truth table task
& Politzer, 2014; Elqayam & Over, 2012; Evans & Over, settings.
2004; Oaksford & Chater, 2009; Pfeifer, 2013; Pfeifer & Both experiments are designed to investigate the fol-
Douven, 2014). This development has influenced how lowing key questions: Are there any differences in the
the quality of human inference has been evaluated. Cor- probabilistic interpretations of conditionals, comparing
respondence between human inference about indica- indicative and counterfactual conditionals in causal and
tive conditionals and the semantics of the conditional non-causal settings? How do people draw inferences
event,1 for example, is nowadays regarded by most psy- from argument forms involving counterfactuals?
chologists of reasoning as rational, whereas the seman-
tics of the material conditional2 was regarded as the Experiment 1: Indicative Conditionals
normative gold standard in the last century. For this
Participants Forty students of Protestant Theology
reason, the majority of human responses in truth ta-
at Augustana-Hochschule Neuendettelsau (Germany)
ble tasks was labeled defective truth table, whereas it is
were assigned equally to a non-causal and a causal con-
broadly regarded as rational today, since this response
ditional task set. Participants were payed 10¤.
is not defective. Rather, it corresponds to the de Finetti
table (i.e., the truth table of the conditional event). Task Materials The materials were adapted from
Conditionals and reasoning about conditionals are the probabilistic truth table tasks used in Fugard,
basic for human reasoning. Among other things, con- Pfeifer, Mayerhofer, and Kleiter (2011). Materials were
ditionals can not only express abstract relationships but presented in two pen and paper task sets. In each task,
1 The conditional event C | A (“C given A”) is true if A ∧ C
a short cover story introduced the domain of the task.
For the non-causal conditions, we used pictures of
(“A and C”) is true, false if A ∧ ¬C (“A and not-C”) is true, and
void if ¬ A is true. six-sided dice with black or white geometric figures.
2 The material conditional A ⊃ C (“A implies C”, i.e., The target sentences had the form “If the side shows
“not A or C”) is false if A ∧ ¬C is true, but true otherwise. white, then the side shows a triangle.” For the causal
651
condition, we used stylized pictures of six medical data
sheets detailing the (purely fictional) name of a drug
10 12 14 16 18 20
and the medication’s effect on a patient’s symptoms.
Target sentences had the form “If a patient takes Ambu-
tal, then the symptoms diminish.” Participants were
Frequency p(C|A)
asked how sure they could be that the target sentence
holds. They responded by ticking boxes in a “x out
of y” format. Also, participants gave a rating for their
confidence in the correctness of their response for each
task, which we gathered to check for possible changes
8
in confidence accompanying shifts of interpretation of Non−causal group
Causal group
the conditional. The target sentence was formulated in
the indicative “If A, then C”-form for the first 19 tasks. 1 3 5 7 9 11 13 15 17 19
Task 20 and 21 formulated a disjunction of the negated
antecedent (¬ A) and consequent (C) of a corresponding Target task number (1−19)
(and logically equivalent) material conditional (A ⊃ C).
Figure 1: Frequency of participants per task giving the
Conditional Event response in the non-causal (n1 = 20)
Procedure Each participant was tested individually. and the causal group (n2 = 20), Experiment 1. The
After the pen and paper tasks, we collected qualitative dashed/solid lines were generated using the locally
data on how they interpreted the conditionals and the weighted scatter plot smoother method (lowess, imple-
respective role of cause and effect by a structured inter- mented in R).
view.
Results After performing Holm-Bonferroni correc- to the Conditional Event within the first 19 tasks and
tions for multiple significance tests, the probability re- 38% of participants reported higher confidence values
sponse patterns of the first 19 tasks did not differ sig- within the three tasks after the shift.
nificantly between both groups. Participants in both In the structured interview at the end of the experi-
groups predominantly chose the Conditional Event in- ment, participants’ responses confirmed the results re-
terpretation (see Figure 1 for details). The probabil- ported above. Thirty-six participants explained their
ity responses according to the three main interpreta- solution by appeal to features of reasoning with the
tions of the conditional for tasks 1-19 (the tasks with Conditional Event interpretation, such as only count-
“If A, then C” target sentences) were distributed as fol- ing the objects mentioned in the antecedent of the tar-
lows: In the non-causal group (n1 = 20), out of 380 re- get sentence and then using this as the relevant set
sponses, 81% were Conditional Event responses, 15% from which to count the objects that fit the consequent.
were Conjunction responses, 1% were Material Condi- When participants were asked to construct a consis-
tional responses, and 4% were “other” responses. In the tent premise set based on a given degree of belief in a
causal group (n2 = 20), 95% were Conditional Event conclusion, 27 participants gave a set that corresponds
responses, 1% were Conjunction responses, 1% were unequivocally to the Conditional Event interpretation.
Material Conditional responses, and 3% were other re- Eleven participants produced sets that could fit either
sponses. Across both groups, 35 participants responded the Conjunction or the Conditional Event interpreta-
by the Conditional Event in at least 78% of the tasks. tion. Only one participant gave a set that corresponds
We observed statistically significant differences be- unequivocally to the Conjunction interpretation.
tween the non-causal and the causal group with regard
75% of participants in the causal group judged the
to the probability responses for the pooled data from
“symptoms diminish” target sentence to be an exam-
tasks 20 and 21 (the tasks with disjunctions as target
ple of a relation of cause and effect, compared to 50% of
sentences), as determined by Fisher’s Exact test (p =
participants for the “no influence” target sentence. By
.04). In the non-causal group, 15% of responses were
comparison, only 30% of participants in the non-causal
consistent with the Conditional Event response, 48%
group judged the “dice” target sentences to be exam-
of responses were consistent with the Material Condi-
ples of a relation of cause and effect. This validates the
tional responses, and 38% were other responses. In the
assumption that the medical task material triggered pri-
causal group, 38% of responses were consistent with
marily causal reasoning whereas the dice task material
the Conditional Event response, 25% of responses were
did not do so.
consistent with the Material Conditional responses, and
38% were other responses. Discussion The findings clearly show that the Condi-
In total, eight participants shifted their interpretation tional Event interpretation was the dominant response
across both groups. Furthermore, participants in the
652
causal group more frequently mentioned “cause” and We also added two questions to the structured inter-
“effect” in the interview, while the non-causal group view, to get an insight into the reasoning process during
did not do so: This can be interpreted as an indicator the uncertain argument form tasks.
for causal reasoning in the causal group. Results As in Experiment 1, participants in both
groups predominantly chose the Conditional Event
Experiment 2: Counterfactual Conditionals interpretation (see Figure 2 for details). Also the
probability response patterns of the first 19 tasks did
Participants Forty students of Protestant Theology not differ significantly between both groups. The
at Augustana-Hochschule Neuendettelsau (Germany) probability responses according to the three main
were assigned equally to a non-causal and a causal con- interpretations of the conditional for tasks 1-19 (the
ditional task set. Participants were paid 15¤ for their tasks with counterfactuals as target sentences) were dis-
time. We ensured that no participant of Experiment 1 tributed as follows: In the non-causal group (n3 = 20),
took part in Experiment 2. out of 380 responses, 77% were Conditional Event
Task Materials We used the same materials as in Ex- responses, 13% were Conjunction responses, 1% were
periment 1, with the difference that the target condi- Material Conditional responses, and 9% were other
tionals were replaced by corresponding counterfactual responses. In the causal group (n4 = 20), 84% were
conditionals, such as “If the patient were to take Rav- Conditional Event responses, 8% were Conjunction
erat, then it would have no influence on the symptoms” responses, 2% were Material Conditional responses,
(“Wenn der Patient Raverat nehmen würde, dann hätte es and 6% were other responses. Across both groups
keinen Einfluss auf die Symptome”). To clearly mark the (n3 + n4 = 40), 30 participants gave the Conditional
target sentences as counterfactual, we added informa- Event response for more than 78% of the tasks.
tion about a factual case to each task’s cover story.
The factual cases diverged from the content of the an-
tecedent of the target sentence, e.g. the factual case
would state that the side of the die that faces up shows
a triangle, and the antecedent would state “If the side
were to show a circle.”
In addition, we investigated ten tasks involving un-
certain argument forms, which—to our knowledge—
have not been investigated experimentally with coun-
terfactual conditionals yet. We designed the tasks to in-
vestigate inference schemes which are valid/invalid in
standard systems of counterfactuals (e.g., Lewis, 1973).
The cover story involved the production of toy build-
ing blocks in different shapes, colours, and materials.
In the Modus Tollens case, an inspector just got a closed
box with a toy block in it (i.e., the factual case) and now Figure 2: Frequency of participants per task giving the
considers two beliefs (i.e., the premises). She is quite Conditional Event response in the non-causal (n3 = 20)
sure that: (A) If the toy block were green, then the toy and the causal group (n4 = 20), Experiment 2.
block would be a cylinder, and she is quite sure that
(B) the toy block is not a cylinder. Participants are then The differences between the non-causal and the
asked to judge how sure she can be, based on these causal group with regard to the probability responses
two sentences, that the conclusion, (C) the toy block for tasks 20 and 21 (the tasks with “not-A or C” as tar-
is not green, holds. Participants could respond by ei- get sentences) approach significance when the data for
ther judging that she cannot or that she can conclude task 20 and 21 is pooled for each group (Fisher’s Ex-
(C) based on (A) and (B) (i.e., is the argument proba- act test: p = .07). In the non-causal group, 13% of
bilistically non-informative or is it informative?). In the responses were consistent with the Conditional Event
latter case, participants additionally gave a response re- response, 35% of responses were consistent with the
garding whether she can be quite sure that the sentence Material Conditional responses, 10% were consistent
(C) holds or whether (C) doesn’t hold (i.e., is the degree with the Conjunction response, and 43% were other re-
of belief in the conclusion high or low?). sponses. In the causal group, 35% of responses were
Procedure The procedure was identical to Experi- consistent with the Conditional Event response, 18%
ment 1, except for the addition of argument form tasks, of responses were consistent with the Material Condi-
which we handed out as a final pen and paper task tional responses, 5% were consistent with the Conjunc-
booklet. Table 1 lists the investigated argument forms. tion response, and 43% were other responses.
653
The number of shifts of interpretation was similar large majority of participants who did not assign a low
to Experiment 1. Within the first 19 tasks, 13 partici- degree of belief in the conclusion of the Negated Modus
pants shifted towards the Conditional Event interpreta- Ponens (NMP). Also the frequency of true responses
tion and 38% reported higher confidence values within to Cut was lower than expected. While—under the
the three tasks after the shift. conditional event interpretation—NMP is probabilis-
In the interview at the end of the experiment, 30 par- tically informative (i.e., here, the coherent conclusion
ticipants explained their solution by appeal to a feature probability is low), Contraposition (CP), Hypothetical
of the Conditional Event interpretation, such as restrict- Syllogism (HS), and Premise Strengthening (PS) are
ing the set of relevant stimuli to those mentioned in the probabilistically non-informative (any conclusion
antecedent. Moreover, when participants were asked to probability in the unit interval [0, 1] is coherent; see
construct a consistent premise set based on a given de- Pfeifer and Kleiter (2006, 2009)). CP, HS, NMP and PS
gree of belief in a counterfactual, 26 participants gave are also invalid in standard systems of counterfactual
a set that corresponds unequivocally to the Conditional conditionals (e.g., Lewis, 1973). Pooling the false and
Event interpretation. void responses gives an indicator for the participant’s
Like in Experiment 1, we observed that 80% of par- evaluation of the validity of the respective argument
ticipants in the causal group judged the symptoms di- form. With the exception of Cut, HS, and PS, the
minish target sentence to be an example of a relation response patterns are also consistent with systems
of cause and effect, compared to 60% of participants of counterfactual conditionals. Moreover, the clear
for the no influence target sentence. By comparison, majority of responses to Negated Reflexivity (NR) and
only 15% of participants in the non-causal group judged both versions of Aristotle’s Thesis are consistent with
the dice target sentences to be examples of a relation of the Conditional Event interpretation (Pfeifer, 2012) and
cause and effect. counterfactuals.
Table 1: Results (N = 40) of the argument form tasks in Discussion As in Experiment 1, the findings clearly
percentages of responses. Predictions derived from the show that the Conditional Event interpretation was
Conditional Event interpretation are printed in bold. the dominant response across both groups. Moreover,
more participants in the causal group associated the
Name Argument form T F V antecedent and the consequent with cause and effect,
NMP A ⇒ B, A ∴ ¬ B 40 15 45 respectively, than in the non-causal group. In the un-
PS A ⇒ B ∴ A∧C ⇒ B 50 5 45 certain argument form tasks, the majority of responses
CM A ⇒ B, A ⇒ C ∴ A∧ B ⇒ C 78 3 20 were consistent with conditional probability and with
Cut A ⇒ B, A∧ B ⇒ C ∴ A ⇒ C 48 5 48 counterfactuals.
HS A ⇒ B, B ⇒ C ∴ A ⇒ C 63 3 35
CP B ⇒ ¬ A ∴ A ⇒ ¬B 30 8 63 General Discussion
MT A ⇒ B, ¬ B ∴ ¬ A 55 3 43 Interpretations of the Conditional Our findings of-
T F CT fer a negative reply to our first main question, whether
there are any differences in the probabilistic interpre-
NR ¬( A ⇒ A) 10 78 13 tations of indicative and counterfactual conditionals in
AT 1 ¬( A ⇒ ¬ A) 68 23 10 causal and non-causal settings: In all four conditions,
AT 2 ¬(¬ A ⇒ A) 70 20 10 the Conditional Event was the dominant response type.
Note: ⇒=counterfactual, ∴=conclusion indicator (“there- One main difference between the results of Experi-
fore”), ¬=negation, ∧=conjunction, T=true, F=false, V=void ment 1 and Experiment 2 is that the counterfactual con-
(i.e., non-informative conclusion probability), CT=can’t tell,
ditional tasks in the latter were arguably more difficult
NMP= Negated Modus Ponens, PS=Premise Strengthen-
ing, CM=Cautious Monotonicity, HS=Hypothetical Syllo- for the participants. Moreover, participants in Exper-
gism, CP=Contraposition, MT=Modus Tollens, NR=Negated iment 1 reported higher confidence in the correctness
Reflexivity, AT=Aristotle’s Thesis. of their responses across the task set of tasks 1-19 (on
a scale from -6 to +6, M = 4.04, SD = 2.06) than in Ex-
Since there are no statistically significant differ- periment 2 (M = 2.50, SD = 2.47). The reason for the
ences between the groups, we pooled the data for the higher difficulty of the counterfactual tasks could stem
uncertain argument form tasks. The majority of the from the counterfactual conditionals themselves—the
responses to the uncertain argument forms involving surface grammar is more complex than in indicative
counterfactuals is consistent with indicative versions conditionals and this might be reflected in the reason-
of these argument forms observed in the literature ing process. Likewise, participants had to evaluate the
(Pfeifer, 2012; Pfeifer & Kleiter, 2010)—see Table 1 for relevance of the stated factual case (which contradicts
detailed results. An exception to this agreement is the the counterfactual antecedent). Across both task types
654
in Experiment 2, 73% of participants (65% in the non- tual conditionals, on the basis of similar psychological
causal group and 80% in the causal group) commented processes.” (Over et al., 2007, p. 83) We submit that this
upon the factual case during the experiment or in the in- is due to the central role of probabilistic reasoning for
terview. In their comments, 43% of participants judged conditional reasoning in all of its modes that we have
the factual case to be irrelevant for solving the task (25% tested (counterfactual, indicative, causal).
in the non-causal group, 18% in the causal group). Uncertain Argument Forms Our second main ques-
Our results vindicate the notion that de Finetti tables tion was: How do people draw inferences from un-
aren’t defective truth tables, and they thus lend further certain argument forms involving counterfactual condi-
credence to the main tenets of the New Paradigm Psy- tionals? The data from the inference tasks suggests the
chology of Reasoning (cf. Pfeifer, 2013). The mental following. As observed by Pfeifer (2012) in the context
models explanation, appealing to the “implicit” men- of indicative conditionals, most participants used the
tal model of the conditional as the conjunction of an- Conditional Event interpretation when reasoning with
tecedent and consequent or the “explicit” model of the Aristotle’s Thesis (AT 1 and AT2) and Negated Reflexiv-
conditional as the material conditional of classical logic ity (NR). This new result for counterfactual conditionals
(cf. Johnson-Laird & Byrne, 2002), were only used by a further confirms the results from the probabilistic truth
small part of all four groups. table tasks and the hypothesis that conditional proba-
Furthermore, the present study contributes to the bility is fundamental for reasoning with uncertain con-
study of shifts of interpretations of the conditional. ditionals.
However, the effect was weaker than reported in The other tasks furthermore provide additional in-
Fugard et al. (2011). Since there was no time pressure formation about inferences from conditionals in more
during the experiment, it is possible, albeit not verifi- complex cases. One main finding is that only few par-
able with the data at hand, that some participants men- ticipants (3–8%) judge the—under the material condi-
tally shifted towards the Conditional Event while solv- tional interpretation—deductively valid (even though,
ing task 1, considering the Conjunction interpretation in several cases, probabilistically non-informative) ar-
or another interpretation before choosing the Condi- gument forms to be invalid. The responses to the
tional Event response. This idea is supported by the Negated Modus Ponens task are atypical in this regard,
fact that between 69% (Experiment 2) and 75% (Experi- also because of the high percentage of “true” and “void”
ment 1) of shifts occurred before task 4, i.e. early on in responses. By comparison, in Pfeifer and Kleiter (2007),
the experiment. the majority of participants gave coherent responses in
Causal Conditionals Although our results are in ac- the Modus Ponens tasks, including Modus Ponens with
cordance with Over et al. (2007), we observed higher a negated conclusion. The unusual responses in the
conditional event response frequencies. This could present study could be attributed to the task’s position
be caused by differences in the experimental mate- in the task set: It was the first task, and the task format
rial. First, their tasks elicited probabilistic judgements was arguably unfamiliar to participants. Also, difficul-
regarding conditional sentences concerning possible ties in processing negations are a well-known psycho-
states of affairs using background knowledge. Second, logical phenomenon (see, e.g., Evans, 1982).
more crucially, Over et al. (2007) asked participants to Even the responses that prima facie don’t fit with the
assign probability ratings to the four truth table cases Conditional Event interpretation don’t actually speak
(T ∧ T, T ∧ F, F ∧ T, F ∧ F) and then compared these val- against it, but rather highlight a pertinent pragmatic is-
ues to the conditional probabilities (the probability of sue in conditional reasoning. The high percentage of
the consequent given the antecedent) that participants participants assigning a high degree of belief to the con-
had given in addition to the four truth table cases. As clusion of the counterfactual Hypothetical Syllogism
pointed out in Fugard et al. (2011), asking for conjunc- can be explained by appeal to the following conversa-
tions could elicit higher frequencies in conjunction re- tional implicature: When stating A ⇒ B as the first
sponses. premise, one sets a frame of reference for the usage of
So, while there are some methodological differences B in the second premise B ⇒ C—such that B ⇒ C
between the present study and Over et al. (2007), their actually means A∧ B ⇒ C, as it is formalized in the
results fit well with the results from our experiment: Cut inference schema (see also Pfeifer & Kleiter, 2010).
Reasoning with causal conditionals can be best ex- The slight dominance of the “classical” response in
plained by appeal to the probability of the causal con- the Premise Strengthening inference can be interpreted
ditionals as conditional probability. Our results regard- analogously; participants might have assumed that the
ing the similarities between reasoning with counterfac- conjunction A∧C wouldn’t have been introduced with-
tual and indicative conditionals furthermore support out a relevant connection between A and C, such as
their hypothesis that “people [...] make similar prob- A ⇒ C. This explanation also fits with the high per-
ability judgments about [...] indicative and counterfac- centage of Conditional Event responses for the Cau-
655
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Acknowledgements tional probability logic. Kybernetika, 42, 391-404.
This work is financially supported by the DFG grant PF Pfeifer, N., & Kleiter, G. D. (2007). Human reason-
740/2-1 (project leader: Niki Pfeifer) as part of the Prior- ing with imprecise probabilities: Modus ponens and
ity Programme 1516 “New Frameworks of Rationality” Denying the antecedent. In G. De Cooman, J. Vej-
and by the Alexander von Humboldt-Foundation. narová, & M. Zaffalon (Eds.), Proceedings of the 5th in-
ternational symposium on imprecise probability: Theories
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