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Abductive, Causal, and Counterfactual Conditionals Under Incomplete Probabilistic Knowledge

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Abstract

We study abductive, causal, and non-causal conditionals in indicative and counterfactual formulations using probabilistic truth table tasks under incomplete probabilistic knowledge (N = 80). We frame the task as a probability-logical inference problem. The most frequently observed response type across all conditions was a class of conditional event interpretations of conditionals; it was followed by conjunction interpretations. An interesting minority of participants neglected some of the relevant imprecision involved in the premises when inferring lower or upper probability bounds on the target conditional/counterfactual ("halfway responses"). We discuss the results in the light of coherence-based probability logic and the new paradigm psychology of reasoning.
Abductive, Causal, and Counterfactual Conditionals
Under Incomplete Probabilistic Knowledge
Niki Pfeifer (niki.pfeifer@lmu.de)
Munich Center for Mathematical Philosophy, LMU Munich, Germany
Leena Tulkki (leena.tulkki@helsinki.fi)
Department of Philosophy, History, Culture and Art Studies, University of Helsinki, Finland
Abstract
We study abductive, causal, and non-causal conditionals in
indicative and counterfactual formulations using probabilis-
tic truth table tasks under incomplete probabilistic knowledge
(N=80). We frame the task as a probability-logical inference
problem. The most frequently observed response type across
all conditions was a class of conditional event interpretations
of conditionals; it was followed by conjunction interpreta-
tions. An interesting minority of participants neglected some
of the relevant imprecision involved in the premises when in-
ferring lower or upper probability bounds on the target con-
ditional/counterfactual (“halfway responses”). We discuss the
results in the light of coherence-based probability logic and the
new paradigm psychology of reasoning.
Keywords: abductive conditionals; causal conditionals; coun-
terfactuals; indicative conditionals; psychological experiment;
uncertain argument form; probabilistic truth table task
Introduction
The probabilistic truth table task was introduced by two in-
dependent studies at the beginning of this millennium (Evans,
Handley, & Over, 2003; Oberauer & Wilhelm, 2003). It
serves to investigate how people interpret conditionals under
uncertainty. Moreover, it is one of the starting points of the
new (probability-based) paradigm psychology of reasoning
(see, e.g. Baratgin, Over, & Politzer, 2014; Elqayam, Bon-
nefon, & Over, 2016; Oaksford & Chater, 2007; Over, 2009;
Pfeifer, 2013; Pfeifer & Douven, 2014), which started to re-
place the old (classical logic-based) paradigm psychology of
reasoning. The probabilistic truth table task was constructed
to investigate how people interpret conditionals (i.e., indica-
tive sentences of the form If A, then C). As its name suggests,
the task consists of inferring the degree of belief in a condi-
tional based on probabilistic information attached to the truth
table cases. What are truth table cases? Let Aand Cdenote
two propositions (i.e., sentences for which it makes sense to
assign the truth values true or false) like a fair die is rolled
and the side of the die shows an even number, respectively.
The four truth table cases induced by Aand Care: AC,
A∧ ¬C,¬AC, and ¬A∧ ¬C, where denotes conjunction
(“and”) and ¬denotes negation. Classical logic is bivalent,
involving only the truth values true and false. Therefore, the
conditional defined in classical logic (i.e., the material con-
ditional, see Table 1) is either true or false. The conditional
event C|Ainvolved in conditional probability (p(C|A)), how-
ever, is not bivalent, as it is void (or undetermined) if its an-
tecedent Ais false (see Table 1). Therefore, it cannot be rep-
resented by the means of classical logic.
Table 1 presents the truth conditions of the most impor-
tant psychological interpretations for adult reasoning about
indicative conditionals (i.e., conditional event, conjunction,
and material conditional). Moreover, it presents the bicondi-
tional and biconditional event interpretations, which were re-
ported in developmental psychological studies (see, e.g. Bar-
rouillet, Gauffroy, & Lecas, 2015).
Psychological evidence for the conditional event interpre-
tation was already observed within the old paradigm psychol-
ogy of reasoning (see, e.g., Wason & Johnson-Laird, 1972).
The response pattern, which is consistent with the conditional
event interpretation was seen as irrational (dubbed “defective
truth table”), as it violates the semantics of the material condi-
tional. The material conditional used to be the gold standard
of reference for the meaning of indicative conditionals in the
old paradigm. However, within the new paradigm psychol-
ogy of reasoning this response pattern is, of course, perfectly
rational (Over & Baratgin, 2017; Pfeifer & Tulkki, 2017).
Table 1: Truth tables for the material conditional AC, the
conjunction AC, the biconditional AC, the biconditional
event C||A(i.e., AC|AC), and the conditional event C|A.
A C A C A C A C C||A C|A
true true true true true true true
true false false false false false false
false true true false false false void
false false true false true void void
While classical truth table tasks require to respond with
truth values, probabilistic truth table tasks require to with
respond degrees of belief. Following Pfeifer (2013), we in-
terpret the task as a probability logical inference problem.
Specifically, it is formalised as a probability logical argument
with assigned degrees of belief in the four truth table cases
as its premises and the degree of belief in a conditional as its
conclusion. The inference problem consists in propagating
the uncertainties of the premises to the conclusion. As an ex-
ample, consider the conditional probability interpretation of
the (p(C|A)) conditional If A, then C as the conclusion. This
argument scheme is formalised by:
(A) From p(AC) = x1,p(A ¬C) = x2,p(¬AC) = x3,
and p(¬A∧ ¬C) = x4, infer p(C|A) = x1/(x1+x2).
In argument scheme (A), the fraction x1/(x1+x2)is the prob-
ability propagation rule for the conditional probability. For
different interpretations of the conditional, the probability
propagation rules differ. Table 2 presents the corresponding
probability propagation rules of the interpretations given in
Table 1.
Table 2: Probability propagation rules for the different in-
terpretations of If A, then C. The premise set {p(AC) =
x1,p(A∧ ¬C) = x2,p(¬AC) = x3,p(¬A∧ ¬C) = x4}en-
tails the respective conclusion (see also Table 1).
Interpretation Conclusion
Material conditional p(AC) = x1+x3+x4
Conjunction p(AC) = x1
Biconditional p(AC) = x1+x4
Biconditional event p(C||A) = x1/(x1+x2+x3)
Conditional event p(C|A) = x1/(x1+x2)
The main empirical result of classical probabilistic truth
table tasks is that the dominant responses are consistent with
the conditional event interpretation of conditionals. More-
over, if people solve the task several times, some people shift
to the conditional event interpretation during the course of the
experiment (see, e.g., Fugard, Pfeifer, Mayerhofer, & Kleiter,
2011a; Pfeifer & St¨
ockle-Schobel, 2015).
A key feature of classical probabilistic truth table tasks
is that they present complete probabilistic knowledge w.r.t.
the truth table cases: all (point-valued) probabilities x1,...x4
are available to the participant (see, e.g., Evans et al., 2003;
Oberauer & Wilhelm, 2003; Fugard et al., 2011a; Pfeifer
& St¨
ockle-Schobel, 2015). When all probabilities involved
in argument scheme (A) are given as point values, for ex-
ample, it is possible to infer a precise (point-valued) prob-
ability of C|A. Of course, x3and x4are irrelevant for cal-
culating p(C|A)in this case. However, as full probabilistic
information is usually not available in everyday life, we ar-
gue for investigating incomplete probabilistic knowledge. If
x1in argument scheme (A) is only available as an imprecise
(i.e., interval-valued) probability, i.e., x
1p(AC)x′′
1, the
probability of the conclusion of (A) is also imprecise, i.e.,
x
1/(x
1+x2)p(C|A)x′′
1/(x′′
1+x2).Table 4 (see below)
presents a numerical illustration of the different interpreta-
tions of conditionals in the probabilistic truth table task with
imprecise premises. Incomplete probabilistic knowledge has
not been investigated within the probabilistic truth table task
paradigm yet (for an exception, where only indicative condi-
tionals were investigated, see Pfeifer, 2013).
With only a few exceptions (i.e., Over, Hadjichristidis,
Evans, Handley, & Sloman, 2007; Pfeifer & St¨
ockle-Schobel,
2015), the important classes of causal conditionals and coun-
terfactuals have not been investigated empirically within the
probabilistic truth table task paradigm yet. Causal condi-
tionals are characterized by connecting cause (i.e., the con-
ditional’s antecedent) and effect (i.e., the conditional’s conse-
quent), like If you take aspirin, your headache will disappear.
Counterfactuals are conditionals in subjunctive mood, where
the grammatical structure signals that the antecedent is false.
For instance, If you were to take aspirin (A), your headache
would disappear (D), which signals that you had not taken
aspirin yet (¬A). This example is a counterfactual version of
the above described causal conditional. Of course, there are
also non-causal versions of counterfactuals, like If the side of
a card were to show an ace, it would show spades.
Traditionally, counterfactuals are interpreted by possible
world semantics (most prominently by Lewis and Stalnaker).
However, we interpret counterfactuals in terms of coherence
based probability logic. In a nutshell, the aforementioned
example of a (causal) counterfactual can by interpreted by
a (right-)nested conditional, where the antecedent represents
the factual statement ¬Aand the consequent represents the
conditional D|A. This representation amounts to (D|A)A,
which is a conditional random quantity (because of the page-
limit we refer for the technical details to Gilio & Sanfilippo,
2013, 2014; Gilio, Over, Pfeifer, & Sanfilippo, 2017, submit-
ted). It can be proved that the degree of belief in the con-
ditional random quantity (D|A)Ais also equal to p(D|A)
(i.e., Prevision((D|A)A) = p(D|A); see Gilio & Sanfilippo,
2013, Example 1, p. 225). Therefore, we hypothesize that the
participants’ degrees of belief in counterfactuals are equal to
corresponding conditional probabilities.
Note that the conditional random quantity (C|A)Adoes
not coincide with C|(A∧ ¬A), because the Import-Export
Principle does not hold (Gilio & Sanfilippo, 2014). There-
fore, as shown by Gilio and Sanfilippo (2014), the counter-
intuitive consequences of the well-known triviality results
(Lewis, 1976) are avoided. By the way, the formula C|(A
¬A)is unintelligible (in terms of the Ramsey test, you cannot
add a contradiction to your stock of beliefs).
To our knowledge, abductive conditionals have not been
empirically investigated in the probabilistic truth table task
paradigm yet. Abductive conditionals can be conceived as
reversed causal conditionals, characterized as follows: the ef-
fect is located in the conditional’s antecedent and the cause
is located in the conditional’s consequent. For example, If
your headache disappeared, then you took aspirin. Abduc-
tive inferences are also known as inferences to the best ex-
planation (for philosophical and psychological overviews on
abduction see, e.g., Douven, 2016a; Lombrozo, 2012, respec-
tively). Like indicative and causal conditionals, abductive
conditionals can be formulated in indicative and in subjunc-
tive mood.
The aim of the present study is to help to fill the above
mentioned research gaps. Specifically, we aim to shed light
on the following questions using probabilistic truth table tasks
under incomplete probabilistic knowledge: Are there reason-
ing strategies for inferring lower and upper bounds in the
context of incomplete probabilistic knowledge? Is the con-
ditional event interpretation dominant for abductive, causal,
and non-causal counterfactuals?
Method
Materials and Design The task material consisted of 18
pen and paper tasks, preceded by 4 examples explaining the
answer format. The task sequence consisted of 9 different
tasks that were presented twice in the same random order (i.e.,
task T10 is a repetition of task T1), resulting in the total of 18
tasks. The tasks were designed to test how participants infer
about uncertain conditional sentences in four different exper-
imental conditions (see Table 3). All conditions had the same
task sequence, with the following variations: For the first
two conditions we used a non-causal scenario in both indica-
tive and counterfactual moods. For the other two conditions
we used two variations of a causal scenario in counterfactual
mood; inference from causes to effects (causal) and inference
from effects to causes (abductive). The material was adapted
from probabilistic truth table tasks, which involved precise
premises (Fugard et al., 2011a; Pfeifer & St¨
ockle-Schobel,
2015).
Table 3: Between participant conditions C1–C4 defined by
the types and formulations of conditionals, and sample sizes.
Type Formulation Sample
C1 non-causal indicative (n1=20)
C2 non-causal counterfactual (n2=20)
C3 causal counterfactual (n3=20)
C4 abductive counterfactual (n4=20)
For the non-causal scenario we used a vignette story about
a six-sided die. The story describes that the die was randomly
thrown so that the participants did not know which of the
sides ended up facing upwards. The sides of the die were
illustrated as six squares. Each side had an image of a black
or white geometric figure. In tasks T1, T2, T10, and T11 all
sides of the die were shown. To introduce incomplete proba-
bilistic knowledge we presented “covered” sides in the rest of
the tasks. Covered sides were indicated by a question mark.
Here is an example of how we presented the six sides of a die
(task T3/T12):
(Die sides) ?
Next, the participants were presented with the question
“How sure can you be that the following sentence holds?”
(Kuinka varma voit olla siit¨
a, ett¨
a seuraava lause pit¨
a¨
a
paikkaansa?). The target sentences were highlighted with a
frame to make the scope of the question clear, for example:
If the figure on the upward facing side of the die is a circle,
then the figure is black.
The answer format had two sets of tick boxes in a “x out of y”
format for responding interval-valued degrees of belief. The
two response boxes were labeled accordingly (“at least” and
“at most”); for instance, as follows:
It was explained in the introduction to give point valued re-
sponses by marking the same numbers in both response boxes
(i.e., lower and upper bounds coincide).
The target sentence in the non-causal tasks was formulated
in terms of “If A, then C”. In all non-causal tasks the an-
tecedent mentioned a form and the consequent mentioned a
color. After completing each task, the participants were asked
to rate their confidence in the correctness of their response on
a 10-step rating scale from “fully confident that your answer
is incorrect” to “fully confident that your answer is correct”.
Apart from the following two differences, the counterfac-
tual task version was identical to the indicative version of
the task: (i) we added a factual statement which contradicted
the antecedent of the target sentence and (ii) the target sen-
tence was formulated in subjunctive mood. “The form of an
upward-facing side of the die is a cube” is an example of a
factual statement and the corresponding target sentence is:
“if the figure on the upward facing side of the die were a cir-
cle, then the figure would be black” (Jos yl¨
osp¨
ain osoittavan
kyljen kuvio olisi ympyr¨
a, niin t¨
am¨
a kuvio olisi musta). The
suffix -isi in the Finnish original indicates the counterfactual
mood.
For the causal and abductive conditions, the tasks were
structurally identical to the (non-causal) dice-scenario. How-
ever, instead of dice, a vignette story about drugs and their
effects created a causal scenario. In the vignette story, six
patient reports were shown to the participants. The patient re-
ports were illustrated as six rectangles having a name of a fic-
tional drug and a result of the medication (either “diminishes
symptoms” or “no impact on the symptoms”). We used ques-
tion marks on some patient reports (like in the dice scenario)
to introduce incomplete probabilistic knowledge. Here is an
example of the patient reports, which contains the same prob-
abilistic information as the above mentioned die-example:
In the causal version of the task material the antecedent
denotes the name of a drug and the consequent denotes an ef-
fect. In the abductive version the order was reversed: first an
effect was presented and then a drug was named. In this way
the tasks called for either causal inferences from causes to ef-
fects, or abductive inferences from effects to causes. As the
material was formulated in counterfactual mood, we added
a factual statement to each task, which contradicted the an-
tecedent of the target sentence.
Participants and Procedure Eighty students from the Uni-
versity of Helsinki (Finland) participated in the experiment.
The students were native Finnish speakers with no previous
academic training in logic or probability. Each participant
was tested individually. The paper and pencil tasks were fol-
lowed by a short structured interview about how the partici-
pants had interpreted the target tasks. Participants were paid
15¤for their participation.
Results and Discussion
We performed Fisher’s exact tests to investigate whether the
four different versions of the task booklets had an impact on
the participants’ degrees of belief in the respective target sen-
tences. After p-value corrections for multiple significance
tests, we did not observe significant differences between the
four conditions and we therefore pooled the data.
Table 5 shows the percentages of responses according to
the different interpretations of conditionals. All tasks differ-
entiate between the conditional probability, the conjunction,
and the material conditional interpretation. A subset of the
tasks differentiates between biconditional and biconditional
event interpretations as well. Conditional probability inter-
pretations marked with indices, however, were patterns iden-
tified from the data and were not anticipated during the con-
struction of the task material. Therefore, not all tasks differ-
entiate among all interpretations. Table 4 shows the norma-
tive answers for each interpretation for the example task dis-
cussed in the previous section (see (Die sides)). Both, lower
and upper bound responses, had to match the normative lower
and upper bounds for the categorization of the responses in
Table 5. Since each response box enables 42 different “X
out of Y” responses, and since both, lower and upper bound
responses needed to match for the classification, the a pri-
ori chance for guessing an interpretation was very low (i.e.,
1/(422) = 0.0006).
Figure 1: Mean confidence values for tasks T1–T18 by con-
dition. C1–C4 denote the four condition as defined in Table 3.
The task material was designed so that the normative pre-
dictions of the three main psychological interpretations of
conditionals (i.e., conditional probability, conjunction, and
Table 4: Example of predicted responses where the task con-
sists in inferring the degree of belief in the conditional “If the
figure on the upward facing side of the die is a circle,then
the figure is black” (i.e., the conclusion), based on the die
presented in (Die sides) above (i.e., (Die sides) contains the
premises). The index l(resp., u) denotes conditional proba-
bility responses where the covered sides are ignored for infer-
ring the lower (resp., upper) bound response. These response
types are the “halfway lower” and “halfway upper” interpre-
tations, respectively. lu denotes conditional probability re-
sponses where covered sides are ignored for inferring both
bound responses, i.e., the “halfway both interpretation”. See
also Table 2.
Interpretation Predictions
at least at most
p(black |circle)1 out of 2 2 out of 2
p(black |circle)l1 out of 1 2 out of 2
p(black |circle)u1 out of 2 1 out of 1
p(black |circle)lu 1 out of 1 1 out of 1
p(circle black)1 out of 6 2 out of 6
p(circle black)5 out of 6 6 out of 6
p(circle black)3 out of 6 4 out of 6
p(circle || black)1 out of 4 2 out of 4
material conditional) were unique for each task. During the
analysis we identified three further response strategies related
to the conditional probability interpretation. In what we call
halfway lower interpretation (denoted by p(·|·)l) the upper
bound is the same as in conditional probability, but the lower
bound response differs in that the covered sides (i.e., sides
marked with question mark) are ignored. Halfway upper in-
terpretation (denoted by p(·|·)u) is the same, but in reverse
order. In a halfway both interpretation the covered sides are
ignored for both bound responses. As these answer strate-
gies are in a sense partial versions of the conditional prob-
ability interpretation, we combined their results with condi-
tional probability answers into grouped conditional probabil-
ity. Notice that the tasks T1,T2, T10 and T11 with full infor-
mation (i.e., no question marks) have the same value for lower
and upper bound answers. Therefore the halfway responses
could not be distinguished from the conditional probability
answers in these tasks.
Of all 1440 responses, 32.1% were consistent with stan-
dard conditional event responses, 29.9% were consistent with
conjunction responses, and 0.2% were consistent with ma-
terial conditional responses. Like the material conditional,
also the biconditional and the biconditional event response
frequencies play a neglectable rˆ
ole in the data (0%–3% in the
four tasks T3, T6, T12, T15 where these interpretations were
differentiated). The predominant response strategy in point-
valued tasks (T1, T2, T10 and T11) was consistent with the
conditional probability interpretation. In nine out of 14 tasks
with incomplete probabilistic information (i.e., tasks involv-
Table 5: Percentages of responses from all four conditions
and all 18 tasks (T1–T18; N=80). “Grouped p(·|·)” denotes
the sum of all conditional probability responses, including
those marked with the indices. The halfway interpretations
(indices land u) and the numerical predictions are explained
in Table 4. “- -” denotes cases where different conditional
probability interpretations cannot be individuated (i.e., in the
point value tasks). Similarly, “[- -]” denotes cases where bi-
conditional and biconditional event interpretations cannot be
distinguished from the other interpretations. The interpre-
tations are explained in Table 2. ” denotes psychological
main interpretations.
Interpretation T1 T2 T3 T4 T5 T6
[p(·|·)][46] [55] [15] [19] [24] [24]
[p(·|·)l][- -] [- -] [5] [13] [18] [11]
[p(·|·)u][- -] [- -] [23] [10] [13] [11]
[p(·|·)lu ][- -] [- -] [0] [3] [1] [0]
Grouped p(·|·)46 55 43 44 55 46
p(· ∧ ·)29 28 34 39 34 31
p(· ⊃ ·)1 0 0 0 0 1
p(· ≡ ·)[- -] [- -] 1 [- -] [- -] 0
p(·||·)[- -] [- -] 3 [- -] [- -] 0
Other 24 18 20 18 11 21
T7 T8 T9 T10 T11 T12
[p(·|·)][24] [28] [26] [44] [55] [25]
[p(·|·)l][10] [15] [10] [- -] [- -] [9]
[p(·|·)u][16] [8] [10] [- -] [- -] [23]
[p(·|·)lu ][0] [0] [0] [- -] [- -] [0]
Grouped p(·|·)46 55 43 44 55 46
p(· ∧ ·)34 29 33 26 29 30
p(· ⊃ ·)0 0 0 1 0 0
p(· ≡ ·)[- -] [- -] [- -] [- -] [- -] 0
p(·||·)[- -] [- -] [- -] [- -] [- -] 0
Other 16 21 21 18 18 14
T13 T14 T15 T16 T17 T18
[p(·|·)][34] [33] [29] [26] [31] [31]
[p(·|·)l][9] [13] [11] [10] [18] [13]
[p(·|·)u][11] [10] [11] [15] [8] [11]
[p(·|·)lu ][0] [0] [1] [3] [0] [0]
Grouped p(·|·)46 55 43 44 55 46
p(· ∧ ·)28 30 26 29 25 28
p(· ⊃ ·)0 0 0 0 0 0
p(· ≡ ·)[- -] [- -] 0 [- -] [- -] [- -]
p(·||·)[- -] [- -] 3 [- -] [- -] [- -]
Other 19 15 19 18 19 18
ing “covered” sides or patient reports) the predominant an-
swer strategy was conjunction. We also observed shifts of
interpretation towards conditional probability: comparing the
first three tasks with incomplete information (i.e., T3–T5) to
the last three (i.e., T16–T18), the number of conditional prob-
ability answers increased from 19% to 30%, and conjunction
answers decreased from 35% to 27%. This replicates shifts of
interpretations reported in the literature (Fugard et al., 2011a;
Pfeifer, 2013).
However, when all the conditional probability response
types are grouped together, the resulting set of response
strategies is clearly the predominant one in all tasks. 51.5% of
all answers are consistent with the grouped conditional prob-
ability responses. 18.1% were “other” responses, that is, re-
sponses that did not fit the grouped conditional probability,
conjunction, biconditional, biconditional event, or material
conditional. Thus, in total 81.9% of the data can be modeled
by the investigated hypotheses concerning the interpretation
of conditionals.
Compared to a previous study that investigated non-causal
indicative conditionals under incomplete probabilistic infor-
mation (Pfeifer, 2013, i.e., similar tasks as in condition C1),
our results show lower level of conditional event responses
(compared to the previous 65.6%), and higher levels of con-
junction responses (compared to the previous 5.6%). The ma-
terial conditional responses were similar (previously 0.3%).
Pfeifer and St¨
ockle-Schobel (2015) investigated condition-
als under complete probabilistic knowledge and used similar
tasks as in our conditions C1, C2 and C3. These authors also
reported higher levels of conditional probability answers and
lower levels of conjunction responses, while material condi-
tional responses were similarly low. The lower levels of con-
ditional probability responses may be explained by the appar-
ent higher complexity of the tasks used in the present experi-
ment. The tasks are more complex (i) because of the combi-
nation of using counterfactuals as target sentences in three of
four conditions and (ii) because of imprecise probabilities in
the premises (i.e., incomplete probabilistic information).
The tendency to give answers that partially coincide with
conditional probability has also been found in a previous
study which tested non-causal cases in indicative mood with
similar task material as we used for condition C1 (Pfeifer,
2013). In that study our halfway lower-interpretation is re-
ferred to as “halfway conditional event strategy”. However,
in the present study we found two completely new strategies:
the halfway upper- and the halfway both-interpretation. The
halfway upper-interpretation is particularly interesting, as it
explains 12.8% of the total 1120 responses, slightly more than
the halfway lower response strategy (i.e., 11.6%). Halfway
conditional probability responses might unburden the work-
ing memory load by ignoring the covered sides (see also
Pfeifer, 2013).
Figure 1 shows the results of the confidence ratings. We
performed analyses of variance to investigate impacts of the
different conditions. After Holm-Bonferroni corrections we
observed significant differences within the three tasks T1, T2
and T10. The corresponding p-values were 0.006, 0.01, and
0.008. In each of these tasks—as well as in all other tasks—
the condition C4 had lower mean confidence values compared
to the other conditions. The lower confidence may be be-
cause of the apparent higher difficulty of the task material in
condition C4 for two reasons: first, the target sentence was
a counterfactual. Because of the inconsistency between the
factual statement and the antecedent, many participants re-
ported counterfactuals as puzzling in the post-test interview.
Second, abductive tasks required “backward” inference from
effects to causes and are incongruent with the more natural if
cause, then effect-direction. In general, backward inferences
are known to be harder to draw compared to forward infer-
ences (Evans & Beck, 1981).
Concluding Remarks
We investigated how people reason about conditionals under
incomplete probabilistic knowledge. The novel features in
our test design were comparisons of causal and abductive
scenarios, as well as counterfactuals under incomplete proba-
bilistic knowledge. One of our main findings is that the dom-
inant response is consistent with the conditional event inter-
pretation of conditionals among all four groups. Moreover,
we discovered two major answer strategies, halfway upper
and halfway lower conditional event responses, which can
be understood as strategies to unburden the working memory
load.
Inferentialist accounts of conditionals propose that there
should be an inferential relation between the antecedent and
the consequent (see, e.g., Douven, 2016b). Thus, when con-
ditionals with inferential relations (e.g., causal or abductive
ones) are compared with conditionals where no apparent in-
ferential relation exists (like in our conditions C1 and C2),
one would expect significant differences. Our data, however,
do not support this inferentialist hypothesis.
The results of our paper broaden the area of inferences
where conditional probability seems to be the best predictor
for how people reason. We have shown that coherence-based
probability logic provides a formalization of the meaning of
counterfactuals and provides a rationality framework for rea-
soning under complete and incomplete probabilistic knowl-
edge. This suggests that it may also be suitable for subclasses
of causal reasoning like abductive reasoning, which is impor-
tant in the studies on (scientific) explanation and learning.
Acknowledgments DFG project PF740/2-2 (SPP1516).
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