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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: A∧C,

A∧ ¬C,¬A∧C, 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 A⊃C, the

conjunction A∧C, the biconditional A≡C, the biconditional

event C||A(i.e., A∧C|A∨C), 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(A∧C) = x1,p(A∧ ¬C) = x2,p(¬A∧C) = 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(A∧C) =

x1,p(A∧ ¬C) = x2,p(¬A∧C) = x3,p(¬A∧ ¬C) = x4}en-

tails the respective conclusion (see also Table 1).

Interpretation Conclusion

Material conditional p(A⊃C) = x1+x3+x4

Conjunction p(A∧C) = x1

Biconditional p(A≡C) = 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′

1≤p(A∧C)≤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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