Conditionals are basic for human reasoning. In our paper, we present two experiments, which for the first time systematically compare how people reason about indicative conditionals (Experiment 1) and counterfactual conditionals (Experiment 2) in causal and non-causal task settings (N = 80). The main result of both experiments is that conditional probability is the dominant response pattern and thus a key ingredient for modeling causal, indicative, and counterfactual conditionals. In the paper, we will give an overview of the main experimental results and discuss their relevance for understanding how people reason about conditionals.
Classical logic used to be the dominating rationality
framework for psychological reasoning research in the
20th century. To deal with the defeasibility and
uncertainty of everyday life inference, probabilistic
rationality norms have gained popularity
Conditionals and reasoning about conditionals are basic for human reasoning. Among other things, conditionals can not only express abstract relationships but 1The conditional event CjA (“C given A”) is true if A ^ C (“A and C”) is true, false if A ^ :C (“A and not-C”) is true, and void if :A is true.
2The material conditional A C (“A implies C”, i.e.,
“not A or C”) is false if A ^ :C is true, but true otherwise.
allow also for representing causal information: If some
cause (e.g., taking aspirin) is present, then an effect
occurs (alleviates headache). Such causal conditionals are
closely related with counterfactuals. When people think
about whether taking aspirin and headache are causally
related, they ask whether the corresponding
counterfactual “If aspirin were taken, headache would be
alleviated” holds. Thus, understanding how people reason
about causal and counterfactual conditionals is crucial
for understanding causal cognition. Compared to the
vast psychological literature on indicative conditionals
Both experiments are designed to investigate the following key questions: Are there any differences in the probabilistic interpretations of conditionals, comparing indicative and counterfactual conditionals in causal and non-causal settings? How do people draw inferences from argument forms involving counterfactuals?
Participants Forty students of Protestant Theology at Augustana-Hochschule Neuendettelsau (Germany) were assigned equally to a non-causal and a causal conditional task set. Participants were payed 10¤. Task Materials The materials were adapted from the probabilistic truth table tasks used in Fugard, Pfeifer, Mayerhofer, and Kleiter (2011). Materials were presented in two pen and paper task sets. In each task, a short cover story introduced the domain of the task. For the non-causal conditions, we used pictures of six-sided dice with black or white geometric figures. The target sentences had the form “If the side shows white, then the side shows a triangle.” For the causal condition, we used stylized pictures of six medical data sheets detailing the (purely fictional) name of a drug and the medication’s effect on a patient’s symptoms. Target sentences had the form “If a patient takes Ambutal, then the symptoms diminish.” Participants were 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 in confidence accompanying shifts of interpretation of the conditional. The target sentence was formulated in the indicative “If A, then C”-form for the first 19 tasks. Task 20 and 21 formulated a disjunction of the negated antecedent (:A) and consequent (C) of a corresponding (and logically equivalent) material conditional (A C). Procedure Each participant was tested individually. After the pen and paper tasks, we collected qualitative data on how they interpreted the conditionals and the respective role of cause and effect by a structured interview.
Results After performing Holm-Bonferroni corrections for multiple significance tests, the probability response patterns of the first 19 tasks did not differ significantly between both groups. Participants in both groups predominantly chose the Conditional Event interpretation (see Figure 1 for details). The probability responses according to the three main interpretations of the conditional for tasks 1-19 (the tasks with “If A, then C” target sentences) were distributed as follows: In the non-causal group (n1 = 20), out of 380 responses, 81% were Conditional Event responses, 15% were Conjunction responses, 1% were Material Conditional responses, and 4% were “other” responses. In the causal group (n2 = 20), 95% were Conditional Event responses, 1% were Conjunction responses, 1% were Material Conditional responses, and 3% were other responses. Across both groups, 35 participants responded by the Conditional Event in at least 78% of the tasks.
We observed statistically significant differences between the non-causal and the causal group with regard to the probability responses for the pooled data from tasks 20 and 21 (the tasks with disjunctions as target sentences), as determined by Fisher’s Exact test (p = .04). In the non-causal group, 15% of responses were consistent with the Conditional Event response, 48% of responses were consistent with the Material Conditional responses, and 38% were other responses. In the causal group, 38% of responses were consistent with the Conditional Event response, 25% of responses were consistent with the Material Conditional responses, and 38% were other responses.
In total, eight participants shifted their interpretation Non−causal group Causal group 1 3 5 7 9 to the Conditional Event within the first 19 tasks and 38% of participants reported higher confidence values within the three tasks after the shift.
In the structured interview at the end of the experiment, participants’ responses confirmed the results reported above. Thirty-six participants explained their solution by appeal to features of reasoning with the Conditional Event interpretation, such as only counting the objects mentioned in the antecedent of the target sentence and then using this as the relevant set from which to count the objects that fit the consequent. When participants were asked to construct a consistent premise set based on a given degree of belief in a conclusion, 27 participants gave a set that corresponds unequivocally to the Conditional Event interpretation. Eleven participants produced sets that could fit either the Conjunction or the Conditional Event interpretation. Only one participant gave a set that corresponds unequivocally to the Conjunction interpretation.
75% of participants in the causal group judged the “symptoms diminish” target sentence to be an example of a relation of cause and effect, compared to 50% of participants for the “no influence” target sentence. By comparison, only 30% of participants in the non-causal group judged the “dice” target sentences to be examples of a relation of cause and effect. This validates the assumption that the medical task material triggered primarily causal reasoning whereas the dice task material did not do so.
Discussion The findings clearly show that the Conditional Event interpretation was the dominant response across both groups. Furthermore, participants in the causal group more frequently mentioned “cause” and “effect” in the interview, while the non-causal group did not do so: This can be interpreted as an indicator for causal reasoning in the causal group.
Participants Forty students of Protestant Theology at Augustana-Hochschule Neuendettelsau (Germany) were assigned equally to a non-causal and a causal conditional task set. Participants were paid 15¤ for their time. We ensured that no participant of Experiment 1 took part in Experiment 2.
Task Materials We used the same materials as in Experiment 1, with the difference that the target conditionals were replaced by corresponding counterfactual conditionals, such as “If the patient were to take Raverat, then it would have no influence on the symptoms” (“Wenn der Patient Raverat nehmen w u¨rde, dann ha¨tte es keinen Einfluss auf die Symptome”). To clearly mark the target sentences as counterfactual, we added information about a factual case to each task’s cover story. The factual cases diverged from the content of the antecedent 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
uncertain argument forms, which—to our knowledge—
have not been investigated experimentally with
counterfactual conditionals yet. We designed the tasks to
investigate inference schemes which are valid/invalid in
standard systems of counterfactuals
Procedure The procedure was identical to Experiment 1, except for the addition of argument form tasks, which we handed out as a final pen and paper task booklet. Table 1 lists the investigated argument forms. We also added two questions to the structured interview, to get an insight into the reasoning process during the uncertain argument form tasks.
Results As in Experiment 1, participants in both groups predominantly chose the Conditional Event interpretation (see Figure 2 for details). Also the probability response patterns of the first 19 tasks did not differ significantly between both groups. The probability responses according to the three main interpretations of the conditional for tasks 1-19 (the tasks with counterfactuals as target sentences) were distributed as follows: In the non-causal group (n3 = 20), out of 380 responses, 77% were Conditional Event responses, 13% were Conjunction responses, 1% were Material Conditional responses, and 9% were other responses. In the causal group (n4 = 20), 84% were Conditional Event responses, 8% were Conjunction responses, 2% were Material Conditional responses, and 6% were other responses. Across both groups (n3 + n4 = 40), 30 participants gave the Conditional Event response for more than 78% of the tasks.
The differences between the non-causal and the causal group with regard to the probability responses for tasks 20 and 21 (the tasks with “not-A or C” as target sentences) approach significance when the data for task 20 and 21 is pooled for each group (Fisher’s Exact test: p = .07). In the non-causal group, 13% of responses were consistent with the Conditional Event response, 35% of responses were consistent with the Material Conditional responses, 10% were consistent with the Conjunction response, and 43% were other responses. In the causal group, 35% of responses were consistent with the Conditional Event response, 18% of responses were consistent with the Material Conditional responses, 5% were consistent with the Conjunction response, and 43% were other responses.
The number of shifts of interpretation was similar to Experiment 1. Within the first 19 tasks, 13 participants shifted towards the Conditional Event interpretation and 38% reported higher confidence values within the three tasks after the shift.
In the interview at the end of the experiment, 30 participants explained their solution by appeal to a feature of the Conditional Event interpretation, such as restricting the set of relevant stimuli to those mentioned in the antecedent. Moreover, when participants were asked to construct a consistent premise set based on a given degree of belief in a counterfactual, 26 participants gave a set that corresponds unequivocally to the Conditional Event interpretation.
Like in Experiment 1, we observed that 80% of
participants in the causal group judged the symptoms
diminish target sentence to be an example of a relation
of cause and effect, compared to 60% of participants
for the no influence target sentence. By comparison,
only 15% of participants in the non-causal group judged
the dice target sentences to be examples of a relation of
cause and effect.
large majority of participants who did not assign a low
degree of belief in the conclusion of the Negated Modus
Ponens (NMP). Also the frequency of true responses
to Cut was lower than expected. While—under the
conditional event interpretation—NMP is
probabilistically informative (i.e., here, the coherent conclusion
probability is low), Contraposition (CP), Hypothetical
Syllogism (HS), and Premise Strengthening (PS) are
probabilistically non-informative (any conclusion
probability in the unit interval [0, 1] is coherent; see
Discussion As in Experiment 1, the findings clearly show that the Conditional Event interpretation was the dominant response across both groups. Moreover, more participants in the causal group associated the antecedent and the consequent with cause and effect, respectively, than in the non-causal group. In the uncertain argument form tasks, the majority of responses were consistent with conditional probability and with counterfactuals.
fer a negative reply to our first main question, whether there are any differences in the probabilistic interpretations of indicative and counterfactual conditionals in causal and non-causal settings: In all four conditions, the Conditional Event was the dominant response type.
One main difference between the results of Experiment 1 and Experiment 2 is that the counterfactual conditional tasks in the latter were arguably more difficult for the participants. Moreover, participants in Experiment 1 reported higher confidence in the correctness 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 Experiment 2 (M = 2.50, SD = 2.47). The reason for the higher difficulty of the counterfactual tasks could stem from the counterfactual conditionals themselves—the surface grammar is more complex than in indicative conditionals and this might be reflected in the reasoning process. Likewise, participants had to evaluate the relevance of the stated factual case (which contradicts the counterfactual antecedent). Across both task types T
Since there are no statistically significant
differences between the groups, we pooled the data for the
uncertain argument form tasks. The majority of the
responses to the uncertain argument forms involving
counterfactuals is consistent with indicative versions
of these argument forms observed in the literature
Our results vindicate the notion that de Finetti tables
aren’t defective truth tables, and they thus lend further
credence to the main tenets of the New Paradigm
Psychology of Reasoning
Furthermore, the present study contributes to the
study of shifts of interpretations of the conditional.
However, the effect was weaker than reported in
Causal Conditionals Although our results are in
accordance with
So, while there are some methodological differences
between the present study and
tion was: How do people draw inferences from
uncertain argument forms involving counterfactual
conditionals? The data from the inference tasks suggests the
following. As observed by
The other tasks furthermore provide additional
information about inferences from conditionals in more
complex cases. One main finding is that only few
participants (3–8%) judge the—under the material
conditional interpretation—deductively valid (even though,
in several cases, probabilistically non-informative)
argument forms to be invalid. The responses to the
Negated Modus Ponens task are atypical in this regard,
also because of the high percentage of “true” and “void”
responses. By comparison, in
Even the responses that prima facie don’t fit with the
Conditional Event interpretation don’t actually speak
against it, but rather highlight a pertinent pragmatic
issue in conditional reasoning. The high percentage of
participants assigning a high degree of belief to the
conclusion of the counterfactual Hypothetical Syllogism
can be explained by appeal to the following
conversational implicature: When stating A ) B as the first
premise, one sets a frame of reference for the usage of
B in the second premise B ) C—such that B ) C
actually means A^B ) C, as it is formalized in the
Cut inference schema
Furthermore, as
We conclude from this that the results of our present investigation into counterfactual conditional reasoning underline the importance of conditional probability not only for reasoning about indicative and causal conditionals but also for reasoning about counterfactual conditionals.
In both experiments and in all four experimental conditions, the Conditional Event interpretation is the dominant response type. This speaks for the ecological validity of the conditional probability hypothesis and indicates that conditional probabiltity is basic to indicative, counterfactual, and causal conditionals.
Finally, we note that probabilistic approaches where
conditional probability (p(CjA)) is defined by the
fraction of the joint (p(A ^ C)) and the marginal probability
(p(A)), cannot deal with zero-antecedent probabilities
(i.e., p(CjA) is undefined if p(A) = 0). However, as
pointed out by
This work is financially supported by the DFG grant PF 740/2-1 (project leader: Niki Pfeifer) as part of the Priority Programme 1516 “New Frameworks of Rationality” and by the Alexander von Humboldt-Foundation.