=Paper=
{{Paper
|id=Vol-2445/paper_1
|storemode=property
|title=Human Syllogistic Reasoning: Towards Predicting Individuals’ Reasoning Behavior based on Cognitive Principles
|pdfUrl=https://ceur-ws.org/Vol-2445/paper_1.pdf
|volume=Vol-2445
|authors=Emmanuelle-Anna Dietz Saldanha,Robert Schambach
|dblpUrl=https://dblp.org/rec/conf/ki/SaldanhaS19
}}
==Human Syllogistic Reasoning: Towards Predicting Individuals’ Reasoning Behavior based on Cognitive Principles==
https://ceur-ws.org/Vol-2445/paper_1.pdf
Copyright c 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
Human Syllogistic Reasoning:
Towards Predicting Individuals’ Reasoning
Behavior based on Cognitive Principles
Emmanuelle-Anna Dietz Saldanha and Robert Schambach⇤
International Center for Computational Logic, TU Dresden, Germany
Abstract. During the last decades, a tremendous effort was put into bring-
ing together the techniques developed within the area of computational
logic and the findings on human reasoning in cognitive science. In 2012,
Khemlani and Johnson-Laird published a meta-study on human syllogis-
tic reasoning, showing that none of the well-established cognitive theories
performed well with respect to human data. Recently, a novel cognitive
theory using techniques from computational logic seemed to outperform
these theories. In this paper we extend this approach by identifying indi-
vidual reasoning behavior and grouping it by clusters. Each cluster can
be characterized by a set of cognitive principles or by additional heuristic
strategies. We evaluate our approach with CCOBRA, a benchmark tool in
which cognitive models can learn from individual human patterns and
adapt their strategies accordingly.
1 Introduction
During the last decades, a tremendous effort was put into bringing together the
techniques developed within the area of computational logic and the findings
on human reasoning in cognitive science. It seems that the effort had effect, as
now terms such as computational cognition, computational cognitive science
or computational psychology get more and more attention within the area of
AI.
The meta-study on human syllogistic reasoning by Khemlani and Johnson-
Laird published in 2012 [1] provides an excellent overview of the most es-
tablished cognitive theories, and their performance on predicting human re-
sponses with respect to six psychological experiments in syllogistic reasoning.
The result was sobering: There are more than twelve different cognitive theo-
ries, but none of them predicted the human data well. This meta-study gave
a good starting point to test own models and to compare them with the other
ones. In 2017, the first syllogism challenge took place, where the performance of
the submitted models was tested on previously unseen data.1 In 2019, the chal-
lenge was further developed, providing the CCOBRA framework, an acronym
⇤
The authors are mentioned in alphabetical order.
1
www.cc.uni-freiburg.de/modelingchallenge/challenge-2017
2
Natural Language First-order Logic Mood Figure Premise 1 Premise 2
all a are b 8X(a(X) ! b(X)) Aab 1 a b b c
some a are b 9X(a(X) ^ b(X)) Iab 2 b a c b
no a are b 8X(a(X) ! ¬b(X)) Eab 3 a b c b
some a are not b 9X(a(X) ^ ¬b(X)) Oab 4 b a b c
Table 1: The four moods and their formalization. Table 2: The four figures.
for Cognitive COmputation for Behavioral Reasoning Analysis, an environ-
ment where models are given the individual participants’ response patterns
and adapt their strategies accordingly. Different than in the meta-study of [1]
and in the syllogism challenge in 2017, not the aggregated results of the psycho-
logical experiment, but the non-aggregated results the individual participants’
response patterns had to be dynamically predicted. In contrast to the challenge
in 2017, the challenge in 2019 considered two additional aspects: (1) Not only
the majority’s response, but the response with respect to an individual partici-
pant had to be predicted, and (2) the model had to dynamically adapt its strat-
egy, depending on the previously given (individual) participants’ responses.
Consider the following pair of syllogistic premises:
All bakers are artists. All chemists are bakers. (1)
Given these two premises, which conclusion on the relation between artists and
chemists necessarily follows? One conclusion that follows according to FOL is
All chemists are artists (Aca). Different to classical First-order Logic, but accord-
ing to the Aristotelian interpretation [2], universally quantified entities are as-
sumed to exist, i.e. Some chemists are artists (Ica) and Some artists are chemists (Iac)
are also valid conclusions. The majority of participants in experimental studies
concluded All artists are chemists (35%) and All chemists are artists (48%) [1]. Con-
sider another pair of syllogistic premises:
No artists are bakers. Some bakers are chemists. (2)
Some chemists are not artists (Oca) follows according to FOL. Again, accord-
ing to [1], this conclusion does not comply with the majority’s responses: 53%
concluded No artists are chemists (Eac) and 19% concluded No valid conclusion
(NVC).
Syllogisms have originally been introduced by Aristotle [3], and each syl-
logism consists of a pair of syllogistic premises, according to the four classical
moods and figures shown in Table 1 and 2, together with one conclusion about
a and c expressed in one of the four moods in Table 1. A conclusion can take
one of the following forms: Aac Eac Iac Oac Aca Eca Ica Oca
There are 42 ⇥ 4 = 64 different pairs of premises that can be uniquely speci-
fied by the abbreviations of the moods and figures in Table 1 and 2: The pairs
of syllogistic premises in (1) and in (2), are uniquely specified as AA2 and EI1,
3
Abbreviation FOL2 Majority of participants
AA2: Aba, Acb Aca, Iac, Ica Aac (35%), Aca (48%)
IE3: Iab, Ecb Oac Oac (20%), Eca (29%), NVC3 (26%)
EI1: Eab, Ibc Oca Eac (53%), NVC (19%)
EA2: Eba, Acb Eac, Eca, Oac, Oca Eac (28%), Eca (51%)
Table 3: Four pairs of syllogistic premises and the majority’s responses from [1].
respectively. Together with the 8 possible conclusions above, 64 ⇥ 8 = 512 dif-
ferent syllogism exist (out of which only 48 are valid according to FOL).
The responses given by the majority of participants’ for AA2 and three other
pairs of syllogistic premises are shown in Table 3. Note that, we understand
majority as in [1], i.e. any conclusion that is drawn by more than 16% of the par-
ticipants, is too high to be chosen randomly. Therefore, these conclusions are
interesting enough to be considered relevant for modeling. Two observations
follow immediately from the responses in Table 3: (i) Participants’ responses
do not always comply with the conclusions that are valid according to FOL,
and (ii) Participants do not agree on a given conclusion, i.e. they seem to rea-
son differently among each other. Furthermore, in psychological experiments,
participants are usually required to give only one response. Thus, even if Aac
and Aca do not need to exclude each other according to FOL, in experiments
participants that concluded Aac did not conclude Aca, and vice versa. As the
above example and recent investigations on psychological experiments show
(cf. [4]), it seems that different humans represent and reason about conditional
information differently. This implies that the human reasoner does not exist,
instead there seem to exist various (human) reasoning clusters. The identifica-
tion of these differences and the specification of these clusters is central for the
development of an adequate cognitive theory. Only recently approaches have
been proposed that account for individual differences (or clusters) in syllogistic
reasoning (cf. [4,5]).
The central requirements for the cognitive theory that we establish here is
that it has to be comparable to other theories and that it should provide qual-
itative justifications for its predictions. In Section 2 we present the underlying
theory together with the relevant cognitive principles. Section 3 introduces the
environment for this year’s syllogism challenge, the CCOBRA framework, and
the implementation of our theory and its learning strategy within this frame-
work. Finally, Section 4 provides an evaluation on the best performing set of
clusters and discusses new findings.
2
First-order logic: Valid conclusions according to FOL when existential import is as-
sumed.
3
No valid conclusion
4
2 Clustering by Principles
In the following, we suggest a principle- and cluster-based cognitive model.
For this purpose we establish the following two hypotheses according to the
previously made observations: (i) Humans make (not necessarily classical logic
valid) assumptions, and (ii) the human reasoner does not exist, instead there
are various reasoners. Accordingly, our goal is two-folded: First, providing one
framework for all (possibly not valid according to FOL) assumptions through
the characterization of cognitive principles, and second, characterizing reasoning
clusters by means of the cognitive principles. As a starting point, we take the
approaches presented in [6,5], which have developed eight principles for quan-
tified statements mostly justified by the cognitive science literature. In the fol-
lowing section we recall the principles and heuristic strategies that are relevant
for the purpose of this paper. Thereafter we briefly discuss the reasoning clus-
ters.
2.1 Principles
In [6,5], eight principles and two heuristic strategies have been formalized un-
der the Weak Completion Semantics [7,8], a logic programming approach that
understands models under the three-valued Łukasiewicz logic [9]. In this sec-
tion, we will not show their formal representation but rather give an intuitive
understanding by means of examples.
We distinguish between basic and advanced principles. Basic principles hold
for all reasoners, i.e. for all participants’ responses, for which we believe they
have been derived through reasoning. On the other hand, advanced principles
might not hold for all reasoners.
There are five basic principles, which are quantified statements as condition-
als (conditionals), licenses for inferences (licenses), existential import (import), un-
known generalization (unknownGen) and converse premise (converse).
For premises that are universally quantified, (conditionals), (licenses) and
(import) apply. Consider the first premise in AA2 from the introduction, All bak-
ers are artists: This natural language statement is to be understood as a con-
ditional statement with a license for inference, which we represent by means
of explicitly stating whether something is abnormal and include it within the
conditional:
All bakers, that are not abnormal, are artists. (conditionals & licenses)
By default, no baker is abnormal. (licenses)
The abnormality predicate within the rule allows for defeasible reasoning and
seems to be adequate for modeling rules within human reasoning [10]. Addi-
tionally, by the (import) principle, which states that entities which are univer-
sally quantified need to exist, we also assume the following statement:
Some bakers exist. (import)
The second premise of AA2 can be represented analogously by the above four
principles.
5
For premises that are existentially quantified, additionally, (unknownGen)
applies. Consider the second premise in EI1 from the introduction, Some bak-
ers are chemists. The idea behind (unknownGen) applies by specifying ab in the
conditional statement as follows:
All bakers, that are not abnormal, are chemists. (conditionals & licenses)
Some bakers exist, that are not abnormal. (licenses & unknownGen)
Some bakers exist, for whom it is unknown, (licenses & unknownGen)
whether they are abnormal.
As (unknownGen) is not symmetric, i.e. when we assume Some bakers are chemists
or Some chemists are bakers, the (possibly) resulting derivations from the above
assumptions might be different. However, when we additionally introduce the
(converse) principle, i.e. Some bakers are chemists implies Some chemists are bakers,
then (unknownGen) applies in both directions. Thus, additionally to the three
assumptions above, the (converse) principle applied to the second premise of
EI1 leads to the following three (symmetric) statements:
All chemists, (converse & conditionals & licenses)
that are not abnormal, are bakers.
Some chemists exist, (converse & licenses & unknownGen)
that are not abnormal.
Some chemists exist, for whom (converse & licenses & unknownGen)
it is unknown whether they are abnormal.
There are also advanced principles, not assumed by all reasoners: Generalization
(generalization), search for alternative conclusions (searchAlt) and contraposition
(contraposition).4
Consider again the pair of syllogistic premises, EI1 from the introduction:
Recall that Some chemists are not artists (Oca) follows according to FOL, by the
contrapositive of the first premise No bakers are artists, shown by the X in the
Venn Diagram in Figure 1. However, it doesn’t seem the case that all humans
reason with the contrapositive when applicable. Therefore, (contraposition) be-
longs to the advanced principles. In fact, about half of the participants con-
cluded No artists are chemists (53%). First, assume that (import) holds for the first
premise in EI1, thus artists exist. By (conditionals) and (licenses), we can derive
that these artists are not bakers. Second, assume that (converse) holds for the sec-
ond premise in EI1, thus Some chemists are bakers. The advanced principle exis-
tential as universal quantification (generalization) captures the idea that some par-
ticipants generalize over the existential quantification. Thus for EI1, some par-
ticipants might understand the second premise and its converse universally as
All bakers are chemists and All chemists are bakers (generalization). Consequently,
all artists are not chemists, i.e. No artists are chemists, which corresponds to what
53% of the participants concluded.
The (searchAlt) principle is motivated by the assumption that, if partici-
pants cannot straightforwardly derive any relation between artists and chemists,
4
(contraposition) is assumed together with (converse) for E.
6
a
b c
X
Fig. 1: In FOL, Some chemists are not artists follows from: No artists are bakers.
Some bakers are chemists.
rather than responding NVC, they might search for explanations. Consider yet
another pair of syllogistic premises, IA2:
Some bakers are artists. All chemists are bakers.
A possible reason why the majority of participants’ responses in [1] were Some
artists are chemists (27%) and Some chemists are artists (52%), can intuitively be
explained as follows: The first premise implies that some bakers exist. This can
be explained backwardly, by the information in the second premise, these bakers
are chemists.
Some conclusions given by a significant amount of participants cannot be
explained by the previously introduced (combinations) of principles. Possibly,
these participants simply guess or apply some strategy that is not explainable
by reasoning. For instance, according to the atmosphere bias [11], some humans
might be affected by the moods of the premises, in the sense that universal
conclusions (Aac, Aca, Eac, Eca) are excluded when one of the premises is ex-
istential (Iac, Ica, Oac, Oca) and affirmative conclusions (Aac, Aca, Iac, Ica) are
excluded when one of the premises is negative (Eac, Eca, Oac, Oca). In the case
of identical moods, the conclusion must have this mood as well. Consider again
EI1 from the introduction: Because the second premise is existential, any con-
clusion with the E and A mood is excluded, and because the first premise is
negative, any conclusion with A and I mood is excluded. The only possible con-
clusions which are left are Oac and Oca, where Oca is sometimes given by par-
ticipants (see participant B in Section 4). Consider yet another pair of syllogistic
premises, EA2:
No bakers are artists. All chemists are bakers.
As none of the premises are existential, universal conclusions are not excluded.
However, the second premise is negative, thus all affirmative conclusions are
excluded. The only conclusions that are left are Eac (28%) and Eca (51%), and
correspond to the majority’s responses in [1].
2.2 Clusters and Profiles
Recall that Cognitive principles (and heuristic strategies) are assumptions made
by humans. The relevant ones for this particular case of human syllogistic rea-
7
soning have been introduced in Section 2.1. Reasoning clusters are sets of cog-
nitive principles, where each cluster, i.e. each set of principles, characterizes a
group of reasoners. The population’s responses that are to be predicted can be
represented by a profile, where profiles are sets of clusters, covering the reason-
ing groups within a population. Khemlani and Johnson-Laird [4] identified the
profile that consisted of three clusters of reasoners in human syllogistic rea-
soning: intuitive, intermediate and deliberative. A clustering approach based on
cognitive principles was first introduced in [5] and was specified by five clus-
ters. Three of these clusters were based on a combination of basic and advanced
principles and two clusters were additionally based on heuristic strategies. This
profile achieved an overall match of 92% with the aggregated data provided
in [1]. In the following, we do not specify the profile beforehand but rather
choose the profile that performed best during training.
3 Modeling Challenge
One of the major goals of the (cognitive) modeling challenges is to provide
a competition among cognitive models that predict human behavior best. In
May 2019, the PRECORE challenge took place, where submissions were asked
to predict individual syllogistic reasoning5 . The modeling task is as follows:
First, the model is given a syllogistic reasoning problem and the participant,
whose response will serve as measure for the predictive accuracy. Second, the
model predicts a single-answer response between the nine different options ac-
cording to a strategy based on the previous responses of that participant (if they
exist). Finally, the models’ response is evaluated by means of its predictive ac-
curacy with respect to that participant. The predictive accuracy of a model with
respect to a modeling task is the probability that its predicted response matches
with the participants’ response. For instance, consider a model that randomly
predicts a response:6 For the above specified modeling task, i.e. for the predic-
tion of a single-answer response given nine different answer possibilities, the
probability that the model will correctly predict the participants’ response is
9 = 11.1%. The goal of the challenge is to provide cognitive models that opti-
1
mize their predictive accuracy.
The CCOBRA Modeling Framework, where CCOBRA is an acronym for
Cognitive COmputation for Behavioral Reasoning Analysis, is a benchmark
tool in which the cognitive model can adapt the optimal prediction strategy
through the pre-train phase, by matching the participants’ responses.7 In par-
ticular, in the pre-train phase, the model is asked to predict the participants’
responses, and then adapt its own strategy according to their responses. So far,
the best performance was just above 50%, which has been achieved through
machine learning techniques.8
5
https://www.cc.uni-freiburg.de/modelingchallenge
6
This model corresponds to the UniformModel in Figure 4 in the following section.
7
https://github.com/CognitiveComputationLab/ccobra
8
https://www.cc.uni-freiburg.de/staff/files/2019-04-dresden
8
Start Prediction phase
Start Main program Start Pre-train phase
for each
for each item
cluster
Input: Trainingset Input: Trainingset
Pre-train phase⇤ Train MFA
Ouput: MFA, best profile Ouput: MFA
Input: item, cluster
Predict answer⇤ Ouput: prediction
Input: Testset, best profile Input: Trainingset, profiles
Prediction phase⇤ Calculate Profile Score
Ouput: predictive accuracy Ouput: best profile
Adapt⇤ Input: item,
correct answer
End End
End
Fig. 2: Flowchart of program (left), pre-train (middle) and prediction (right).
3.1 Clustering by Principles
The idea of our approach is to match the participants’ responses with the re-
sponses given by a certain set of cognitive principles during the pre-train phase.
These sets of principles (clusters) should then help as a guideline for the gener-
ation of responses in the prediction phase.
Figure 2 (left) shows an overview of our implementation within the CCO-
BRA framework, which consists of two main phases: The pre-train phase and
the prediction phase, both specified in more details in Figure 2 (middle) and
Figure 2 (right), respectively. During the pre-train phase, for each pair of syl-
logistic premises, the most frequent answer (Train MFA) and the predictive ac-
curacy of each profile (Calculate Profile Scores) is computed. Note, that given
n
n principles, there are 2n clusters, and thus there are 22 profiles to be tested.
The responses given by MFA and the profile with the best score, will then be
used during the prediction phase. How the answers are predicted and adapted
is shown in Figure 3.
In Figure 3 (left) the response(s) for a pair of syllogistic premises (item) with
respect to the given cluster is predicted. In case only one response is predicted,
this one is chosen, and we are done. As the modeling task is specified such that
a single-answer response is required we need to make a choice when a cluster
predicts more than one response. This is done by taking the intersection of the
predictions of the chosen cluster and the predictions of the cluster with the next
highest accuracy (get response from cluster with next best score). If the intersec-
tion is non-empty, the process is repeated until at most one predicted response
is left. If the intersection is empty, then the predicted responses of a (new) clus-
ter, possessing the next highest predictive accuracy, is checked. If there is such
a cluster, the process is repeated again. If not, then the first response from the
initially chosen cluster is selected (select first response).
Figure 3 (right) specifies the adapt strategy. First, for each cluster that pre-
dicts the participants’ response correctly during the adapt phase, its correspond-
ing predictive accuracy (score) is increased dynamically, similar to MFA (most
frequent answer). Initially, when no information about the participants’ response
9
Start Adapt
Start Predict answer
Increase score of
clusters in profile Input: correct response
Get response(s)
Input: item for correct profile, item
from selected
cluster prediction
cluster
Adjust MFA Input: item, correct response
if # no
responses > 1
no if
yes # adaptations
MOD 5 = 0
get response from
cluster with
next best score yes
Select cluster
with highest score
yes if
select intersection intersection
non-empty
if
no
next
no
participant
yes
if prediction
yes not yet no select first
considered response Update
exists score of profile
End Return response
End
Fig. 3: Flowchart for answer prediction (left) and adaption of predictions (right).
behavior is known, the predicted answer will be the one corresponding to MFA.
After each five predictions, the cluster that matched the last responses of the
participants best, is chosen for the next five predictions, until the next partici-
pants’ responses are to be predicted. After all 64 responses for one participant
are predicted, the predictive accuracy (Update score of profile) of the profile is
updated.
4 Evaluation
For each of the 32 participants, the test set used in the prediction phase had
64 entries.9 Figure 4 gives an overview of the predictive accuracy (above) and
the respective box plot (below) of the benchmark models on the data provided
by CCOBRA,7 compared to Clustering by Principles. From left to right the figure
shows the results of the following models: uniform (when the responses are
9
https://github.com/CognitiveComputationLab/ccobra (Veser2018.csv)
10
Fig. 4: Pred. accuracy of benchmarks and Clustering by Principles (rightmost).
chosen randomly), matching bias [12], probability heuristics model [13], atmo-
sphere bias [11], mental model theory [14], NVC (when the chosen responses
are always no valid conclusion), (logic-based) PSYCOP model [15], illicit conver-
sion heuristics [16], verbals models theory [17], and clustering by principles. In
the respective box plots in Figure 4 (below), min and max refer to the worst
and best participants’ prediction, where the dots refer to the average predictive
accuracy of the individual participants.10
The profile of the Clustering by Principles model that predicted best the par-
ticipants’ responses achieved a predictive accuracy of just above 40%, which is
the highest score, compared to the performance of other well known cognitive
models.9 This profile consists of the following six clusters (1) Basic and Abduc-
tion, (2) (variations of) Basic and Abduction and Contraposition, (3) (variations
of) Basic and Abduction and Generalization, (4) (variations of) Atmosphere, (5)
Most frequent answer and (6) NVC. From a qualitative point of view, the first
four clusters are the most interesting ones, as they explain the participants’ re-
sponses. In particular, during the prediction phase, these clusters’ predictive
accuracy were as follows:
Basic and Abduction predicted well at least 58% of 16% (of the responses).
Basic and Abduction and Contraposition predicted well at least 58% of 19%.
Basic and Abduction and Generalization predicted well at least 41% of 16%.
Atmosphere predicted well at least 27% of 31%.
In total, this profile predicted between 27 and 58% of the responses 82% of
the participants correctly. One participant (participant D in Table 4) always an-
swered NVC, for which the NVC cluster achieved a predictive accuracy of 100%.
Consider the four pairs of syllogistic premises from the introduction again.
Table 4 provides the answers of four participants of the test set provided in the
CCOBRA framework. For AA2, all three participants agree on the conclusion,
All chemists are artists, which is also valid in Classical Logic. For EI1 however,
participants differ in their responses. In the case of IE3, only participant B and
C agree on Some artists are not chemists, whereas participant A generalizes the
10
Both figures have been computed by CCOBRA.
11
Participant11 AA2 IE3 EI1 EA2 Cluster
A Aca Eca Eac Eca ) Basic, Abduction and Generalization
B Aca Oac Oca Eac ) Atmosphere
C Aca Oac NVC Eca ) Basic, Abduction and Contraposition
D NVC NVC NVC NVC ) No valid conclusion
Table 4: Pairs of premises of the introduction and four participants’ responses.
conclusion to No chemists are artists. On the other hand, in the case of EA2, only
participant A and C agree on the conclusion.
5 Conclusions and Future Work
We presented clustering by principles, an approach that predicts individual hu-
man behavior for syllogistic reasoning. Beforehand, we specified a set of prin-
ciples, motivated by the literature. Depending on which set of principles is con-
sidered, different responses are generated. We then specified different clusters,
each through a set of principles. With the help of the CCOBRA Modeling frame-
work, in which cognitive models can evaluate their performance on real (hu-
man) data, and dynamically change their predictions, we suggest an adaptive
clustering by principles approach. During the pre-train phase, the best perform-
ing profile was computed, which was then chosen for the prediction phase. For
the training data provided by CCOBRA, the best performing profile consists of
six clusters, from which four are based on principles and heuristics. Compared
with the benchmark models, this profile reached the highest score with a pre-
diction of just above 40%. Even though the machine learning techniques still
achieve a higher score (50%),12 we believe that clustering by principles has the
advantage of explaining its predictions with the underlying principles.
6 Acknowledgments
Many thanks to Nicolas Riesterer for the helpful and quick replies on our ques-
tions concerning CCOBRA. The authors are grateful to the reviewers who pro-
vided many useful suggestions to improve the presentation.
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