Syllogisms and syllogistic reasoning has been the subject of research and scientific discourse for more than two millennia. Syllogisms sum quantified assertions into an overall statement, usually consisting of two premises and one conclusion. While syllogistic reasoning can be modeled by classical first-order logic in a straightforward manner, it is an open question which of the possible syllogisms are accepted as valid by human reasoners. In this paper, we present an approach that models the reasoning process with seven spaces of a set diagram. It can easily be implemented by constraint logic programming. We distinguish several assumptions that humans may make during their reasoning process, in particular that all used categories are non-empty. In contrast to pure logic-based approaches, the proposed procedure allows to represent diverse models human reasoners may follow. The results show good correlation and coincidence with psychological investigations.
Syllogistic argumentation can be traced back to Aristotle [
Example 1. Another example with the concrete categories ‘bakers’, ‘artists’, and
‘chemists’ (cf. [
Some artists are bakers.
All bakers are chemists.
∴ Some artists are chemists.
For this example, most people and also classical logic accept the last assertion as a consequence of the two premises.
It is easy to express a syllogistic statement in classical logic, namely by the logic of monadic assertions which is the subset of first-order logic where predicates, which correspond to categories in this context, always have exactly one argument. For instance, the assertion ‘All A are B’ can be expressed in monadic logic as:
∀x A(x) → B(x) The overall syllogistic statement then consists of three such assertions ϕ1, ϕ2, and ϕ3. It corresponds to the following logical implication:
ϕ1 ∧ ϕ2 → ϕ3 By means of a theorem prover or other kind of logical deduction system, valid syllogisms – in the sense of classical logic – may be determined automatically.
However, human reasoning does not strictly follow the rules of classical logic. Explanations for this may be incomplete knowledge, incorrect beliefs, or inconsistent norms. From the very beginning of artificial intelligence research, there has been a strong emphasis on incorporating mechanisms for such kind of rationality into reasoning systems. Rationality may be bounded, simply because it may be difficult for humans to have all possible cases in mind.
In addition, at latest since the famous Wason selection task [
In the following, we first make the terminology more precise and define among others the meaning of the notions category, mood, and syllogism (Section 2). Then, we briefly discuss related approaches, including works from cognitive psychology (Section 3). After that, we state our approach that models the reasoning process with seven spaces of a simple set diagram (Section 4), which can easily be implemented by constraint logic programming. We present some results (Section 5) which show good correlation with other suited theories of syllogistic reasoning and also with psychological investigations and end up with some conclusions (Section 6). 2
Let us now define more precisely what a syllogism is. For this, we refer to mathematical notions like sets and probabilities. Clearly, we cannot assume that people without much training in mathematics and logic have knowledge on this. Even trained people probably do not apply strict formal rules while deriving logical consequences. Nevertheless in the following we try to capture the required notions and terminology reasonably formally. On the one hand, this helps us to analyze and model underlying assumptions made by human reasoners more precisely. On the other hand, the definitions lay a clear basis for the implementation of the proposed set-based approach with seven spaces presented in this paper.
We start with the notion category: A category C stands for a set of individuals or objects. It is associated with a monadic predicate C(x), saying that x is an individual of the respective category. A classical syllogism combines in total three categories: A, B, and C. A space S is one of the seven possible subsets of A ∪ B ∪ C in the set diagram for the three involved categories (see Figure 1). The set of all such spaces is: M = {A ∩ B ∩ C, A ∩ B ∩ C, A ∩ B ∩ C, A ∩ B ∩ C, A ∩ B ∩ C, A ∩ B ∩ C, A ∩ B ∩ C}
A categorical assertion or just assertion for short for two categories A and B is a statement of the form ‘A certain quantity of A is (not) B’. Classically, four different moods of assertions are distinguished. They correspond to the quantifiers ‘All’ or ‘Some’ and are optionally combined with the negation of the second category involved in the assertion. In addition to the four classical moods, which are abbreviated A, E, I, and O, we also consider a fifth one, which we call U. Every mood can be directly interpreted by means of conditional probabilities P (B|A) as follows: A: ‘All A are B’
This means P (B|A) = 1. Since P (B|A) = P (A ∩ B)/P (A) and A is the disjoint union of A ∩ B and A ∩ B, this is equivalent to P (A ∩ B)/ P (A ∩ B) + P (A ∩ B) = 1 and hence P (A ∩ B) = 0. E: ‘All A are not B’
This means P (B|A) = 1. Since P (B|A) = 1 − P (B|A), this is equivalent to P (A ∩ B) = 0.
I: ‘Some A are B’
This means P (B|A) > 0, which is equivalent to P (A ∩ B) > 0.
O: ‘Some A are not B’
This means P (B|A) > 0, which is equivalent to P (A ∩ B) > 0.
U: ‘Some but not all A are B’
This means 0 < P (B|A) < 1. It corresponds to the conjunction of mood I and the negation of mood A which is equivalent to mood O. Therefore we have P (A ∩ B) > 0 and P (A ∩ B) > 0.
We consider the mood U here, because it allows to state more specific categorical assertions than with the mood I: Human reasoners or the experimental setting may assume that at most one assertion should be accepted as consequence of the premises, but from a logical point of view, there can be more than one valid conclusion, in particular if the classical mood I is used. For instance, ‘All A are B’ (mood A) implies ‘Some A are B’ (mood I) provided that the categories A and B are non-empty, but not ‘Some but not all A are B’ (mood U). In addition, mood I is a symmetric operator, i.e. ‘Some A are B’ and ‘Some B are A’ are logically equivalent. Thus for mood I the order of the categories in the conclusion does not matter, whereas in general it does for mood U. Interestingly, the variation of mood U ‘Some but not all A are not B’ is logically equivalent with mood U. To see this, just replace B by B and vice versa.
Now, three categorical assertions ϕ1, ϕ2, and ϕ3 are combined to an overall statement, called syllogistic statement or just syllogism for short. In this context, ϕ1 and ϕ2 are the premises and ϕ3 is called conclusion of the syllogism. If ϕ1 and ϕ2 together logically imply ϕ3, then we speak of a valid syllogism. The categorical assertion ϕ1 relates the categories A and B with each other, ϕ2 the categories B and C, and ϕ3 the categories A and C.
In each of the assertions, the two involved categories can occur in two orders, e.g. A–B or B–A for ϕ1. Therefore the premises ϕ1 and ϕ2 can have four different orders in total, called figures (cf. Figure 2). Hence, a problem can be completely specified by the moods of the first and second statement and the figure. The conclusion ϕ3 is also a quantified statement where the categories A and C can appear in either order.
Assertion Figure 1 Figure 2 Figure 3 Figure 4 ϕ1 A–B B–A A–B B–A ϕ2 B–C C–B C–B B–C
How many syllogisms are there to worry about? Actually, there is only a small, finite number of cases: Each categorical assertion may be in one of the classical n = 4 or extended n = 5 moods. The categories in each assertion may be in one of two orders. Hence, for the three categorical assertions ϕ1, ϕ2, and ϕ3, we have altogether
N = (n · 2)3 different syllogisms to consider. This yields us N = 512 or N = 1000 for n = 4 or n = 5 moods, respectively. Although conjunction is a commutative operator in classical logic, the presentation of the order of premises may play a role for human reasoners. However, since we already consider both possible orders of the categories A and C in the conclusion, this aspect is already covered and thus needs no special treatment. 3
Syllogistic reasoning has been extensively investigated by cognitive psychologists
(cf. the meta-study in [
Analyses of syllogisms based on first-order logic formalizations cause
several problems (cf. [
Heuristic theories of syllogistic reasoning assume more or less that human
reasoners do not apply any formal rules: According to the so-called atmosphere
theory [
According to [
From a set-theoretic point of view, [
In summary, there are many different theories trying to explain the
experimental findings. However, probably no single theory provides an adequate
account (cf. [
In the literature and in the corresponding psychological experiments, often several prerequisites are assumed for human reasoners, e.g. that all categories are non-empty. This is done implicitly or explicitly, namely as part of the instructions to the test persons in psychological experiments. These assumptions have also to be modeled. We do this using probabilistic notions:
Let us consider the probability space where an event E may be a set in M , i.e. one of the seven spaces (cf. Section 2), or a finite union thereof. The cardinality of the whole probability space is thus 27 = 128. Let P be a probability measure, assigning probabilities to the events. The following assumptions may be considered: 1. If a space S is non-empty, it has a non-zero probability, i.e. P (S) > 0 for
S = ∅. In the following, we always adopt this assumption. 2. Quantification with ‘Some’ (moods I and O) induces the existence of individuals or objects of the respective category. Furthermore it can be understood non-inclusive, i.e. as ‘Some but not all’ (mood U). 3. All categories are non-empty, i.e. A, B, C = ∅ and thus P (A) > 0, P (B) > 0, and P (C) > 0. 4. All categories are non-equivalent, i.e. A = B = C = A. Each of these three inequalities can be interpreted probabilistically, e.g. A = B by P (A\B ∪ B\A) > 0, because A = B is equivalent to A B or B A and hence to A\B ∪ B\A = ∅.
Since assumption 1 is of more or less purely technical nature, we always
adopt it hereinafter. The other assumptions may hold or not and can therefore
be switched on or off explicitly in our implementation, although human
reasoners may not even be aware whether they apply them. Assumptions 2 and 3
can be traced back to Aristotle’s original works on analytics (Analytica priora
and Analytica posteriora, cf. [
Syllogistic reasoning (including these assumptions) can now be implemented
in a more or less straightforward manner by constraint logic programming [
As one can see, every assumption from above and every mood of an assertion can be expressed by one or more conditions of the form P (E) = 0 or P (E) > 0 for some event (set) E. By assumption 1 this eventually means, for a specific case, we simply have to distinguish which spaces are empty and which are not, where a case is defined by its corresponding set of non-empty spaces S ∈ M . If a case satisfies the assumptions 3 and 4, we call it admissible. There are 27 − 3 · 23 + 3 + 2 = 109 cases satisfying assumption 3 and 109 − (1 + 3 · 4) = 96 admissible cases, i.e. where in addition assumption 4 holds.
In [
[S1, S2, S3, S4, S5, S6, S7] ins 0..1 – As said above, every mood can be interpreted by one or more conditions of the form P (E) = 0 or P (E) > 0 for some event (set) E. For example, the assertion ‘All A are B’ (mood A) is equivalent to P (A ∩ B) = 0. It holds A ∩ B = S1 ∪ S5 (cf. Figure 1). Thus this allows us – together with assumption 1 – to implement the condition simply by the constraint S1+S5 = 0, in Prolog syntax:
S1 + S5 #= 0 – Similarly, constraints of the form P (E) > 0 can be implemented: For example, ‘Some A are B’ (mood I) leads to the condition P (A ∩ B) > 0. Because of A ∩ B = S3 ∪ S7 we obtain the constraint S3 + S7 > 0, in Prolog syntax: S3 + S7 #> 0 S4 + S5 + S6 + S7 #> 0
S1 + S2 + S5 + S6 #> 0
– Assumption 3 can also be expressed easily by means of constraint logic
programming. To see this, we consider the example that category C is
nonempty, i.e. P (C) > 0. Because of C = S4 ∪ S5 ∪ S6 ∪ S7, we arrive at the
constraint S4 + S5 + S6 + S7 > 0, in Prolog syntax:
– Last but not least, assumption 4 is also feasible. For example, A = B is
equivalent to A\B ∪ B\A = ∅. Because of A\B = S1∪S5 and B\A = S2∪S6,
we obtain the constraint S1 + S5 + S2 + S6 > 0, in Prolog syntax:
Our constraint logic program for syllogistic reasoning provides
implementations of all assumptions in the above-mentioned manner and also of all 30
categorical assertions (not listed here in detail) that are possible with n = 5
moods involving two of the three categories in any order. By means of our
implementation, we can now find all solutions for every syllogism, simply by
means of Prolog built-in set predicates like bagof [
Example 2. Let us illustrate this by the following example with abstract categories:
All A are B.
All B are C. ∴ All C are A.
Here, all assertions are in mood A, the premises have the form of the syllogistic figure 1 (cf. Figure 2), and in the conclusion, C is related to A, in this order. The premises induce that S1, S2, S3, and S5 must be empty (cf. Figure 1). The remaining three spaces S4, S6, and S7 may be empty or not. Thus 23 = 8 cases satisfy both premises, at least if we allow empty and equivalent categories. The conclusion enforces that S4 and S6 must be empty, too. This means all three categorical assertions hold only if all spaces are empty except possibly S7. Therefore 21 = 2 cases remain. Hence the ratio, i.e. the fraction of valid cases is only 2/8 = 25% for this example.
The situation changes if we adopt one or more of the assumptions from above: For instance, if all categories are non-empty, then space S7 must be non-empty. Hence the syllogism of Example 2 holds in one of four cases, if assumption 3 holds, which again leads to a ratio of 1/4 = 25%. If in addition assumption 4 holds, then spaces S4 and S6 must be non-empty, because otherwise some of the sets are equivalent. The ratio is then 0/1 = 0%. 5
Table 1 shows the complete numbers and ratios for all possible syllogisms with
the classical four moods, i.e. without mood U. Each three consecutive columns
show the number of cases that satisfy all three assertions or only the premises
of the respective syllogism and their ratio in percent. Since in the literature and
in many psychological experiments it is normally given that all categories are
non-empty, we adopt assumption 3 here. Nevertheless we allow that some of the
category sets may be identical, i.e. assumption 4 does not necessarily hold. The
syllogisms that are valid in 100% of the cases are the logically valid syllogisms.
The tables for the other settings can be found in the technical report [
Since the ratio may vary and is not just 0 or 1, we can build a fine-grained
model for modeling syllogistic reasoning and try to predict human responses. In
psychological experiments, the setting often is that that two premises are given
and then the user has to draw the valid conclusions or find out that there are
no valid conclusions. This procedure is reported e.g. in [
First evaluation results for this hypothesis are encouraging: A simple correlation analysis between the percentage values computed by the constraint logic implementation and the empirical findings reported in [5, Table 6] yields, as desired, moderate positive correlation for all possible settings (cf. Table 2). Here the case that there are no valid conclusions is ignored. Interestingly, the maximum is attained if none of the assumptions 2, 3, and 4 is adopted – although the test persons might be instructed that they should hold. For instance, with assumption 3 the assertions in mood I with A and C in any order are also valid conclusions for Example 2 but are not without them. Only the latter coincides with the empirical finding that most people conclude (only) the assertion ‘All A are C’ (mood A). Taking also the case of no valid conclusions into account, we modify Hypothesis 1 as follows: Hypothesis 2. People draw a specific conclusion only if the percentage of its computed ratio is 100% and no valid conclusions otherwise.
If we repeat our comparison under Hypothesis 2, then we gain a coincidence of almost 90% over all 64 · 9 = 576 possible syllogisms including the case that two premises have no valid conclusions (see also Table 2). This is rather good. Here, a conclusion is adopted if the implemented model yields a percentage of 100% or the majority of people in the psychological experiments draw this conclusion, respectively. We thus map the real percentage values of the model and the psychological studies to just two values: yes or no. The coincidence with respect to no valid conclusions alone is about 70% (see again Table 2). This is significantly above chance. Clearly, this point needs further investigation and is still work in progress.
As already been noted (in Section 2), from a logical point of view, mood I is a symmetric operator. The same holds for mood E, i.e. ‘All A are not C’ and ‘All C are not A’ are logically equivalent. It follows that the approach for syllogistic reasoning in seven spaces presented here cannot distinguish the two categorical assertions for the moods I and E. Nonetheless, in the psychological experiments [5, Table 6], human reasoners definitely prefer the order A–C for the conclusion in 76,6% (mood I) and 70.3% (mood E) of the possible 64 cases (corresponding to the rows in Table 1). Furthermore, in all 16 cases where the premises have the form of figure 1, 100% of the human reasoners prefer this order or, at least, do not prefer the reverse order. 6
In this paper, we have introduced a set-based approach to syllogistic reasoning that allows a fine-grained analysis of syllogistic reasoning. It is cognitively simple, because it makes use of only seven spaces. In addition, there is a straightforward implementation by means of constraint logic programming, that is presented here for the first time. Last but not least, several assumptions humans implicitly may have can be tested, e.g. that all categories are non-empty.
Further work will concentrate on a more thorough analysis of syllogistic reasoning and comparison with the empirical findings. Possibly no single, monolithic theory can explain the whole picture. In consequence, the individual behavior of a test person may be derived from sample conclusions taken by that person. In addition, techniques from machine learning like case-based reasoning, clustering, decision tree learning, or neural networks may be employed.