In a recent meta-analysis, Khemlani & Johnson-Laird (2012) showed that the conclusions drawn by human reasoners in psychological experiments about syllogistic reasoning are not the conclusions predicted by classical rst-order logic. Moreover, current cognitive theories deviate signi cantly from the empirical data. In this paper we show how human syllogistic reasoning can be modelled under the weak completion semantics, a new cognitive theory.
The way of how humans ought to reason correctly about syllogisms has already been investigated by Aristotle. A syllogism consists of two quanti ed statements using some of the four quanti ers all (A), no (E), some (I), and some are not (O)1 about sets of entities which we denote in the following by a, b, c. An example is: Some a are b
No b are c (IE1) and the task is to draw a conclusion. Implicitly, most experiments expect to draw a logical consequence from these so-called premises, e.g., `some a are not c' in classical rst-order logic (FOL). The four quanti ers and their formalization in FOL are given in Table 1. The entities can appear in four di erent orders called gures as shown in Table 2. Hence, a problem can be completely speci ed by the quanti ers of the rst and second premise and the gure. E.g., the example discussed above is denoted by IE1.
Altogether, there are 64 syllogisms and, if formalized in FOL, we can compute
their logical consequence in classical logic. However, a meta-analysis [
Natural Language all a are b some a are b no a are b some a are not b 8X(a(X) ! b(X)) 9X(a(X) ^ b(X)) 8X(a(X) ! :b(X)) 9X(a(X) ^ :b(X))
Short Aab Iab Eab Oab
In recent years, a new cognitive theory based on the weak completion
semantics (WCS) has been developed. It has its roots in the ideas rst expressed
by Stenning and van Lambalgen [
In the following, three important cognitive approaches are presented and
compared to predictions made by WCS. Firstly, we show how to adequately
represent the syllogisms in logic programs and, secondly, we demonstrate how
to reason with respect to them. Afterwards we compare the results under the
WCS with the results of FOL, the syntactic rule based theory PSYCOP [
The predictions of the theories FOL, PSYCOP, Verbal, and Mental Models
for the syllogisms OA3, EA3, and AA4 and those of the participants, taken
from [
Oca
Eac AA4
Aac, NVC
Oca Eac, Eca Oac, Oca Iac, Ica
Oca, Ica, Iac
Eac, Eca Oac, Oca
Iac, Ica
Ocs, NVC NVC, Eca
NVC, Aca Oca, Oac, NVC
Eac, Eca Aca, Aac,
Iac, Ica
the conclusion to be chosen randomly.3 The interested reader is referred to [
FOL and the other three cognitive theories make di erent predictions. In particular, each theory provides at least one prediction which is correct with respect to classical FOL and provides an additional prediction which is false with respect to classical FOL. Currently, the best overall results are achieved by the Verbal Models Theory which predicts 84% of the participants responses, closely followed by the Mental Model Theory with 83%, whereas PSYCOP predicts 77% of the participants responses. 2
The general notation, which we will use in the paper, is based on [
We assume the reader to be familiar with logic and logic programming, but
recall basic notions and notations. A (logic) program is a nite set of (program)
clauses of the form A >; A ? or A B1 ^ : : : ^ Bn; n > 0, where A is
an atom, Bi, 1 i n, are literals and > and ? denote truth and falsehood,
respectively. A is called head and >, ? as well as B1 ^ : : : ^ Bn are called body
of the corresponding clause. Clauses of the form A > and A ?4 are called
3 The threshold for the percentage to be signi cant is determined as follows: Given
that there are nine di erent conclusion possibilities, the chance that a conclusion has
been chosen randomly is 1=9 = 11:1%. A binomial test shows that if a conclusion is
drawn in more than 16% of the cases by the participants it is unlikely that is has
been chosen by just random guesses. The statistical analysis is elaborately explained
in [
gP denotes the set of all ground instances of clauses occurring in P, where
a ground instance of clause C is obtained from C by replacing each variable
occurring in C by a term not containing any variables. A ground atom A is
de ned in gP i gP contains a clause whose head is A; otherwise A is said to be
unde ned. def (S; P) = fA Body 2 gP j A 2 S _ :A 2 Sg is called de nition
of S in P, where S is a set of ground literals. S is said to be consistent i it does
not contain a pair of complementary literals. The set of atoms occurring in gP
is denoted as atoms(P).
We consider the three-valued Lukasiewicz logic (L-logic, [
A Body1; : : : ; A Bodym occurring in gP by A Body1 _ : : : _ Bodym. 2. Replace all occurrences of by $.
The obtained ground program is called weak completion of P or wcP.5
It has been shown in [
P (I) = hJ >; J ?i, where
J > = fA j A Body 2 def (A; P) and Body is true under hI>; I?ig J ? = fA j def (A; P) 6= ; and
Body is false under hI>; I?i for all A Body 2 def (A; P)g Weak completion semantics (WCS) is the approach to consider weakly completed logic programs and to reason with respect to the least L-models of these programs. We write P j=wcs F i formula F holds in the least L-model of wcP. In the remainder of this paper, MP denotes the least L-model of wcP.
The correspondence between weak completion semantics and well-founded
semantics [
A set of integrity constraints IC consists of clauses of the form U Body,
where Body is a conjunction of literals and U denotes the unknown.6 Hence, an
interpretation maps an integrity constraint to > i Body is either mapped to U
or ?. This understanding is similar to the de nition of the integrity constraints
for the well-founded semantics in [
We will apply four principles in developing a logical form for the representation of syllogisms. 3.1
Stenning and van Lambalgen [
Humans do understand quanti ers di erently due to a pragmatic
understanding of language. For instance, in natural language we normally do not quantify
over things that do not exist. Consequently, for all implies there exists. This
appears to be in line with human reasoning and has been called the Gricean
Implicature [
Logic programs do not allow negative literals as heads of clauses. In order to represent a negative conclusion :p(X) an auxiliary formula p0(X) is used together with a clause p(X) :p0(X) and the integrity constraint U p(X) ^ p0(X). This is a widely used technique in logic programming. Together with the principle discussed in Section 3.1, the additional clause becomes p(X) :p0(X) ^ :abnpp(X), and its weak completion is p(X) $ :p0(X) ^ :abnpp(X), stating that p0 is the negation of p i nothing abnormal is known with respect to that clause. The integrity constraint states that an object cannot belong to both, p and p0. We call this principle negation by transformation. 3.4
Humans seem to distinguish between `some y are z ' and `some z are y ', as the
results reported in [
In order to prevent such unwanted conclusions we introduce the following principle: if we know that `some y are z ' then there must not only be an object o1 which belongs to y and z (by Gricean implicature) but there must be another object o2 which belongs to y and for which it is unknown whether it belongs to z. We call this principle unknown generalization. 4
Based on the principles presented in the previous section, we can now represent the syllogisms by logic programs. The programs will be speci ed using the predicates y and z and depending on the gures shown in Table 2: yz must be replaced by ab, ba, cb or bc. 4.1 `All y are z ' is represented by the program PAyz which consists of the following clauses: z(X) abyz(X) y(o) y(X) ^ :abyz(X)
? > The rst two clauses are obtained by applying the principle of using licenses for inferences. The last clause follows by the principle of Gricean implicature, where o is the object which is assumed to exist for y. The least L-model for wcPAyz is hfy(o); z(o)g; fabyz(o)gi: 4.2 `No y are z ' is represented by the program PEyz which consists of the following clauses: z0(X) abynz(X) z(X) y(o) > abnzz(o) y(X) ^ :abynz(X) The rst two clauses are obtained by applying the principle of using licenses for inferences, where z0 is an auxiliary predicate symbol used to denote the negation of z. This auxiliary predicate is formally related to z by the third clause applying the principle of negation by transformation. In addition, this principle enforces the integrity constraint. The fourth clause of PEyz follows by the principle of Gricean implicature.
The last clause cannot be generalized to all X, because otherwise we allow conclusions by double negation. For example, consider the case where there is some o0 for which y(o0) is false due to some clause being part of another premise. In this case, the rst clause will enforce the falsehood of z0(o0). Now, if abnzz(X) ? would hold, then abnzz(o0) would be false and, consequently, by the third clause, z(o0) would be true. In other words, z(o0) follows by the negation of y(o0), which in turn is responsible for the negation of z0(o0). Even though this might be logically reasonable, the empirical results indicate that participants do not infer conclusions based on double negation. Therefore, we decided to restrict abnzz(o) ? to the objects occurring in PEyz.
The least L-model for wcPEyz is
hfy(o); z0(o)g; fabynx(o); abnzz(o); z(o)gi: `Some y are z ' is represented by the program PIyz which consists of the following clauses: z(X) abyz(o1) y(o1) y(o2) y(X) ^ :abyz(X)
? > > The rst two clauses are again obtained by the principle of using licenses for inferences. However, the abnormality predicate is restricted to the object o1, which is assumed to exist by the principle of Gricean implicature (see third clause). The fourth clause is obtained by the principle of unknown generalization. The least L-model of wcPIyz is
hfy(o1); y(o2); z(o1)g; fabyz(o1)g: Note that nothing is stated in PIyz about abyz(o2). Accordingly, z(o2) stays unknown in the least L-model. 4.4
`Some y are not z ' is represented by the program POyz which consists of the following clauses: In addition, we need the integrity constraint z0(X) abynz(o1) z(X) y(o1) > y(o2) > abnzz(o1) abnzz(o2) ? ? The rst four clauses as well as the integrity constraints are derived as in the program PEyz except that object o1 is used instead of o and abynz is restricted to o1 like in PIyz. The fth clause of POyz is obtained by the principle of unknown generalization. The last two clauses are again not generalized to all objects for the same reason as previously discussed in Section 5.2 for the representation of E. The least L-model for wcPOyz is
hfy(o1); y(o2); z0(o1)g; fabynz(o1); abnzz(o1); abnzz(o2); z(o1)gi:
We can now combine the proposed representations with respect to the gures in Table 2. In doing so, we replace yz by ab, ba, cb or bc. In addition, we may need to rename objects such that di erent objects are referred to in the representations of di erent syllogisms. Thereafter, we compute the least L-model of the obtained programs and check which syllogisms hold in this model.
However, we have not yet de ned when a syllogism holds given a program P. These de nitions will be developed in this section.
One should observe that [
The existence of an object o belonging to y is due to the principle of Gricean implicature. Moreover, all objects belonging to y must belong to z. The requirement that :abyz(o) is also entailed is a technical one which is based on the principle of licences for inferences. We may omit these abnormalities to obtain the following alternative de nition.
P j=0 Ayz i there exists an object o such that P j=wcs y(o) and for all objects o we nd that if P j=wcs y(o) then P j=wcs z(o). P j= Eyz i there exists an object o such that P j=wcs y(o) and for all objects o we nd that if P j=wcs y(o) then P j=wcs z0(o) ^ :z(o) ^ :abynz(o) ^ :abnzz(o).
The existence of an object o belonging to y is again due to the principle of Gricean implicature. Moreover, all objects o belonging to y must not belong to z and, hence, :z(o) must be entailed. The remaining entailed formulas are due to the principles of negation by transformation and of licenses for inferences. We may omit these formulas to obtain the following alternative de nition.
P j=0 Eyz i there exists an object o such that P j=wcs y(o) and for all objects o we nd that if P j=wcs y(o) then P j=wcs :z(o). P j= Iyz i there exists an object o1 such that P j=wcs y(o1) ^ z(o1) ^ :abyz(o1) and there exists an object o2 such that P j=wcs y(o2) and P 6j=wcs z(o2) ^ :abyz(o2).
The existence of an object o1 belonging to y is again due to the principle of Gricean implicature. This object o1 must also belong to z. In addition, there must be another object o2 belonging to y for which it is unknown whether it belongs to z due to the principle of unknown generalization. The abnormality predicates :abyz(o1) and :abyz(o2) are again due to the principle of licenses for inferences. We may omit these abnormalities to obtain the following alternative de nition.
P j=0 Iyz i there exists an object o1 such that P j=wcs y(o1) ^ z(o1) and there exists an object o2 such that P j=wcs y(o2) and P 6j=wcs z(o2). 5.4
P j= Oyz i there exists an object o1 such that P j=wcs y(o1) ^ z0(o1) ^ :z(o1) ^ :abynz(o1) ^ :abnzz(o1) and there exists an object o2 such that P j=wcs y(o2) and P 6j=wcs z0(o2) ^ :z(o2) ^ :abynz(o2) ^ :abnzz(o2).
This case combines E and I. First of all, there must be an object o1 which belongs to y and does not belong to z. Moreover, there must be another object o2 which belongs to y and for which it is unknown whether it belongs to z. If we omit the abnormalities and the auxiliary predicate z0 the following alternative de nition is obtained.
P j=0 Oyz i there exists an object o1 such that P j=wcs y(o1) ^ :z(o1) and there exists an object o2 such that P j=wcs y(o2) and P 6j=wcs :z(o2). 5.5
When we can not conclude any of the previous moods, then we derive from P that no valid conclusion holds. 6
Combining the syllogisms representation and entailment rules explained before we accomplished an average of 85% accuracy in our predictions. In 9 cases we have a perfect match with the answers given by the participants. In 29 cases the match is 89% and in 21 cases the match is 78%. The ve cases left, have a match of 67%.
First, we will explain how the accuracy of the predictions is computed in
general. After that we will show the logic program representation of three
syllogisms with di erent predictions accuracy, and compare their results with the
data from human experiments taken from [
Aac, Eac, Iac, Oac, Aca, Eca, Ica, Oca and NVC.
For every syllogism, we de ne a list of length 9 for the predictions of the weak completion semantics, where the rst element represents Aac, the second element represents Eac, and so forth. When Aac is predicted under the weak completion semantics for a given syllogism, then the value of the rst element of this list is a 1, otherwise it is a 0, and the same holds for the other eight elements in the list (representing the other eight answer possibilities). Analogously, for every syllogism we de ne a list of the participants' conclusions of length 9 containing either 1 or 0 for all nine answer possibilities, depending on whether the majority concluded Aac, Eac, and so forth. For each syllogism we then simply compare each element of both lists as follows, where i is the ith element of both lists: comp(i) = if both lists have the same value for the ith element otherwise The matching percentage of this syllogism is then computed by P9 i=1 comp(i)=9.
Note that the percentage of the match does not only take in account when the weak completion semantics correctly predicts a conclusion, but also whenever it correctly rejected a conclusion. The average percentage of accuracy is then simply the average of the matching percentage of all 64 syllogisms. 6.2
The syllogism OA4 is obtained by combining the last and the rst mood in Table 1 according to the gure 4 in Table 2. It can be read as:
Second Premise Abc `All b are c'
The program POA4 representing the two premises is obtained as the union of the programs POba (obtained from POyz by replacing y and z by b and a, respectively) and PAbc (obtained from PAyz by replacing y and z by b and c, respectively). In addition, the constant o occurring in PAbc has been replaced by o3. POA4 consists of the following clauses:
? :a0(X) ^ :abnaa(X) b(X) ^ :abbna(X) a0(X) abbna(o1) a(X) b(o1) > b(o2) > abnaa(o1) ? abnaa(o2) ? c(X) b(X) ^ :abbc(X) abbc(X) ? b(o3) The least L-model of POA4 is h fb(o1); b(o2); b(o3); abca(o1); a0(o1); c(o1); c(o2); c(o3)g;
fabbna(o1); abnaa(o1); abbc(o1); abbc(o2); abbc(o3); a(o1)g i: This model entails only the conclusion `some c are not a'. We have an object that is in c and not in a (POA4 j=wcs c(o1) and POA4 j=wcs :a(o1)) and an object that is in c and it is not known if it is not in a (POA4 j=wcs c(o2) and POA4 6j=wcs :a(o2)). This prediction matches perfectly with the answers from the participants. We will discuss now the syllogism EA3, one of the syllogisms with the lowest match (67%). EA3 can be read as:
The program PEA3 representing the two premises is obtained as the union of the programs PEab (obtained from PEyz by replacing y and z by a and b, respectively) and PAcb (obtained from PEyz by replacing y and z by c and b, respectively). In addition the constant o occurring in PEab and PAcb have been replaced by o1 and o2, respectively. PEA3 consists of the following clauses: b0(X) abanb(X) b(X) abnbb(o1) a(o1) b(X) abcb(X) c(o2) > a(X) ^ :abanb(X)
? :b0(X) ^ :abnbb(X) > c(X) ^ :abcb(X)
? ? The least L-model of PEA3 is h fa(o1); c(o2); b0(o1); b(o2)g;
fabanb(o1); abnbb(o1); abanb(o2); abnbb(o2); abcb(o1); abcb(o2)g i: This model does not entail any conclusion between a and c, so our prediction is NVC. Participants concluded both Eac and Eca.
Even though none of the entailed conclusions predicted by the weak completion semantics match to the participants' answers, a match of 67 % is computed. The reason is that we also take into account the correct rejections, i.e. the conclusions that are not entailed, as we have explained in Section 6.1 For AA4 we got an accuracy of 78% in our prediction. This syllogism combines two premises in the rst mood according to the gure 4. It can be read as:
The program PAA4 representing the two premises is obtained as the union of programs PAba (obtained from PAyz by replacing y and z by b and a, respectively) and PAbc (obtained from PAyz by replacing y and z by b and c, respectively). In addition the constant o occurring in PAba and PAbc have been replaced by o1 and o2, respectively. PAA4 consists of the following clauses: a(X) b(o1) abba(X) c(X) abbc(X) b(o2) > > b(X) ^ :abba(X)
? b(X) ^ :abbc(X)
? The least L-model of PAA4 is h fb(o1); b(o2); a(o1); a(o2); c(o1); c(o2)g;
fabba(o1); abba(o2); abbc(o1); abbc(o2)g i: This model entails both `all a are c' and `all c are a'. We have an object that is in a, and for all objects that are in a (P j=wcs a(o1) and P j=wcs a(o2)) it holds that they are also in c (P j=wcs c(o1) and P j=wcs c(o2)). Analogously this also holds for `all c are a'. This prediction matches partially with the answers from participants who concluded Aac and NVC. 7
We discussed three examples, which we formalized under WCS. The results are summarized and compared to FOL, PSYCOP, the Verbal, and the Mental Model Theory in Table 5. The selected examples are typical in the sense that for some syllogisms the conclusions drawn by the participants and WCS are identical, for some syllogisms the conclusions drawn by the participants and WCS overlap, and for some syllogisms the conclusions drawn by the participants and WCS are disjoint. Moreover, WCS di ers from the other cognitive theories. The overall result with respect to the 64 syllogisms under WCS shows that we can predict 85% of the participants responses. Compared to the other cognitive theories, we achieve the best performance, which closely followed by the Verbal Models Theory (84%) and the Mental Model Theory (83%). It seems natural to compare these theories in more detail and see where their similarities are and where they di er.
OA4 EA3 AA4
Oca Eac, Eca Aac, NVC
Oca Eac, Eca Oac, Oca Iac, Ica Oca, Ica, Iac
Eac, Eca Oac, Oca
Iac, Ica
Ocs, NVC NVC, Eca NVC, Aca
Oca, Oac, NVC Oca
Eac, Eca Aca, Aac,
Iac, Ica
NVC Aac, Aca Another aspect that might be interesting to investigate is whether the combinations of the moods in uence how the participants perceive the syllogisms. For instance, it seems that participants give di erent answers when they consider syllogistic premises of the same mood, especially in the cases for AA and EE, than when they consider syllogistic premises of two di erent moods. However, this observation needs to be further examined, and can possibly give us more insight about the human reasoning process.