A Syllogistic Reasoning Theory and Three Examples

Abstract

A recent meta-study shows that the conclusions driven 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 the following, three important cognitive approaches are presented and compared to predictions made by a new approach to model human reasoning tasks, viz. the weak completion semantics. Open questions and implications are discussed.

Introduction

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 entities like 'some a are b' and 'no b are c', and is questioning about the logical consequences of these statements. E.g., 'some a are not c' is a logical consequence of the given two statements in classical rst-order logic (FOL). The four quanti ers and their formalization in FOL are given in Table 1. The entities can only 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 so far is IE1.

Altogether, there are 64 syllogisms and, if formalized in FOL, we can compute their logical consequences in classical logic. However, a meta-study [ 24 ] based on six experiments has shown that humans do not only systematically deviate from the predictions of FOL but from any other of at least 12 cognitive theories. In the case of IE1, besides the above mentioned logical consequence, a signi cant number of humans answered no a are c which does not follow from IE1 in FOL.

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 [ 32 ], but is mathematically sound [ 17 ], and has been successfully applied { among others { to the suppression task [ 8 ], the ? The authors are mentioned in alphabetical order. 1 We are using the classical abbreviations. A rmative universal (A) A rmative existential (I) Negative universal (E) Negative existential (O)

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 selection task [ 9 ], the belief bias e ect [ 28,29 ], to reasoning about conditionals [ 5,7 ] and to spatial reasoning [ 6 ]. Hence, it was natural to ask whether WCS is competitive in syllogistic reasoning and how it performs with respect to the cognitive theories considered in [ 24 ]. This paper gives some preliminary results by considering FOL, the syntactic rule based theory PSYCOP [ 31 ], and two model-based theories that performed well in the meta-study: the verbal model theory [ 30 ] and the mental model theory2 [ 19 ]. 2

Predictions of Cognitive Theories

Due to space limitations we will refer for the assumed operations and underlying cognitive processes of the other theories to [ 24 ]. The predictions of the theories FOL, PSYCOP, verbal, and mental models for the syllogisms IE1, EA3, and AA4 and those of the participants are depicted in Table 3. For the statistical analysis, the reader is refered to [ 24 ].

FOL and the other three cognitive theories make di erent predictions. Additionally, 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. 3

Syllogisms

Various theories have tried to explain this phenomenon. Some conclusions can be explained by converting the premises [ 2 ] or by assuming that the atmosphere of the premises in uences the acceptance for the conclusion [ 34 ]. Johnson-Laird and Byrne [ 21 ] proposed the mental model theory [ 20 ], which additionally supposes the search for counterexamples when validating the conclusion. Evans et 2 http://mentalmodels.princeton.edu/models/mreasoner/

IE1 EA3 AA4

Eac, Oac

Oac Eac, Eca

Eac, Eca Aac, NVC

Iac, Ica verbal models

mental models Oac, Iac,

Ica Eac, Eca, Oac, Oca Iac, Ica

Oac NVC, Eca NVC, Aca

Eac, Eca, Oac,

Oca, NVC Eac, Eca Aca, Aac,

Iac, Ica al. [ 12,11 ] proposed a theory which is sometimes referred to as the selective scrutiny model [ 14,1 ]. First, humans heuristically accept any syllogism having a believable conclusion, and only check on the logic if the conclusion contradicts their belief. Adler and Rips [ 1 ] claim that this behavior is rational because it e ciently maintains our beliefs, except in case if there is any evidence to change them. It results in an adaptive process, for which we only make an e ort towards a logical evaluation when the conclusion is unbelievable. It would take a lot of e ort if we would constantly verify them even though there is no reason to question them. As people intend to keep their beliefs consistent, they invest more e ort in examining statements that contradict them, than the ones that comply with them. However, this theory cannot fully explain all classical logical errors in the reasoning process. Yet another approach, the selective processing model [ 13 ], accounts only for a single preferred model. If the conclusion is neutral or believable, humans attempt to construct a model that supports it. Otherwise, they attempt to construct a model, which rejects it.

As summarized in [ 14 ], there are several stages in which a belief bias can take place. First, beliefs can in uence our interpretation of the premises. Second, in case a statement contradicts our belief, we might search for alternative models and check whether the conclusion is plausible. 4

Weak Completion Semantics

The general notation, which we will use in the paper, is based on [ 26,16 ]. 4.1

Logic Programs 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, 3 156 participants have been asked where the population ranges from highschool to university students.

F :F ^ > U ? _ > U ? > U ? $ > U ? > ? ? > U U

Bi, 1 i n, are literals and > and ? denote truth and falsehood, resp. 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 positive and negative facts, respectively. We restrict terms to be constants and variables only, i.e. we consider data logic programs. Throughout this paper, P denotes a program. We assume for each P that the alphabet consists precisely of the symbols occurring in P and that non-propositional programs contain at least one constant. When writing sets of literals we will omit curly brackets if the set has only one element.

gP denotes the set of all ground instances of clauses occurring in P. 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. We consider the three-valued Lukasiewicz Semantics [ 27 ], for which the corresponding truth values are >, ? and U, which mean true, false and unknown, respectively. A three-valued interpretation I is a mapping from formulas to a set of truth values f>; ?; Ug. The truth value of a given formula under I is determined according to the truth tables in Table 4. We represent an interpretation as a pair I = hI>; I?i of disjoint sets of atoms, where I> is the set of all atoms that are mapped to > by I, and I? is the set of all atoms that are mapped to ? by I. Atoms which do not occur in I> [ I? are mapped to U. Let I = hI>; I?i and J = hJ >; J ?i be two interpretations: I J i I> J > and I? J ?: I(F ) = > means that a formula F is mapped to true under I. M is a model of gP if it is an interpretation, which maps each clause occurring in gP to >. I is the least model of gP i for any other model J of gP it holds that I J . 1. For each A where def (A; P) 6= ;, replace all

A Body1; : : : ; A Bodym 2 def (A; P) by A 2. Replace all occurrences of by $. 4 We consider weak completion semantics and, hence, a clause of the form A turned into A $ ? provided that this is the only clause in the de nition of A. ? is Body1 _ : : : _ Bodym.

The obtained ground program is called weak completion of P or wcP.5

It has been shown in [ 18 ] that logic programs as well as their weak completions admit a least model under L-logic. Moreover, the least L-model of wcP can be obtained as the least xed point of the following semantic operator, which is due to Stenning and van Lambalgen [ 32 ]: P (hI>; I?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 [ 33 ] for tight programs, i.e. those without positive cycles, is shown in [ 10 ]. 4.4

Integrity Constraints A set of integrity constraints IC comprises clauses of the form ? Body, where Body is a conjunction of literals. Under a three-valued semantics, there are several ways on how to understand integrity constraints [ 23 ], two of them being the theoremhood view and the consistency view. Consider the IC ?

:p ^ q The theoremhood view requires that a model only satis es the set of integrity constraints if for all its clauses, Body is false under this model. In the example, this is only the case if p is true or if q is false in the model. In the consistency view, the set of integrity constraints is satis ed by the model if Body is unknown or false in it. Here, a model satis es IC already if either p or q is unknown.

In this paper we adopt the consistency view. Formally, given P and set IC, P satis es IC i there exists I, which is a model for gP, and for each ? Body 2 IC, we nd that I(Body) 2 f?; Ug.

Reasoning Towards an Appropriate Logical Form 5

5.1

Existential Import: Modeling Gricean Implicature We assume that humans understand quanti ers with existential import, i.e. for all implies there exists. This is a reasonable assumption { called the Gricean Implicature [ 15 ] { as in natural language we normally do not quantify over things that do not exist. Furthermore, Stenning and van Lambalgen [ 32 ] have shown that humans require existential import for a conditional to be true. The program for A in Table 1 together with existential import is

PA = fb(X) a(X); a(o) >g 5 Note that unde ned atoms are not identi ed with ? as in the completion of P [ 3 ]. where the rst clause represents `all a are b' and the second clause states that there exists a constant, viz. o, for which a(o) is true. The least L-model of PA is hfa(o); b(o)g; ;i which maps Aab, Iab, Iba and Aba to true, i.e. the programs which represent Aab, Iab, Iba and Aba are true under the least L-model of PA. We will now show how we can represent the corresponding programs for Iab and Iba. 5.2

Positive and Negative Facts The second and the third mood in Table 1, I and E, each implies two facts about something, e.g., about some constant o. The program for I in Table 1 is PI = fa(o) >; b(o) >g where o is a constant for which it holds that a(o) and b(o) are true. Its least L-model is

hfa(o); b(o)g; ;i which maps Iab, Iba, Aab, and Aba to true. Section 5.4 explains why whenever Iab is mapped to true, Iba is mapped to true as well, and vice versa. As o is the only object for which a(o) and b(o) is true, we can generalize over all constants. Accordingly, Aab and Aba hold as well. Similarily, the program for E is where o is a constant for which a(o) is true and b(o) is false. Its least L-model is PE = fa(o) >; b(o)

?g hfa(o)g; fb(o)gi which maps Eab and Oab to true. Like in the case of I, as o is the only object for which a(o) is true and b(o) is false, we can generalize over all constants. Therefore, Eab holds as well. 5.3

Negative Conclusions The consequence in the third mood E is the negation of b(X). As the weak completion semantics does not allow negative heads in clauses, we cannot represent this inference straightaway. Therefore, for every negative conclusion :p(X) we introduce an auxiliary formula p0(X) together with the clause p(X) :p0(X). Accordingly, the program of the example for E in Table 1 together with the assumption of existential import, is

PE = fb0(X) which maps Eab and Oab to true. With the introduction of these auxiliary atoms, the need for integrity constraints arises. A model I = hI>; I?i for P that contains both b(c) and b0(c) in I>, where c is any constant in P, should be invalidated. This condition can be represented by the integrity constraint IC = f?

b(X) ^ b0(X)g and is to be understood as discussed in Section 4.4. For the following examples, whenever there exists a p(X) and its p0(X) counterpart in P, we implicitly assume IC = f? p(X) ^ p0(X)g. 5.4

Symmetry The results of the psychological experiments presented in [ 25 ] show that participants distinguish between the cases `some a are c' or `some c are a' (Iac and Ica). However, for the mood I, a and c can be interchanged in FOL, because by commutativity, 9X(a(X) ^ c(X)) is semantically equivalent to 9X(c(X) ^ a(X)). Likewise, the formalizations of Iac and Ica under WCS, i.e. fa(o) >; c(o) >g and fc(o) >; a(o) >g are semantically equivalent. Thus, neither FOL nor WCS can distinguish between Iac and Ica.

In FOL, 8X(a(X) ! b(X)) is semantically equivalent to 8X(:b(X) ! :a(X)) by modus tollens. Likewise, 8X(a(X) ! :b(X)) is semantically equivalent to 8X(b(X) ! :a(X)). For the representation under WCS and, in particular, given the additional fact representing the existential import, these two formulas are not semantically equivalent anymore. Eab is represented by whereas Eba is represented by fb0(X) fa0(X) In this section we present the three problems IE1, AA3, and EA3 and show the generated conclusions under the weak completion semantics. f a(o1) $ >; b(o1) $ >; c0(o1) $ b(o1); c(o1) $ :c0(o1); b(o2) $ >; c0(o2) $ b(o2); c(o2) $ :c0(o2) g

hfa(o1); b(o1); c0(o1); b(o2); c0(o2)gfc(o1); c(o2)gi; which maps Oac and Eac to true. These are exactly the conclusions drawn by the participants and, hence, there is a perfect match in this example. One should observe that Oac is the only valid conclusion in classical FOL, whereas Eac is not a valid conclusion in classical FOL. Likewise, neither PSYCOP nor verbal models nor mental models match the participant's choices (see Table 3). 6.2

Syllogism AA4 The program representing syllogism AA4 assuming existential import is PAA4 = fa(X) b(X); b(o1) >; c(X) b(X); b(o2) >g The weak completion of gPAA4 is f a(o1) $ b(o1); c(o1) $ b(o1); b(o1) $ >;

a(o2) $ b(o2); c(o2) $ b(o2); b(o2) $ > g Its least L-model is

hfb(o1); c(o1); a(o1); b(o2); c(o2); a(o2)g; ;i and maps Iac, Ica, Aac, and Aca to true. The majority of the participants concluded Aac and NVC. Hence, there is a partial overlap in this example. One should observe that Iac and Ica are the only valid conclusions in classical FOL. From Table 3 we observe that in this example PSYCOP computes a perfect match, the conclusions computed by the mental model theory overlap, and there is no overlap between the participant's choices and the conclusions computed by the verbal model theory. 6.3

Syllogism EA3 The program representing syllogism EA3 assuming existential import is PEA3 = fb0(X) which does not map any statement involving A, I, E, or O to true. Hence, WCS leads to NVC. The majority of the participants concluded Eac and Eca. Hence, there is no overlap in this example. One should observe that Eac and Eca are the only valid conclusions in classical FOL. Inspecting the results depicted in Table 3 we observe that PSYCOP, the verbal model theory as well as the mental model theory compute solutions which overlap with the participants choices. IE1

Eac, Oac EA3

Eac, Eca Oac

Oac, Iac,

Ica Eac, Eca Eac, Eca,

Oac, Oca

Oac

NVC, Eca participants

PSYCOP verbal models mental models

Eac, Eca, Oca, Oac,

NVC Eac, Eca Aca, Aac,

Iac, Ica Eac, Oac

NVC Aca, Aac,

Iac, Ica AA4

Aac, NVC

Iac, Ica

Iac, Ica

NVC, Aca We have formalized three examples under WCS and have compared them to FOL, PSYCOP, the verbal, and the mental model theory. The results are summarized 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 particpants and WCS are disjoint. Moreover, WCS di ers from the other cognitive theories. 7

Discussion

Our goal is to compare WCS to existing cognitive theories on syllogistic reasoning. To this end, we need to evaluate the predictions of WCS concerning all 64 syllogisms and compare it to all the cognitive theories mentioned in [ 24 ].

We did not consider abnormalities in the speci cation of implications as suggested in [ 32 ]. If each abnormality is mapped to false, then the speci cation with abnormalities is semantically equivalent to the speci cation given in this paper. On the rst sight it appears that all abnormalities are indeed mapped to false, but we should take a second look. In particular, because we need to break the symmetry between our current speci cations of Iac and Ica. Furthermore, in the reported meta-study participants dealt with abstract reasoning problems. This may explain why we did not need abnormalities here. However modeling the belief bias in syllogistic reasoning can require the abnormality predicate [ 28,4 ].

Rules like b(X) :b0(X) have been introduced as a technical means to deal with implications whose conclusion is negative. Such negative conclusions cannot be directly modelled in WCS as the model intersection property would be lost. This technical reason might be justi ed by the principle of truths [ 22 ], which states that only true items can be represented. Please note that these additional rules come without existential import. Adding such import will introduce a new constant for each of these rules, which may lead to di erent conclusions.

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