On the analysis of set-based fuzzy quantified reasoning using classical syllogistics
NNOTICE: this is the author’s version of a work that was ac-cepted for publication in
Fuzzy Sets and Systems . Changes re-sulting from the publishing process, such as peer review, editing,corrections, structural formatting, and other quality control mech-anisms may not be reflected in this document. Changes may havebeen made to this work since it was submitted for publication. Adefinitive version was subsequently published in “Pereira-Fari˜na,M., D´ıaz-Hermida, F., Bugar´ın, A. (2013). On the analysis of set-based fuzzy quantified reasoning using classical syllogistics.
FuzzySets and Systems , vol. 214(1), 83-94. DOI: 10.1016/j.fss.2012.03.015” a r X i v : . [ c s . A I] N ov n the analysis of set-based fuzzy quantified reasoningusing classical syllogistics M. Pereira-Fari˜na, F. D´ıaz-Hermida, A. Bugar´ın
Centro de Investigaci´on en Tecnolox´ıas da Informaci´on (CITIUS), University ofSantiago de Compostela, Campus Vida, E-15782, Santiago de Compostela, Spain
Abstract
Syllogism is a type of deductive reasoning involving quantified statements.The syllogistic reasoning scheme in the classical Aristotelian framework in-volves three crisp term sets and four linguistic quantifiers, for which themain support is the linguistic properties of the quantifiers. A number offuzzy approaches for defining an approximate syllogism have been proposedfor which the main support is cardinality calculus. In this paper we ana-lyze fuzzy syllogistic models previously described by Zadeh and Dubois et al.and compare their behavior with that of the classical Aristotelian frameworkto check which of the 24 classical valid syllogistic reasoning patterns (calledmoods) are particular crisp cases of these fuzzy approaches. This allows us toassess to what extent these approaches can be considered as either plausibleextensions of the classical crisp syllogism or a basis for a general approachto the problem of approximate syllogism.
Keywords: syllogistic reasoning, fuzzy quantifiers
1. Introduction
Syllogistic inference or syllogism is a type of deductive reasoning in whichall the statements involved are quantified propositions. A well-known exam-ple of a syllogism is the one shown in Table 1, where P denotes the majoror first premise, P the minor or second premise, and C the conclusion. Email addresses: [email protected] (M. Pereira-Fari˜na), [email protected] (F. D´ıaz-Hermida), [email protected] (A. Bugar´ın)
Preprint submitted to Elsevier October 5, 2018 P ) All human beings are mortal( P ) All Greeks are human beings( C ) All Greeks are mortal Table 1: Aristotelian syllogistic inference.
A: All S are P E: No S is PI: Some S is P O: Not all S are PContrariesSubcontraries S ub a lt e r n a ti on s S ub a lt e r n a ti on s Contradictories
Figure 1: The classical logic square of opposition.
A quantified proposition involves two main elements: a quantifier (suchas all, many, 25, 25%, double. . . that of. . . ) and terms, usually interpretedas sets. In the typical binary quantified statement, the subject is the “re-striction” of the quantifier and the predicate its “scope”. For instance, for P in the example in Table 1, human beings is the restriction and mortal isthe scope of the all quantifier.The classical approach to syllogism was developed by Aristotle [1], whoconsidered four crisp quantifiers: all ( A ), none ( E ), some ( I ), and not all ( O ). These quantifiers are defined accordingly to the linguistic properties inthe logic square of opposition (LSO) [2] shown in Figure 1.With respect to the structure of inferences, the classical syllogism com-prises reasoning patterns involving two premises with a term in common (theso-called middle term) and a conclusion that involves the terms that are notshared between the premises (major term and minor term). The positionof the middle term in the premises allows the definition of four differentstructures known as figures (Table 2).Considering the four crisp quantifiers and the four figures together leads to24 correct inference schemes called Aristotelian moods . All these valid cases3 igure I Figure II Figure III Figure IV Q DTs are MTs Q MTs are DTs Q DTs are MTs Q MTs are DTs Q NTs are DTs Q NTs are DTs Q DTs are NTs Q DTs are NTs——————– ——————– ——————– ——————– Q NTs are MTs Q sNTs are MTs Q NTs are MTs Q NTs are MTs
Table 2: The four figures of Aristotelian syllogistics. DT denotes middle term, MTdenotes major term · and NT denotes minor term. Figure I Figure II Figure III Figure IVAAA EAE AAI AAIEAE AEE EAO AEEAII EIO IAI IAIEIO AOO AII EAOAAI EAO OAO EIOEAO AEO EIO AEO
Table 3: The four figures and 24 valid inference schemes or moods of Aristotelian syllo-gistics. are shown in Table 3. They are described using the classical notation in whicheach mood is denoted by the symbols for the quantifiers involved, which arethe major premise, the minor premise and the conclusion, respectively. Forexample, mood
AAA for Figure I refers to the reasoning scheme for theexample in Table 1.Two alternative approaches have been used in the literature to endowthe classical syllogism with more expressive capabilities: (i) addition of newcrisp [3] or fuzzy [4, 5, 6] quantifiers (but preserving the number of premises)or (ii) addition of more statements to the set of premises [7] (but only con-sidering the four classical quantifiers).We focus our analysis on models that follow the first alternative, in whichquantifiers are extended for inclusion of fuzzy expressions, such as many,most, a few, between approximately 25% and 30%,. . . , since this is the closestapproach to classical Aristotelian models from the point of view of bothsyntax and set-based interpretation of the terms.Dubois et al. [5, 8, 9] proposed a framework in which fuzzy quantifiersare represented as intervals, as shown in the example in Table 4.40 . , .
1] people who have children are single[0 . , .
2] people who have children are young[0 , .
1] people who have children are young and single
Table 4: Interval fuzzy syllogism.
Using earlier results [5, 8], Dubois et al. took the first step in developinga fuzzy linguistic syllogism that is closer to the Aristotelian viewpoint thanother approaches [9].Zadeh [4] defined fuzzy syllogism as “an inference scheme in which themajor premise, the minor premise and the conclusion are propositions con-taining fuzzy quantifiers.” Thus, fuzzy syllogism comprises two premises anda conclusion involving quantifiers that are different to those in the LSO. Atypical example of Zadeh’s fuzzy syllogism is shown in Table 5. The quantifierof the conclusion,
M ost := M ost ⊗ M ost , is calculated from the quantifiersin the premises by applying the quantifier extension principle (QEP) [10],in this case using the fuzzy arithmetic product.( P ) Most students are young( P ) Most young students are single( C ) M ost students are young and single Table 5: An example of Zadeh’s fuzzy syllogism.
Closely related to this interpretation, Yager proposed a number of syl-logistic schemes that are extensions or variants of Zadeh’s framework [11].These approaches are very similar and therefore the conclusions of our anal-ysis also directly apply to them.Both approaches have been analyzed in detail in the literature [12, 13].Spies considered all the models proposed by Zadeh from the point of view oftheir basic definitions [12]. These models are syllogisms with a middle termthat can be categorized into two classes:1. Property inheritance (asymmetric syllogism): the link between the sub-ject and the predicate of the conclusion is semantic. Thus, a term set X and a term set Z are linked via concatenation of X with a term set Y and Y with Z .2. Combination of evidence (symmetric syllogism): the link between thesubject and the predicate of the conclusion is syntactic, not semantic.5hus, the links between X and Z and between Y and Z are calculatedseparately and they are joined in the conclusion by a logic operator(conjunction/disjunction).However, neither the disadvantages that QEP presents nor their compat-ibility with classical syllogisms are considered. Liu and Kerre analyzed theapproaches of Zadeh and Dubois et al. in depth considering their multipledimensions [13].In this study we evaluate the capability of both fuzzy frameworks tocomprise and reproduce Aristotelian moods as particular (crisp) cases. Weexpand the analysis of Pereira-Fari˜na et al. [14] and describe each of thesyllogistic patterns in detail in terms of both their inferences schemes andthe problems that arise when considering the classical inferences patterns.The remainder of the paper is organized as follows. Section 2 analyzesthe reasoning patterns of Dubois et al. Section 3 analyzes the behavior ofZadeh’s patterns. In Section 4, we summarize our conclusions and proposefuture work.
2. Interval syllogistics: the Dubois et al. framework
Dubois and co-workers proposed an approach to syllogistic reasoning thatis based on the interpretation of linguistic quantifiers as intervals; that is,a quantifier Q = (cid:2) q, q (cid:3) , where q denotes the lower bound and q denotesthe upper bound [5, 8]. Since this model focuses on managing proportionalquantifiers (such as most, many, some, between 25% and 34%,. . . ) we havethat q, q ∈ [0 , Three quantifier types are considered:6. Imprecise quantifiers: quantifiers whose values are not precisely knownbut have precise bounds; for example, between and of studentsare young and less than of young people are blond . They arerepresented as an interval Q = (cid:2) q, q (cid:3) .2. Precise quantifiers: quantifiers whose values are precisely known andhave precise bounds; for example, 10% of animals are mammals and30% of young students are tall . In this case, the lower and upper boundsof the interval coincide ( q = q ).3. Fuzzy quantifiers: quantifiers whose bounds are ill defined and areimprecise and fuzzy; for example, most Spanish cars are new and afew elephants are pets . Fuzzy quantifiers are represented using a fuzzynumber defined through the usual four-point trapezoidal representation Q i = (cid:110) q ∗ i , q i , q i , q ∗ i (cid:111) , where SU P Q i := [ q ∗ i , q ∗ i ] represents the supportof Q i and KER Q i := (cid:104) q i , q i (cid:105) its core. The calculation procedure is based on minimization and maximization ofthe quantifier in the conclusion. This quantifier can be modeled as an intervalor a as a trapezoidal function and is calculated by taking the quantifiers of thepremises as restrictions. The main aim is to obtain the most favorable andmost unfavorable proportions among the terms of the conclusion accordingto the proportions expressed in the premises.From an operational point of view, researchers deal with the previouslyindicated pair of intervals
SU P Q i and KER Q i . Letting A, B and C be thelabels of the term sets and Q , Q (cid:48) , Q , Q (cid:48) , Q and Q (cid:48) the quantifiers, which canbe precise, imprecise or fuzzy, three different reasoning schemes or patternsare proposed. We refer in more detail to Pattern I, since this has the samesyntactical structure as the classical Aristotelian figures. Table 6 (left) shows the linguistic expression of this pattern for the threeterm sets A , B and C , where Q denotes the quantifier of the first premiseand Q (cid:48) its converse, Q denotes the quantifier of the second premise and Q (cid:48) its converse, and Q and Q (cid:48) denote quantifiers for the conclusions. Table 6(right) shows an illustrative example of this pattern.Fig. 2 shows a graphic representation of the Pattern I syllogistic scheme.The circles denote term sets A, B and C ; arrows with solid lines denote the7 inguistic expression Q As are Bs Q (cid:48) Bs are As Q Bs are Cs Q (cid:48) Cs are Bs Q As are Cs Q (cid:48) Cs are As
Example [0 . , .
95] students are young[0 . , .
35] young people are students[0 . ,
1] young people are single[0 . , .
80] single people are young[0 . ,
1] students are single
Table 6: The Pattern I syllogistic scheme of Dubois et al. Intervals in the example denoterelative quantifiers.
A BC Q '1 Q2Q Q'2Q Q'
Figure 2: Graphic representation of Pattern I. links between two term sets in a premise, where Q i is the quantifier for thatpremise; arrows with discontinuous lines denote the links between two termsets in the conclusion, where Q is the quantifier for the conclusion.Pattern I is an asymmetric syllogism since the link between the subject( A ) and the predicate ( C ) of the conclusion is via concatenation of A with B and B with C . It is relevant to note that the links among all three predicatesinvolved have to be fully known (i.e., the relationships between A and B andvice versa and between B and C and vice versa are needed to infer as aconclusion the relationships between A and C and vice versa).For precise quantifiers, the quantifier of the conclusion Q := (cid:2) q, q (cid:3) iscalculated from Q := (cid:2) q , q (cid:3) and Q := (cid:2) q , q (cid:3) and the corresponding Q (cid:48) Q (cid:48) using the following expressions: q = q · max (cid:18) , − − q q (cid:48) (cid:19) (1) q = min (cid:18) , − q + q · q q (cid:48) , q · q q (cid:48) · q (cid:48) , q · q q (cid:48) · q (cid:48) [1 − q (cid:48) + q ] (cid:19) . (2)Management of imprecise quantifiers involves minimizing (1) and maxi-mizing (2). The extension to fuzzy quantifiers is made by directly and inde-pendently calculating SU P Q and KER Q as a pair of imprecise quantifiers. Dubois et al. proposed other reasoning schemes based on three terms[8]. Pattern II involves two linguistic expressions (Table 7) called the generalversion and the particular version. The main difference between them isthe information available for the set of premises. The general version isapplied when the information available for the terms is enough to completethe six premises; the particular version can be used when the informationavailable allows completion of four corresponding statements. Table 8 showsan illustrative example of a particular Pattern II.
General version Q As are Bs; Q (cid:48) Bs are As Q Bs are Cs; Q (cid:48) Cs are Bs Q As are Cs; Q (cid:48) Cs are As Q As and Bs are Cs
Particular version Q As are Bs; Q (cid:48) As are Bs Q Bs are Cs Q As are Cs Q As and Bs are Cs
Table 7: Linguistic schemes of Pattern II.
Between 70% and 80% of students are women More than 35% of women are studentsAt least 70% of women are youngBetween 80% and 90% of students are young Q of female students are young Table 8: Example of a particular Pattern II. Calculation of Q (cid:48) is identical. For details of the proof, refer to [5, 8]. General version Q As are Bs; Q (cid:48) Bs are As Q Bs are Cs; Q (cid:48) Cs are Bs Q As are Cs; Q (cid:48) Cs are As Q Cs are As and Bs
Particular version Q (cid:48) Cs are Bs Q (cid:48) Cs are As Q Cs are As and Bs
Table 9: Linguistic schemas of Pattern III.
Between 5% and 10% of people who have children are singleLess than 5% of people who have children are young Q people who have children are young and single Table 10: Example of a particular Pattern III.
We can observe from the linguistic schemes for both patterns that theyhave a symmetric-type structure [12]; that is, the core of the reasoning processis the and logic operator. Therefore, they do not share but extend theAristotelian inference scheme.
Patterns II and III of Dubois et al. cannot be considered for comparisonwith the Aristotelian framework, since their structure is in fact a syntacti-cal extension of the classical approach and therefore it cannot be used toreproduce classical syllogisms. Nevertheless, Pattern I can be considered forcomparison, since it is an asymmetric syllogism, as are all the Aristotelianmoods.Regarding the capability of Pattern I to reproduce the four classical fig-ures, only Figure I is compatible, as shown in Table 11, when we take intoaccount the conclusion Q , because of the position of the middle term. Asa consequence, even though the classical Figures II, III and IV fit this ap-proach from a syntactical point of view, they cannot be modeled within thisframework. 10attern Major premise Minor premiseFigure I Subject PredicateFigure II Predicate PredicateFigure III Subject SubjectFigure IV Predicate SubjectPattern I Subject Predicate Table 11: Position of the middle term in the Aristotelian figures and Pattern I.
AAA EAE AII EIO AAI EAO
Pattern I
Yes Yes Yes Yes Yes Yes
Table 12: Behaviour of Pattern I with respect to the six moods in Figure I.
This limits the scope of this model and therefore comparison with thesix Aristotelian moods in Figure I. To proceed with the analysis, consistentdefinitions for the four classical quantifiers should be provided within thisframework: A ( all ) := [1 , E ( none ) := [0 , I ( some ) := [ (cid:15), O ( notall ) := [0 , − (cid:15) ], with (cid:15) ∈ (0 ,
3. Fuzzy syllogism: Zadeh’s approach
Table 13 shows the general scheme of the fuzzy syllogism proposed byZadeh [4], where Q , Q and Q are fuzzy quantifiers ( most, many, some,around 25,. . . ) and A , B , C , D , E and F are interrelated fuzzy propertiesor terms. Linguistic schema Q As are BsQ Cs are DsQ Es are
F s
Table 13: Zadeh’s general fuzzy syllogistic scheme.
Two linguistic quantifier types are described in this framework: absolute( around five, many, some,. . . ) and proportional ( many, more than a half,most,. . . ). Thus, for instance, in “ around ten students are blond”, aroundten is an absolute quantifier because it denotes the absolute quantity ten ;in “ most students are blond”, most is a proportional quantifier because itrefers to a proportional quantity relative to the cardinality of the set denotedby the subject (“ students ”). Some linguistic quantifiers (e.g. some ) can beeither absolute or proportional, depending on the context.Zadeh interpreted linguistic quantifiers as fuzzy numbers [10]. Therefore,absolute quantifiers are identified with absolute fuzzy numbers and propor-tional quantifiers with proportional fuzzy numbers.The procedure proposed for combining fuzzy numbers with the corre-sponding fuzzy sets represented in the properties is the well-known Σ
Count scalar cardinality for fuzzy sets [10, 15]. It is relevant to note that the Aris-totelian approach and Zadeh’s approach exhibit methodological differences.As mentioned in Section 2, Aristotle manages and proves all his moods usingnatural logic [7], while Zadeh’s inference process is based on fuzzy arithmetic,since the linguistic quantifiers are interpreted as fuzzy numbers. Zadeh’s approach manages the usual quantified statements “
Q As are Bs ”, where Q is a proportional fuzzy number (equivalent to the correspond-ing linguistic proportional quantifier) and A and B (fuzzy or crisp) sets. Ac-cording to the general scheme shown in Table 13, they can only be combinedinto inferences with two premises and a conclusion. Liu and Kerre discuss the most relevant problems generated by the use of QEP andfuzzy arithmetic in Zadeh’s framework [13]. C = f ( P ; P ; . . . ; P n ), then Q = φ f ( Q ; Q ; . . . ; Q n ) , (3)where C is the conclusion, P ; P ; . . . ; P n are the premises, f is a function, Q is the quantifier of the conclusion, Q ; Q ; . . . ; Q n are the quantifiers of thepremises and φ f is an extension of f obtained using the extension principle.The main idea of QEP is to apply the extension principle to f to obtaina fuzzy function φ f that can be directly applied to the corresponding fuzzynumbers. Since fuzzy numbers are managed, the corresponding arithmeticoperations must be performed using fuzzy arithmetic.Now we analyze each of the syllogistic inference patterns proposed byZadeh that emerge from the general pattern (Table 13). We focus on patternsthat have the same syntactical structure as the classical Aristotelian figures:multiplicative chaining and major premise reversibility chaining. Table 14 (left) shows the typical linguistic expression for the multiplica-tive chaining (MC) reasoning pattern [4]. In this syllogistic pattern, Q denotes the quantifier of the first premise, Q the quantifier of the secondpremise and Q the quantifier of the conclusion. Table 14 (right) presents anexample of use of this pattern. Figure 3: Graphic representation of chaining syllogism.
Fig. 3 presents a graphic representation of MC syllogism. Three termsare involved that play the following roles: A and C constitute the conclusion,where A is the subject and C is the predicate, and B is the “link” betweenthese two terms that makes it possible to infer a coherence conclusion. Since13he terms of the conclusion are “chained” by a middle term, this pattern iscalled a chaining pattern.Furthermore, it is worth noting that there is a constraint between A and B ; that is, B ⊆ A , i.e. µ B ( u i ) ≤ µ A ( u i ) , u i ∈ U, i = 1 , . . . . This constraintcan be expressed as an additional quantified statement: “all Bs as As ”.This constraint is relevant because it allows us to know the distribution ofthe elements in the sets A and B ; without this information, the conclusioncannot be calculated.Taking all the previous considerations into account, the procedure forcalculating the conclusion is shown in Equation (4), where ⊗ denotes a fuzzyproduct and thus the chaining pattern is multiplicative. Q ≥ ( Q ⊗ Q ) . (4) Linguistic schema Q As are Bs (all Bs are As) Q Bs are Cs Q As are Cs
Example
Most American cars are bigMost big cars are expensive
M ost American cars are expensive
Table 14: Zadeh’s multiplicative chaining syllogism (left) and an example (right).
Table 15 (left) shows the general structure of the MPR syllogistic pattern,where Q denotes the quantifier of the first premise, Q the quantifier of thesecond premise and Q the quantifier of the conclusion. Table 15 (right)presents an example of the use of this pattern. Linguistic scheme Q Bs are As Q Bs are Cs Q As are Cs
Example
Most big cars are AmericanMost big cars are expensive ≥ ∨ (2 most (cid:9)
1) American cars are expensiveTable 15: Zadeh’s major premise reversibility chaining syllogism (left) and an exampleof its use (right). igure 4: Graphic representation of MPR chaining syllogism. Fig. 4 is a graphic representation of the structure of the MPR chainingreasoning pattern [4]. We can consider this model as a variant of the MCpattern in Section 3.2.1 where the constraint “ B ⊆ A ” is substituted by thereversibility of the first premise; i.e., “ Q As are Bs ↔ Q Bs are As”. Zadehpoints out that this semantic equivalence is approximate rather than exact;the calculation of how “approximate” it can be remains a nontrivial openquestion [16].Nevertheless, there is a remarkable obstacle for this constraint. For pro-portional quantifiers, it does not hold that a quantified sentence (e.g., “mostAmerican cars are big”) and another sentence with the arguments inter-changed (“most big cars are American”) are semantically equivalent. It istrue, when talking about absolute quantifiers, that such semantic equivalenceholds (e.g., “around a hundred thousand American cars are big” is semanti-cally equivalent to “around a hundred thousand big cars are American”).The procedure for calculating the conclusion, shown in Equation (5),involves fuzzy addition ( ⊕ ) and subtraction ( (cid:9) ). q ≥ max (0 , q ⊕ q (cid:9) . (5) Apart from the models described above, Zadeh proposed an asymmet-ric reasoning scheme ( intersection/product , Table 16) and two symmetricschemes ( antecedent conjunction/disjunction , Table 17; consequent conjunc-tion/disjunction , Table 18).The difference between the antecedent and consequent patterns lies in theposition of the logic operator in the conclusion: if it occurs for the subject,15 inguistic schema Q As are Bs Q As and Bs are Cs Q As are Bs and Cs
Example
Most students are youngMany young students are single Q students are young and single Table 16: Intersection/product syllogism.
Linguistic scheme Q As are Cs Q Bs are Cs Q As and/or Bs are Cs
Example
Most students are youngAlmost all single people are young Q single people or students are young Table 17: Antecedent conjunction/disjunction syllogism. we obtain the antecedent pattern ( and for the conjunction; or for the dis-junction); if it occurs for the predicate, we obtain the consequent pattern (asin the previous pattern, the and operator is used for the conjunction and or for the disjunction). We focus the analysis on the capability of Zadeh’s framework to manageand reproduce Aristotelian syllogistics. First, it is worth noting that all themoods are of the property inheritance type (asymmetric syllogisms); there-fore, the structure of Zadeh’s symmetric syllogistic patterns is not adequate.Furthermore, the intersection/product scheme involves logic operations inthe second premise and the conclusion that do not appear in the classicalmoods. As a consequence, none of the previously indicated patterns can beconsidered for this analysis.
Linguistic scheme Q As are Bs Q As are Cs Q As are Bs and/or Cs
Example
Most students are youngAlmost all students are single Q students are young and single Table 18: Consequent conjunction/disjunction syllogism.
Table 19: The position of the middle term in Aristotelian figures and Zadeh’s MC andMPR patterns
Zadeh’s patterns AAA EAE AII EIO AAI EAOMC No No No No No NoMPR No No Yes No No No
Table 20: Behavior of Zadeh’s patterns with respect to Figure I.
Therefore, only the MC and MPR patterns can be considered for com-parison with classical syllogisms. Regarding their capability to reproducethe four classical figures, only Figure I is compatible, as shown in Table 19,because of the position of the middle term.As a consequence, none of the classical Figures II, III and IV can bemodeled using this approach. This limits the scope of the model and thereforecomparison to the six Aristotelian moods in Figure I. Zadeh’s patterns arecompared with these six moods in Table 20. Only the AII mood is compatiblewith the MPR scheme (i.e., one mood out of 12).The problem in the MC pattern is in the constraint B ⊆ A in the firstpremise (minor premise in Aristotelian terms), which is much too restrictive.In this set of Aristotelian moods, the converse of the minor premise in lin-guistic terms is, “some Bs are A”; that is, A ∪ B (cid:54) = ∅ ; for instance, in the AIImood, B ⊆ A cannot be inferred from the statement I: some As are Bs . Thiscondition is less restrictive than Zadeh’s and therefore many of the possibleinferences dismissed by Zadeh’s pattern can be solved using the Aristotelianframework.The MPR pattern is only valid for the AII mood since the statementand its converse in the minor premise have the same quantifier (“some”).For the other moods, those that include A in the minor premise present thesame problem as in the MC pattern. The solution of the EIO mood, which17s compatible, presents a problem: the value obtained for Q is 0 instead of[0 , − (cid:15) ) with (cid:15) ∈ (0 ,
4. Conclusions and future work
We have shown that the two most relevant fuzzy approaches behave verydifferently with regard to management of Figure I of Aristotelian syllogistics,which is one of the classical approaches to this type of reasoning. While themodel of Dubois et al. is consistent with all six moods in Figure I, Zadeh’sMPR scheme is only consistent for one mood out of six and his MC approachis consistent with none of the six moods. Furthermore, none of the models iscompatible with the other classical Figures II, III and IV, which involve 18moods.Therefore, from this point of view, the proposal of Dubois et al. canbe properly considered as a plausible fuzzy extension that comprises someof the classical cases as particular instances. Therefore, this proposal couldbe considered as a basis for defining a more general approach to syllogisticreasoning that involves, for example, the other moods currently excludedand/or other types of quantifier used in the linguistics field [17], such ascomparative (e.g., there are approximately three more tall people than blondpeople ) and exception (e.g., all except three students are tall quantifiers).
Acknowledgment
This work was supported in part by the Spanish Ministry of Science andInnovation (grant TIN2008-00040), the Spanish Ministry for Economy andInnovation and the European Regional Development Fund (ERDF/FEDER)(grant TIN2011-29827-C02-02) and the Spanish Ministry for Education (FPUFellowship Program). We also wish to thank the anonymous referees fortheir useful and constructive comments and insightful suggestions on previousversions of this paper. 18 eferenceseferences