Abstract Peterson’s Intermediate Syllogisms, generaliz-ing Aristotelian syllogisms by intermediate quantifiers‘Many’, ‘Most’ and ‘Almost all’, are studied. It is demon-strated that, by associating certain values V, W and U onstandard Łukasiewicz MV-algebra with the first and secondpremise and the conclusion, respectively, the validity of a cor-responding intermediate syllogism is determined by a simpleMV-algebra (in-)equation. Possible conservative extensionsof Peterson’s system are discussed. Finally it is shown thatPeterson’s bivalued intermediate syllogisms can be viewedas fuzzy theories in Pavelka’s fuzzy propositional logic, i.e. afuzzy version of Peterson’s Intermediate Syllogisms is intro-duced.
Aristotelian syllogisms deal with categorical propositionsand forms of inference on such propositions. For example
If ‘All M are non-P’and ‘All M are S’then ‘Some S are non-P’
Communicated by V. Loia.
E. TurunenResearch Unit Computational Logic, Institute of DiscreteMathematics and Geometry, University of Technology, Vienna,Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria
E. Turunen (B)Tampere University of Technology, Tampere, Finlande-mail: [email protected]
is such a syllogism called Felepton since ancient times. Thereare 256 possible Aristotelian syllogisms, and 24 valid ones.Until the mid 1800’s, Aristotelian syllogisms were the cen-tral conformation of valid reasoning, however, the arise ofmodern mathematical logic took syllogisms sidelined. Sci-entific research in the field of syllogisms, however, is con-tinually taking place. Peterson writes (page 47 in Peterson2000) ‘In the traditional doctrine of syllogism there areonly two quantifiers involved—the universal (all) and par-ticular (some). But additional quantifier words exist—‘few’,‘many’, ‘most’, etc.—over and above expressions for uni-versal and particular quantity.’ In his book (Peterson 2000)Peterson presents a detailed linguistic analysis of interme-diate quantifiers ‘Almost-all’, ‘Most’ and ‘Many’ (and theircomplements), and a list of all the valid (intermediate) syl-logisms, 105 in all, related to these quantifiers. There are3,895 intermediate syllogisms that are not valid. However,Peterson continues, (pages 157–158) ‘... the inference pat-terns do not constitute a theory, or theoretical approach atall. Rather, the (valid syllogism) patterns provide additionalsemantic data ... that any theory (of syllogisms) must accountfor. Peterson calls valid syllogisms also ‘heavily confirmedempirical hypothesis’.
We give an MV-algebraic semantics for Peterson’s inter-mediate categorical propositions (such as ‘Many X are Y’)and show how the validity/invalidy of Peterson’s Intermedi-ate Syllogisms (written with capital letters, in order to dis-tinguish them from other approaches) can be calculated bymeans of these semantics. In particular, we prove that syllo-gisms of type (in what is called Figure III)
If ‘A quantity Q1 of M are P’and ‘A quantity Q2 of M are S’then ‘Some S are P’
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are valid if, and only if the corresponding statements Q1 andthe complement statement of Q2 form a pair of contradictory,that is, contradict each other. We do this by first interpreting‘contradictioness’ in Venn diagrams and then giving a moregeneral algebraic definition for a pair of statements to becontradictory with each other. Validity of all the other syl-logisms (in Figures I, II and IV) depends ultimately on therelationship of mutual strengths of the intermediate categor-ical propositions, a concept which we investigate.
In other words, we show that the empirical facts on inter-mediate syllogisms examined by Peterson form an alge-braic structure, known as MV-algebra.1 The situation canbe vaguely compared with the following event describedin Livio (2005, pages 183–184): ‘An extremely complexkinship–marriage system that was discovered among theKariera, a tribe of Australian Aboriginals, left anthropol-ogists baffled. Each Kariera belongs to one of four classesor clans: Banaka, Karimera, Burung, and Palyeri. Marriageand the association of descendants to classes were foundto obey certain strict rules.’ Finally a French mathemati-cian Andre Weil showed that permitted marriages and thestatus of children is totally determined by a group opera-tion of an abstract four element group. Likewise, do we notdevelop any new theory of intermediate syllogisms, rather weshow that valid syllogisms are determined in an MV-algebrastructure.
We also study possible generalizations of intermediatesyllogisms from an algebraic point of view. Naturally, suchextensions must be first linguistically justified. Contrary toPeterson’s own general theory, which is based on fractionalquantities, our approach is conservative in a sense that, byincreasing or decreasing the number of quantifiers, the valid-ity or invalidity of the already existing syllogisms does notchange.
Finally, we deal briefly with intermediate syllogisms inrelation with fuzzy logic. We propose how they can be inter-preted as fuzzy theories in Łukasiewicz–Pavelka style fuzzypropositional logic.
2 Aristotelian syllogisms
Complete sentences usually contain two parts: a subject S anda predicate P. The subject is what (or whom) the sentence isabout, while the predicate tells something about the subject.In particular, ‘Most sailors are not politicians’ or ‘Many soupsare poisonous’ and alike are such sentences; Peterson calls
1 An interesting historical detail is the fact that Jan Łukasiewicz whosemultivalued logic’s algebraic counterpart MV-algebras are, was a lead-ing figure in studies of syllogisms in the early twentieth century.Łukasiewicz published in 1951 his now famous monograph Aristotle’sSyllogistic from the Standpoint of Modern Formal Logic. It is unknownif Łukasiewicz himself ever investigated intermediate syllogisms.
them intermediate categorical propositions. For simplicity,we will call them statements. A general frame and impor-tant features of Aristotelian categorical propositions can bepresented as follows:
Aristotle’s squareAffirmative statements: Negative statements: Generic term
A: All X are Y ↔ E: No X are Y universal⇓ ⇓I: Some X are Y · · · O: Some X are not Y particular
Meaning of the symbols ‘↔’, ‘· · · ’ and ‘⇓’ will be clarifiedin the Sect. 3. Syllogisms can be seen as special kinds ofrules of inference. If we know a quantified relation betweenM (a middle term) and S and a quantified relation betweenM and P, we infer something about the a quantified relationbetween S and P. A syllogism is valid if, whenever somerelation between M and S is assumed to hold and anotherrelation between M and P is assumed to hold, then we cannotdeny that certain relation between S and P would not hold.If this is not the case, then a syllogism is invalid. Syllogismsare grouped into the following four subgroups, traditionallycalled Figures.
Figure IA quantity Q1 of M are P (premise 1)A quantity Q2 of S are M (premise 2)A quantity Q3 of S are P (conclusion)
Abbreviated MPSM
Figure IIA quantity Q1 of P are M (premise 1)A quantity Q2 of S are M (premise 2)A quantity Q3 of S are P (conclusion)
Abbreviated PMSM
Figure IIIA quantity Q1 of M are P (premise 1)A quantity Q2 of M are S (premise 2)A quantity Q3 of S are P (conclusion)
Abbreviated MPMS
Figure IVA quantity Q1 of P are M (premise 1)A quantity Q2 of M are S (premise 2)A quantity Q3 of S are P (conclusion)
Abbreviated PMMS
For example an Aristotelian syllogism
All P are MNo S are M (i.e. All S are non-M)Some S are not P
called Camestros is a valid syllogism. Syllogisms, Aris-totelian as well as the intermediate (to be introduced in theSect. 3), can be set-theoretically presented, justify the validones and reject the invalid ones by following Venn diagrams;we refer them a number of times.
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Natural numbers a, b, c, d, e, f , and g refer to the amountof elements in the corresponding restricted area. All sets andsubsets under considerations are assumed to be non-empty;for example if all S are M then a + b = 0 while d + e > 0.This is called Principle of imported existence. In particular,a +b+d +e �= 0, b+c+e+ f �= 0 and d +e+ f +g �= 0.For example, to validate Camestrop we reason that if ‘AllP are M’ are then b = c = 0, and if ‘No S are M’ thene = d = 0. Thus e = b = d = 0. Since S is non-void (byimported existence) it is of necessity that a �= 0. This provesthat there is some (at least one!) S that are not P.
3 Peterson’s Intermediate Syllogisms
Peterson’s Intermediate Syllogisms contain five quantifiersand related statements; the traditional ‘All X are Y’ and‘Some X are Y’, three new ones, and their complements,all presented in the following diagram:
Peterson’s squareAffirmative statements: Negative statements: Generic term
A: All X are Y ↔ E: No X are Y universal⇓ ⇓P: Almost-all X are Y ↔ B: Almost-all X are not Y predominant⇓ ⇓T: Most X are Y ↔ D: Most X are not Y majority⇓ ⇓K: Many X are Y · · · G: Many X are not Y common⇓ ⇓I: Some X are Y · · · O: Some X are not Y particular
By Peterson’s linguistic analysis, a statement ‘Almost-all Xare not Y’ has the same meaning as ‘Few X are Y’ and astatement ‘No X are Y’ and has the same meaning than ‘AllX are non-Y’.
Statements A and E, P and B, T and D compose contrarypairs, denoted by X ↔ Y. They cannot be simultaneouslytrue, however, they can be simultaneously false. We also saythat the complement of A, P, and T is E, B and D, respec-tively. Universal, predominant and majority statement willbe called preponderance statements; if a statement QXY is
a preponderance statement then more than half of X is Y.This concept will be crucial when we subsequently define ageneral algebraic semantics for intermediate syllogisms. Wehave selected the phrase preponderance statement instead ofmajority statement as majority is already in use. Similarly, toavoid misunderstandings, we call the premises ‘first premise’and ‘second premise’ instead of the classical ‘Majority’ and‘Minority’ premise. Statements K and its complement G aswell as I and its complement O compose sub-contrary pairs,denoted by ‘· · · ’. They are not preponderance statements; itis possible that ‘Some/Many X are Y’ is true and simultane-ously ‘Some/Many X are not Y’ is also true.
By Peterson’s linguistic analysis, statements pairs (A, O),(I, E), (P, G) and (K, B) are pairs of contradictory state-ments. If the first one is true, then the second one is false andconversely. For example, it is not possible that ‘Almost-allX are Y’ holds and simultaneously also ‘Many X are not Y’would hold.—We note here the following fact, the meaningof which Peterson has hardly noticed. If P and G are con-tradictory statements, i.e. in conflict with each other, thenobviously so are (even more contradictory statements) P andD as well as P and E.
The downward pointing arrows ⇓ indicate sub alternations(immediate entailment): if e.g. a statement ‘All X are Y’ istrue then also a statement ‘Almost-all X are Y’ is true (butnot necessarily vice verse), and so on. This principle is togeneralize the classical state of affairs: if ‘All X are Y’ thennecessarily ‘Some X are Y’. This principle is also closelyconnected to the semantic meaning of the quantifiers. Simi-larly to the classical I: ‘Some X are Y’ which means that ‘atleast one or more X is Y’, a statement T: ‘Most X are Y’ is tobe interpreted ‘Most X or more are Y’. Moreover, we say thatstatement A is stronger than statement P, which is strongerthan statement T, etc, and equivalently, e.g. statement I isweaker than statement K. Similarly for negative statements.Thus, the meaning of phrases ‘strengthen a premise’ and‘weaken a conclusion’ should be clear.
An important ingredient in Peterson’s well-founded analy-sis is the Principle of existential import; empty entities are notconsidered. Statements under considerations never includeempty set. Moreover, based on a profound and detailed studyon the English quantifiers ‘Almost-all’, ‘Most’ and ‘Many’,Peterson sums up several principles related to intermediatesyllogisms. We list here those of them that we will need later.
1◦ At least one premise must be affirmative.2◦ The conclusion is negative if, and only if one of the
premises is negative.3◦ At least one of the premise must have a quantity of pre-
ponderance.4◦ If any premise is non-universal, then the conclusion must
have a quantity that is less than or equal to that premise.
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Intermediate syllogisms are presented briefly by XYZ-N, for example KAI-III refers to the following syllogism inFigure III
Many M are P (premise 1 is a statement of form K)All M are S (premise 2 is a statement of form A)Some S are P (conclusion is a statement of form I)
On the bases of such analysis and notions, Peterson intro-duces a special (new) Venn Diagram method for validatingintermediate syllogisms. This method is quite complex andrather unintuitive. Finally, 105 syllogisms are valid in Peter-son’s approach.Valid intermediate syllogisms in Peterson’s 5 quantitiesapproach
Aristotelian (classical) syllogisms are printed by bold font.Notice that all the intermediate syllogisms ‘fall in betweenthe classical ones’.
Figure I AffirmativeAAAAAP APPAAT APT ATTAAK APK ATK AKKAAI API ATI AKI AII
Figure I NegativeEAEEAB EPBEAD EPD ETDEAG EPG ETG EKGEAO EPO ETO EKO EIO
Figure II Negative Case 1AEEAEB ABBAED ABD ADDAEG ABG ADG AGGAEO ABO ADO AGO AOO
Figure II Negative Case 2EAEEAB EPBEAD EPD ETDEAG EPG ETG EKGEAO EPO ETO EKO EIO
Figure III AffirmativeAAI PAI TAI KAI IAIAPI PPI TPI KPIATI PTI TTIAKI PKIAII
Figure III NegativeEAO BAO DAO GAO OAOEPO BPO DPO GPOETO BTO DTOEKO BKOEIO
MV-algebras are the algebraic counterparts of Łukasiewiczinfinite-valued logic in the same sense as Boolean alge-bras are the algebraic counterparts of classical logic. In thispaper we will need the following definitions and proper-ties (cf. Cignoli et al. 2000; Turunen 1999). An MV-algebraL = 〈L ,⊕,∗ , 0〉 is a structure such that 〈L ,⊕, 0〉 is a com-mutative monoid, i.e.,
x ⊕ y = y ⊕ x, (1)
x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, (2)
x ⊕ 0 = x (3)
holds for all elements x, y, z ∈ L and, moreover,
x∗∗ = x, (4)
x ⊕ 0∗ = 0∗, (5)
(x∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x . (6)
Denote x � y = (x∗ ⊕ y∗)∗ and 1 = 0∗. Then 〈L ,�, 1〉 isanother commutative monoid and hence
x � y = y � x, (7)
x � (y � z) = (x � y) � z, (8)
x � 1 = x (9)
holds for all elements x, y, z ∈ L . It is obvious that x ⊕ y =(x∗�y∗)∗, thus the triple 〈⊕,∗ ,�〉 satisfies De Morgan laws.A partial order on the set L is introduced by
x ≤ y iff x∗ ⊕ y = 1 iff x � y∗ = 0. (10)
By setting
x ∨ y = (x∗ ⊕ y)∗ ⊕ y, (11)
x ∧ y = (x∗ ∨ y∗)∗[= (x∗ � y)∗ � y] (12)
for all x, y, z ∈ L the structure 〈L ,∧,∨〉 is a lattice. More-over, x ∨ y = (x∗ ∧ y∗)∗ holds and therefore the triple〈∧,∗ ,∨〉, too, satisfies the De Morgan laws. However, theunary operation ∗ called complementation is not a latticecomplementation. By stipulating
x → y = x∗ ⊕ y (13)
the structure 〈L ,≤ ∧,∨,�,→, 0, 1〉 is a residuated latticewith the bottom and top elements 0, 1, respectively. In partic-ular, a residuation (also somewhat misleadingly called Galoisconnection)
x � y ≤ z iff x ≤ y → z (14)
holds for all x, y, z ∈ L . The couple 〈�,→〉 is an adjointcouple. Notice that the lattice operations on L can beexpressed also via
x ∨ y = (x → y) → y, (15)
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An algebraic study of Peterson’s Intermediate Syllogisms
x ∧ y = x � (x → y). (16)
The standard example of an MV-algebra is the standardŁukasiewicz structure L; the underlying set is the real unitinterval [0, 1] equipped with the usual order and, for eachx, y ∈ [0, 1],x ⊕ y = min{x + y, 1}, (17)
x∗ = 1 − x . (18)
Moreover,
x � y = max{0, x + y − 1}, (19)
x ∨ y = max{x, y}, (20)
x ∧ y = min{x, y}, (21)
x → y = min{1, 1 − x + y}, (22)
3.2 Observations on Peterson’s 5 quantities approach
To begin with, consider Peterson’s square. Contradictorypairs (X, Y) are closely connected with valid syllogisms inFigure III, thus let us examine them in detail. First notice thatexactly one of X, Y is a preponderance statements while theother one is not, and if (X, Y) is a contradictory pair, so is thecomplement pair. Therefore it is enough to focus on such con-tradictory pairs (X, Y) that X is an affirmative preponderancestatement and Y is a negative non-preponderance statement.Next focus on (P, G). It is a border line contradictory pair inthe sense that
1◦ if P is substituted by a weaker affirmative statement P’(here by T), then (P’, G) is not a contradictory pair, or
2◦ if G is substituted by a weaker negative statement G’(here by O), then (P, G’) is not a contradictory pair.
Let us now associate with each statement S an elementq ∈ (0, 1], called grade, in the the following way
Affirmative Grade Negative GradeA 1 E 0P p B 1 − pT t D 1 − tK k G 1 − kI ε O 1 − ε
such that (a) 0 < ε < 1 − p < k < 12 < t < p < 1, (b)
k + p > 1 and (c) t + k ≤ 1. Notice that conditions (a)–(c)can be expressed by
0 < ε < 1 − p < k ≤ 1 − t <1
2< t ≤ 1 − k < p < 1,
however, we divide these conditions into three parts as wewant to clarify their meaning. Indeed, by associating value 1to statement A we express ‘all X are Y’ and by associating asmall positive value ε to statement I we express a statement‘some (at least one) X are Y’, and by associating a value
ε < q < 1 to a statement S we express that S is an interme-diate statement. Condition k < 1
2 < t < p indicates that Pand T (and of course A) are preponderance statements whileK is not (and naturally neither I is not). The condition (b),equivalent to 1 − k < p, express the fact that (P, G) con-stitute (a border line) intermediate contradictory pair, whileby (c) we express the fact that (K, D) does not constitute acontradictory pair.
We remark that the choice of the elements ε, k, t, p isof course not unique. Indeed, we can first fix the elements0 < ε < 1 − p < 1
2 < p < 1, then select k such that1− p < k < 1
2 and finally, choose t such that 12 < t ≤ 1−k.
Any such choice satisfies conditions (a)–(c) which, in turn,will turn out to be necessary and sufficient conditions todetermine valid and invalid Peterson’s Intermediate Syllo-gisms. Now we examine each Figure of valid syllogisms inthe light of these concepts. The Venn diagram arguments wepresent, with the exception those concerning Figure III, areonly indicative and instructive, details can found in Peter-son’s book.
3.3 Valid syllogisms in Figure I
First focus on the negative syllogisms in Figure I These syl-logisms can be justified by the following Venn diagram rea-soning: Since all M are non-P by the first premise, we havee = f = 0. Moreover, a d
a+b+d -portion of S are M by thesecond premise and, by existential import principle d �= 0. Iffact, a d
a+b+d -portion of S are M and are non-P (i.e. a sub caseof non-P). On the other hand, we see in the Venn diagramthat a a+d
a+b+d -portion of S are non-P. Now a ≥ 0 thus, the
premises entail that at least a da+b+d -portion of S are non-P
holds and consequently, the conclusion is justified.We observe the following:
1◦ The first premise in each of the negative syllogisms inFigure I is the classical E: ‘All M are non-P’.
2◦ Denote the grade of the second premise by W (=1, ε orsome q) and the grade of the conclusion by U (=0, 1 − ε
or some 1−q). Then the corresponding syllogism is validif, and only if W ⊕ U = 1, where ⊕ is the Łukasiewiczsum. Indeed, see the following Cayley table:
The condition W ⊕ U = 1 is equivalent to U∗ ≤ W .Then focus on the affirmative syllogisms in Figure I. These
syllogisms can be justified by the following Venn diagramreasoning: Since all M are P by the first premise, we haved = g = 0. Moreover, a e
a+b+e -portion of S are M by the
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E. Turunen
second premise where, by existential import principle e �= 0.If fact, a e
a+b+e -portion of S are M and are P (i.e. a sub caseof P). On the other hand, we see in the Venn diagram that a
b+ea+b+e -portion of S are P. Now b ≥ 0 thus, the premises entailthat at least a e
a+b+e -portion of S are P is valid, therefore theconclusion is justified.
We observe that the affirmative syllogisms in Figure I areobtained by replacing the first premise and the conclusionwith their complements, in other words:
1◦ The first premise in each of the (affirmative) syllogismsin Figure I is the classical A ‘All M are P’.
2◦ Denote the grade of the second premise by W (=1, ε orsome q) and the grade of the conclusion by U (=1, ε orsome q). Then the corresponding syllogism is valid if,and only if U ≤ W , or equivalently U∗ ⊕ W = 1.
3.4 Valid syllogisms in Figure II
All valid syllogisms in Figure II are negative. Figure II isobtained from Figure I by changing places of P and M witheach other in the first premise. Consequently, a large sym-metry follows. Indeed, the negative Case 2 in Figure II hasexactly the same valid syllogisms that the negative syllogismsin Figure I. Moreover, we observe that
1◦ The first premise in each of the syllogisms in negativeCase 2 is the classical E ‘All M are non-P’.
2◦ Denote the grade of the second premise (affirmative) byW (=1, ε or some q) and the grade of the conclusion byU (=0, 1 − ε or some 1 − q). Then the correspondingsyllogism is valid if, and only if W ⊕ U = 1.
The valid syllogisms in the negative Case 1 in Figure II areobtained by substituting both premises by their complementsin Case 2. Moreover, if we denote the grade of the secondpremise (negative) by W (=0, 1 − ε or some 1 − q) and thegrade of the conclusion by U (=0, 1−ε or some 1−q). Thenthe corresponding syllogism is valid if, and only if W ≤ U ,or equivalently, if W ∗ ⊕ U = 1.
3.5 Valid syllogisms in Figure IV
All the intermediate valid syllogisms fall in between pairs oftraditional ones, e.g. TAI falls in between AAI and IAI. Anintermediate syllogism in Figure IV is valid if, and only ifit is obtained from a classical syllogism by replacing one ofthe premises by a stronger one or the conclusion by a weakerone, where the notions of stronger and weaker are presentedin Peterson’s square by the symbol ⇓. For example AEG isobtained by AED by replacing the conclusion D by a weakerG and ETO is obtained from EKO by replacing the secondpremise K by a stronger T.
3.6 Valid syllogisms in Figure III
Valid syllogisms in Figures I, II and IV can be justified byrelatively simple Venn diagram examinations, presented inPeterson’s book, we do not repeat all of them here. Validsyllogisms in Figures III are more problematic and arguable.First consider the following valid affirmative syllogisms inFigure III on Peterson’s list:
Figure III AffirmativeAAI PAI TAI KAI IAIAPI PPI TPI KPIATI PTI TTIAKI PKIAII
Observe that the only possible conclusion is I ‘Some S are P’.If the first premise is the universal A ‘All M are P’, then in thecorresponding Venn diagram d = g = 0. This informationcombined with the second premise implies that there is a
ee+ f -portion of M in P, where by existential import principlee �= 0. This implies that the intersection S ∩ P is non-empty.Thus, I ‘Some S are P’ is justified. In a similar manner wevalidate those syllogisms whose second premise is A ‘All Mare S’.
Then focus on the remaining four syllogisms PPI, TPI, PTIand TTI. Notice that in all of them both premises are prepon-derance statements in the sense of majority or plurality. Con-sider the corresponding Venn diagrams. By the first premisee + f > d + g and by the second premise d + e > f + g,thus by summing up 2e + d + f > 2g + d + f . If it wouldbe the case that e = 0 then d + f > 2g + d + f , a con-tradiction. Hence the intersection S ∩ P is non-empty. Thus,the conclusion I ‘Some S are P’ is justified.
Finally, consider the two underlined syllogisms PKI anfKPI in Figure III by the corresponding Venn diagram. Noticethat they are the only affirmative valid syllogisms in Figure IIIwhere none of the premises is the universal A statement,exactly one of the premises is a preponderance statementwhile the other premise is not. Since S, P and M can bedistributed such that b = g = 0 independently of other para-meters a, c, d, e and f , a necessary and sufficient conditionfor S ∩ P to be non-empty (i.e. that the sets M ∩ P and M ∩ Soverlap) is that
e + f
d + e + f + g+ d + e
d + e + f + g> 1, (23)
where e+ fd+e+ f +g is the portion of M that are P and d+e
d+e+ f +gis the portion of M that are S. Indeed, assume e = 0. Then
d+ fd+ f +g > 1, a contradiction. Thus, PKI and KPI in FigureIII are valid syllogisms if, and only if (23) holds. We observethat (23) holds if, and only if
e + f
d + e + f + g+ d + e
d + e + f + g− 1 > 0 iff
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An algebraic study of Peterson’s Intermediate Syllogisms
(e + f
d + e + f + g� d + e
d + e + f + g
)> 0.
where � is the Łukasiewicz product. These observations leadus to the following
Theorem 1 An affirmative syllogism in Figure III is validand the conclusion is I ‘Some S are P’ if V � W �= 0,where V and W are the grades associated with the first andsecond premise, respectively. All other affirmative syllogismsin Figure III are invalid.
Proof Consider the following Cayley table:� (A, 1) (P, p) (T, t) (K , k) (I, ε)
Lastly consider the valid negative syllogisms in Figure IIIon Peterson’s list. We observe that they are obtained fromthe affirmative ones by replacing the first premise and theconclusion by their complements, respectively. We have
Theorem 2 A negative syllogism in Figure III is valid withconclusion O ‘Some S are not P’ if V ∗ � W �= 0, whereV ∗ and W are the grades associated with the complementof the first and to the second premise, respectively. All othernegative syllogism in Figure III are invalid.
Proof Consider the following Cayley table:� (A, 1) (P, p) (T, t) (K , k) (I, ε)
Remark 3 Returning now to the question of contradictorystatements, an Aristotelian A ‘All M are P’ indicates in Venndiagram notation that d = g = 0 and e+ f
d+e+ f +g = 1.Thus, in particular, it is conflicting to assume simultane-ously that d
d+e+ f +g > 0 should hold, i.e. that an instanceof Aristotelian O ‘Some M are non-P’ would be true. Since
d+ed+e+ f +g ≥ d
d+e+ f +g we have, regardless of the choice of
parameter e, that e+ fd+e+ f +g + d+e
d+e+ f +g > 1, which is condi-tion (23). This is the origin of defining grades p, t, k in theway we defined them at the beginning of this section.
4 Introducing new quantities: how many validsyllogisms there are?
An interesting problem is how Peterson’s results can be gen-eralized. Two principles are obvious. First, in the same wayas the 24 valid Aristotelian syllogisms remain valid in Peter-son extension—no new syllogism containing only classicalstatement appears nor disappears in the list of Peterson’s 105
valid syllogisms—introducing a sixth, seventh, etc. quantityshould leave the 105 intermediate syllogisms untouched. Forexample, KKI in Figure III in invalid, thus, it should remaininvalid in all extended systems of syllogisms, similarly, PPI inFigure III is valid, so it should remain valid in all extensions.If an intermediate syllogism is ‘an empirical fact’, then itsvalidity does not depend on the other quantifiers we have. Inshort, any generalization has to be conservative. The secondevident starting point for extending Peterson’s system is tolinguistically analyze new quantities and their relation to theold ones, the ‘some’, many’, ’most’, ‘almost-all’ and ‘all’.In practice, this means extending Peterson’s square. Theseobservations are consistent with Peterson’s thoughts. Basedon our algebraic analysis, only the order (mutual strength) ofstatements and contradictory pairs are important.
Peterson deals with the issue of generalization in the fol-lowing way (page 109 in Peterson 2000) ‘If we introduce asixth quantity ... of a type Half-or-more ... Then or-more issuperfluous ... Half would require the or-more rider, too.’To our understanding ‘Half’ is problematic. We return to thesubject soon. Moreover, on pages 118–119 Peterson writes‘Adding Half (denoted by F and the complement statementdenoted by V) would occur ... new valid argument forms’,e.g.
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Figure I Affirmative (+‘Half’)AAAAAP APPAAT APT ATTAAF APF ATF AFFAAK APK ATK AFK AKKAAI API ATI AFI AKI AII
and
(73)
Figure IV (+‘Half’)AAI AEE EAOPAI AEB EPOTAI AED ETOFAI AEV EFOKAI AEG EKOIAI AEO EIO
‘Now we have not given the rules for generating these validforms nor another new Venn Diagram method ... Rather, weclaim that sufficient study of the structure of the arrays in(Peterson’s Square) — and of the reasons for those structuresvia the ‘intermediate’ nature of the new quantifiers ... willmake it obvious that any extension of the 5-quantity rules(and of the Venn Diagram method ...) to 6 quantities willhave to validate argument forms in the new positions givenin (72) and (73) as well as in analogous new position forthe other figures’.
One can show by Venn diagram interpretation that addingnew rows and columns to the sets of valid syllogisms in Fig-ures I, II and IV, as done above, correspond to new validsyllogisms. For example, the validity of AFF in Figure I canbe seen by the fact that if ‘All M are P’ then d = g = 0 andif ‘At least half of S are M’ then e ≥ a + b, thus b + e ≥ a.Consequently, ‘At least half of S are P’.
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Figure III is more challenging: is FFI-III a valid syllogism?Peterson does not give a direct answer to the question, but bystudying the diagram (71) on page 118, it is found that F andV do not constitute a contradiction pair; by our analysis thiswould indicate that the answer is negative. This conjecturecan be proved by Venn diagram method. We observe that ifS and P are two disjoint sets and M is such that exactly halfof M are P and half of M are S, then premises in FFI-III hold,however, no S are P.
By a formula k2+k2 , k = 6 (on page 121 in Peterson 2000),
we calculate that there should be 21 valid affirmative syllo-gisms in Figure III. This is an indirect argument that Peter-son’s list of affirmative syllogisms would look as follows
(F-III+ 12 )
Figure III Affirmative (+‘Half’)AAI PAI TAI FAI KAI IAIAPI PPI TPI FPI KPI ×ATI PTI TTI FTI × ×AFI PFI TFI × × ×AKI PKI × × × ×AII × × × × ×
where ‘×’ means that at that position the corresponding syllo-gism is not valid. Of course, the situation would be differentby agreeing that FXY means ‘more than half of X are Y’.Then FFI-III would be valid, however, Peterson’s ‘Half-or-more’ interpretation would be violated.
On page 120–121 Peterson writes ‘... how to constructsound and complete syllogistic system for k quantities, wherefinite k is as high as you like. You do this by selecting addi-tional fractional quantifiers for insertion in new fractionallyquantified categoricals and by inserting new intermediate(fractional) syllogisms in appropriate positions in the pat-terns for valid syllogisms—the patterns first introduced in(the list of 105 valid) syllogisms. Some simple equations char-acterize the results ... For Figures I–III there are six triangu-lar arrays of valid forms. The number of forms in each array
is k2+k2 .’ At this point Peterson makes the jump from empir-
ical data to a theory of fractional quantifiers. In our viewthis theory is problematic as it is not conservative. Indeed,assume that, in addition to the statements F and V, we adda new affirmative statement M: ‘Only a couple of X are Y’and its negative complement N: ‘Only a couple of X are notY’ with an interpretation that M is in between I and K.
By the method of fractional quantifiers the list of 28(= 72+7
2 ) valid affirmative syllogisms in Figure III would beas follows:
Figure III Affirmative (+‘Half’ +‘Couple’)AAI PAI TAI FAI KAI MAI IAIAPI PPI TPI FPI KPI MPI?ATI PTI TTI FTI KTI×AFI PFI TFI FFI×AKI PKI TKI×AMI PMI?AII
However, TKI-III and KTI-III are invalid (by Peterson’s ownargument) as well as FFI-III by the argument we presentedabove. Moreover, MPI and PMI are ambiguous cases, that is,it is not clear that we should accept
Almost-all M are POnly a couple of M are SSome S are P
andOnly a couple of M are PAlmost-all M are SSome S are P
to be valid intermediate syllogisms. There is no pure mathe-matical answer, rather the answer is related to the question:are (P, N) and (M, B) pairs of contradictory statements? Thequestion needs to be solved by a linguistic research, only thena mathematical approach can be used. Our example shows,however, that Peterson’s formula of number of valid syllo-gisms is incorrect; there are 23 or 25 valid syllogisms, not28.
Take another example. Let us leave out one of the threeoriginal Peterson’s intermediate quantifiers, say K ‘Many Xare Y”. It is clear that the corresponding list of valid syllogismis achieved by excluding those in which K, or its complementG occurs. Thus, the list of valid syllogisms in Figure III,affirmative cases, would look as follows:
Figure III Affirmative −‘Many’AAI PAI TAI IAIAPI PPI TPIATI PTI TTIAII
This list, 11 valid syllogisms on it, can be justified also by aVenn diagram consideration and by observing that in all inter-mediate cases of them both premises are preponderance state-ments. Here k = 4, so by Peterson’s formula there should be42+4
2 = 10 valid syllogisms, a contradiction.Finally, let us leave out the intermediate quantifier P
‘Almost-all X are Y’. Then the corresponding list of validsyllogism is achieved by excluding those in which P, or itscomplement B occurs. Thus, the list of valid syllogisms inFigure III, affirmative cases, is as follows:
Figure III Affirmative −‘Almost-all’AAI TAI KAI IAIATI TTIAKIAII
Instead of 10 valid syllogisms there are just 8 on the list.The fractional syllogism method would include TKI and KTIwhich, however, are invalid.
5 Algebraic semantics for n-quantity syllogisms
As we have seen in our analysis of Peterson’s 5 quantitiessystem, valid syllogisms in Figures I, II and IV are relativelysimple to characterize; indeed, in all of them at least one of thepremises is a classical one (A or E) and validity ultimatelydepends only on the mutual strength of the statements: aconclusion cannot be stronger than any of the premises. In
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Figure III the situation is more challenging; there are 12 validsyllogisms where both premises are intermediate, in 8 of thempremises being preponderance statements and 4, the mostinteresting cases, one premise is a preponderance statementwhile the other premise is not. The 8 valid syllogisms caneasily be justified by realising that if ‘more than half of Mare P’ and ‘more than half of M are S’ then necessarily S and Pmust overlap: at least one S must be P. Peterson pores over therest 4 valid syllogisms; KPI, PKI, GPO and BKO, and writes(page 61 in Peterson 2000) ‘Certainly some proportion equalto or less than 25% should fail to count as many. And someproportion equal to or less than 75% should fail to count asalmost-all. But what percentages exactly? To that questionwe think an exact answer in isolation should be avoided.Rather, with such borderline examples, we recommend thatthey be allowed to remain simply borderline questions thatother considerations can bear upon to resolve.’
Our solution is the following: the state of affairs that KPI,PKI, GPO and BKO are valid syllogisms in Figure III isstrongly related to the fact that K and B (the complementof P), P and G (the complement of K), G and B (the com-plement of P) and the last, B and G (the complement ofK) constitute contradictory pairs. By our algebraic analysis,two statements X and Y are contradictory if, and only if theŁukasiewicz product V � W of the corresponding grades Vand W is positive. In fact, this condition defines all the valid(affirmative) syllogisms in Figure III. We introduceAn algorithm to produce valid intermediate syllogismsOn the basis of a linguistic analysis of intermediate state-ments and their mutual relation,
1◦ Place new statements R1, · · · , Rn and the correspondingcomplements S1, . . . , Sn on the Peterson’s square.
2◦ Attach the (potential new) contradictory pairs (contradic-tory pairs already existing on Peterson’s square have toremain contradictory pairs).
3◦ Associate grades ε, k, t, p, 1 with I, K, T, P and A,respectively, such that (a) 0 < ε < 1 − p < k < 1
2 <
t < p < 1, (b) k + p > 1 and (c) t + k ≤ 1.4◦ Associate grades r1, . . . , rn to the corresponding new
statements between ε and 1 such that the strength of thestatements is respected and, if (a) (X, Y) constitute a bor-derline contradictory pair, where X is either an originalstatement in Peterson’s list or a new statement and Y is anoriginal or new complement statement, then x > 1 − yand (b) (X, Y) does not constitute a contradictory pairthen x ≤ 1 − y, where x and 1 − y are the corre-sponding grades of X and Y, respectively. Notice thatthe grades s1, . . . , sn of the complements S1, . . . , Sn arealso fixed. Statements F (‘Half’) and its complement Vare always associated with the value 1
2 . If such gradesε < r1, . . . , rn < 1 on the unit real interval that sat-isfy these conditions do not exist, then the new linguistic
analysis is not compatible with that of Peterson, other-wise
5◦ Valid syllogisms are exactly the following ones:
Figure I, affirmative
1◦ The first premise in each of the affirmative syllogisms isthe classical A.
2◦ Denote the grade of the second premise by W (=1, ε orsome q or r j ) and the grade of the conclusion by U (=1, ε
or some q or r j ). Then the corresponding syllogism isvalid if, and only if U ≤ W or, equivalently, U∗⊕W = 1,where ⊕ is the Łukasiewicz sum.
Figure I, negativeValid negative syllogisms are obtained from the affirmativeones by replacing the first premise and the conclusion bytheir complements, respectively.
Figure IIAll valid syllogisms in Figure II are negative. The first set ofvalid syllogisms are exactly the same that the negative validsyllogisms in Figure I.
Another set of valid syllogisms is obtained by substitutingboth premises by their complements, respectively, in the firstset.
Figure III, affirmativeAn affirmative syllogism is valid and the conclusion is I iffV � W �= 0, where V and W are the grades related to thefirst and second premise and � is the Łukasiewicz product.
Figure III, negativeValid negative syllogisms are obtained from the affirmativeones by replacing the first premise and the conclusion bytheir complements, respectively.
Figure IVAn intermediate syllogism is valid if, and only if it is obtainedfrom a classical syllogism by replacing one (and only one)of the premises by a stronger statement or the conclusion bya weaker statement (but not both).
Remark 4 Related to valid affirmative syllogisms in Figure I(notice that the first premise is always A), there is the follow-ing residuation (Galois connection)
V � U ≤ W iff V ≤ U → W, (24)
where V, W and U are the grades of the first and secondpremise and the conclusion, respectively. This Galois con-nection completely describes the valid affirmative syllogismsin Figure I. Since the valid negative syllogisms in Figure Iare obtained by symmetry from the affirmative ones, and allthe valid syllogisms in Figure II are determined by the corre-sponding valid syllogisms in Figure I, we conclude that andthe Galois connection (24) together with symmetry, deter-mines all the valid intermediate syllogisms in Figure I andFigure II.
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The logic of ‘Some’, ‘Just a couple’, ‘Many’, ‘Half’,‘Most’, ‘Almost-all’, ‘All with the exception of a fewpathological cases’ and ‘All’Suppose that we have, in addition to above defined state-ments M, N and F, V, two other statements S: ‘All X withthe exception of a few pathological cases are Y’ (‘All-but’in short) and its complement Z: ‘All X with the exceptionof a few pathological cases are not Y’. Assume further thaton the basis of a linguistic analysis2 all the 16 statementscan be located on the Extended Peterson’s square. Intuitivelyit is easy to accept that (T, V) constitute a border line con-tradictory pair. (P, G) is a borderline contradictory pair byPeterson’s approach, so (S, G) should also constitute a contra-dictory pair as S is stronger than P. Finally assume, althoughthis would require a more detailed linguistic analysis, that (S,N) is not a contradictory pair. Associate with each statementa grade q in the following order:
1◦ A �→ 1, F �→ 0.5; these values are always fixed, moreovertake I �→ 0.01, a small positive ε,
2◦ P �→ 0.8, T �→ 0.6, K �→ 0.4; these choices respectthe strength, contradictory pairs and preponderance state-ment assumption in Peterson’s square,
3◦ S �→ 0.9 and M �→ 0.05; these choices respect theextended strength, contradictory pairs and preponderancestatement assumption. Thus we have
A: All X are Y 1.0 ↔ 0.0 E: No X are Y⇓ ⇓S: All-but X are Y 0.9 ↔ 0.1 Z: All-but X are not Y⇓ ⇓P: Almost-all X are Y 0.8 ↔ 0.2 B: Almost-all X are not Y⇓ ⇓T: Most X are Y 0.6 ↔ 0.4 D: Most X are not Y⇓ ⇓F: Half X are Y 0.5 · · · 0.5 V: Half X are not Y⇓ ⇓K: Many X are Y 0.4 · · · 0.6 G: Many X are not Y⇓ ⇓M: Couple X are Y 0.05 · · · 0.95 N: Couple X are not Y⇓ ⇓I: Some X are Y 0.01 · · · 0.99 O: Some X are not Y
Based on these algebraic semantics of the intermediate state-ments we determine the valid (and invalid) intermediate syl-logisms.
In Figure I the valid affirmative syllogisms are determinedby the equation U∗ ⊕ W = 1, thus consider the followingCayley table
2 This example is not based on the real linguistic analysis, only on ourintuition.
1. premise 2. premise A S P T F K M Iis A grade W↘
which yields the following list of affirmative syllogisms.
(F1-A)
AAAAAS ASSAAP ASP APPAAT AST APT ATTAAF ASF APF ATF AFFAAK ASK APK ATK AFK AKKAAM ASM APM ATM AFM AKM AMMAAI ASI API ATI AFI AKI AMI AII
To write the list of negative syllogisms in Figure I is a rou-tine task to replace in each affirmative case the first argumentby E and the conclusion by this complement. They are thefollowing
This implies the following list of affirmative syllogisms
(F3-A)
AAI SAI PAI TAI FAI KAI MAI IAIASI SSI PSI TSI FSI KSIAPI SPI PPI TPI FPI KPIATI STI PTI TTI FTIAFI SFI PFI TFIAKI SKI PKIAMIAII
Negative valid syllogisms in Figure III are those that sat-isfy V ∗ �W �= 0, where V ∗ and W are the grades associatedwith the complement of the first and with the second premise,respectively, and the conclusion is always O. Alternatively,valid negative syllogisms are obtained from the affirmativeones by replacing the first premise and the conclusion bytheir complements, respectively. Both methods produce thefollowing list
(F3-N)
EAO ZAO BAO DAO VAO GAO NAO OAOESO ZSO BSO DSO VSO GSOEPO ZPO BPO DTO VTO GTOETO ZTO BTO DPO VPOEFO ZFO BFO DFOEKO ZKO BKOEMOEIO
Finally, valid syllogisms in Figure IV are the following
Remark 5 The above eight-quantifier extension of Peter-son’s original approach is conservative in the sense wedefined conservativity. Also notice that other numerical val-ues than those given above but respecting the order of thequantifiers and contradictory statements would yield thesame valid syllogisms. Notice that SMI-III is not valid; thisis due to the fact that we accepted S: ‘All P with the exceptionof a few pathological cases are M’ and N: ‘Just a couple ofS are not M’ to be simultaneously true. For the same reasonNSO-III, ZMO-III and MSI-III are invalid.
6 Peterson’s Intermediate Syllogisms and fuzzy logic
6.1 Some (fuzzy) logic approaches to intermediatesyllogisms
Peterson’s Intermediate Syllogisms are Boolean in a sensethat they are either valid or invalid, there is no third alterna-tive. However, they are linked to many-valued logic; we haveshown above the function of an MV-algebra in Peterson’sIntermediate Syllogisms; if the grades associated with inter-mediate categorical propositions are interpreted as values onthe Łukasiewicz MV-algebra, then the validity or invalidityof these syllogisms are determined by simple MV-equations.
Syllogisms and intermediate categorical propositions havea kinship with fuzzy logic. Indeed, fuzzy logic aims at mod-eling states of affairs that are not black and white, but wheretruth can be a matter of degree. A similar situation can beseen in intermediate syllogisms, though from a slightly dif-ferent point of view. It is therefore quite surprising that a linkbetween fuzzy logic and intermediate syllogism has beenstudied relatively little. Next, we look at some of the few(fuzzy) logic oriented approaches to intermediate syllogisms,namely those of Zadeh (1983), Murinová and Novák (2012),Novák (2008) and Vetterlein (2012). The main differencebetween these approaches and that of Peterson is the factthat Peterson examines syllogisms mainly from a linguis-tic point of view; his mathematical arguments are based onVenn diagrams, while the others are mathematically orientedand use classical or fuzzy logic arguments, mathematical innature.
Zadeh cites in his 1983 paper (Zadeh 1983) Peterson’s1979 work (Peterson 1979) and writes ‘The generic termfuzzy quantifier is employed in this paper to denote the collec-tion of quantifiers in natural languages whose representativeelements are: several, most, much, not many, very many, notvery many, few, quite a few, large number, small number, closeto five, approximately ten, frequently, etc. In our approach,such quantifiers are treated as fuzzy numbers which may bemanipulated through the use of fuzzy arithmetic and, moregenerally, fuzzy logic. A concept which plays an essential rolein the treatment of fuzzy quantifiers is that of the cardinality ofa fuzzy set.’ However, Zadeh does not focus on Petersons’slist of valid syllogisms. With the exception of the startingpoint the two approaches are rather different in nature. It isan open and interesting research problem whether the the-ory of fuzzy numbers can be used in validation of Peterson’sIntermediate Syllogisms.
More recent and still ongoing studies on Peterson Interme-diate Syllogisms are by Murinová and Novák (2012), Novák(2008). Murinová writes in Murinová and Novák (2012) ‘thebasic idea consists in the assumption that intermediate quan-tifiers are just classical quantifiers ∀ and ∃ but the universe ofquantification is modified’; this approach is obviously related
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to Mostowski’s pioneering work Mostowski (1957) on gen-eralized quantifiers. Novák and Murinová work in a verygeneral mathematical framework called Łukasiewicz fuzzytype theory, Ł-FTT in short. Since the approach of Novákand Murinová is related to fuzzy logic, i.e. it is not Boolean,they talk about generalized intermediate syllogisms insteadof intermediate syllogisms. Their goal is ambitious, how-ever, rather complicated. For example, a simple statementG: ‘Many B are not A’ is formally presented in Ł-FTT by
(Q)∀¬(Smv)(B,¬A)
≡ (∃z)(�(z ⊆ B)∧¬(Smv)((μB)z)∧(∀x)(zx ⇒¬Ax)),
and similarly the other intermediate statements. One mayargue that such a formalism is rather heavy, a factor whichhowever, does not detract the theoretical value of theirapproach. The authors show that all the 105 valid Peterson’sIntermediate Syllogisms are valid also in Ł-FTT; however byMurinová (Personal communication), Ł-FTT might also val-idate some such intermediate syllogism which are invalid inPeterson’s approach; for example KKI-III is such an invalidPeterson’s Intermediate Syllogism (to verify that KKI-III isinvalid, take b = e = g = 0 and a = d = c = f �= 0in Venn diagram); the problem in Ł-FTT is open. In orderfor the Ł-FTT approach to be considered as a mathemati-cal formalism of Peterson’s Intermediate Syllogisms, Ł-FTTshould prove invalid those syllogisms that are invalid byPeterson and, if this is not the case, it would be interestingto know which of Peterson’s invalid syllogism are valid inŁ-FTT and analyze what causes the differences. Moreover,by Murinová (Personal communication), another challengingresearch problem in Ł-FTT is the definition of a proper con-cept of contradictory intermediate categorical propositions.We believe that the results presented in this study, a kind ofMV-nature of Peterson’s Intermediate Syllogisms, may beuseful in solving this problem; after all Ł-FTT is based onLukasiewicz’ logic. Another interesting topic for discussionis also the following fact ‘The validity (of Peterson’s Interme-diate Syllogisms) is in most cases weak (the provability of theconclusion C follows from the provability of both premisesP1, P2) but in some cases it is even strong which means thatthe implication P1&P2 ⇒ C is provable and so, the truthvalue of C in any model is greater or equal to the truth valueof P1&P2. (cf. Murinová and Novák 2012). We will returnto this issue briefly in the Sect. 6.2.
The most recent research related to intermediate syllo-gisms is that of Vetterlein (2012). Vetterlein introduces atheory which he calls the first-order Theory of SyllogisticReasoning, TSR in short. The basic idea is to use com-mon (Boolean) first-order language describing relationshipsbetween sets, omitting however, the top element. Basic toolsin TSR are a binary relation ‘∼’ to denote equal cardi-nalities of sets, and particular subset hood relations like
‘Amany⊂ B’ with a meaning (Vetterlein, Personal communica-
tion) ‘Among the B’s, there are many A’s’ or simply ‘Many Bare A’; this corresponds to a statement K in Peterson’s square.Based on these notions, Vetterlein introduces a set of axiomsrelated to intermediate syllogisms. He writes, however, ‘Aswe will model certain expressions of natural language, ouraxioms do not have a definite character; we do not claim thatthese axioms are the only acceptable ones.’ The syllogisticsystems of Peterson and those of Vetterlein share many validintermediate syllogisms; e.g. ATK-I and EAD-II are such(as proved explicitly by Vetterlein). Moreover, if Vetterlein’s‘Nearly all’ is understood to have the same meaning thanPeterson’s ‘Almost all’ then the following Vetterlein’s (valid)intermediate syllogism
No P are MAll S are MNearly all S are not P
corresponds to Peterson’s (valid) EAB-II. However, there arealso important differences between these two approaches.The following is a valid intermediate syllogisms in Vetter-lein’s approach
Nearly all M are PAll M are SSome S are not P
while in Peterson’s approach it would correspond to PAO-IIIwhich is invalid (as it violates the configured rule: the con-clusion is negative if and only if exactly one of the premisesis negative). The difference is obviously due to the fact that inVetterlein’s interpretation ‘Nearly all M are P’ means ‘Nearlyall M are P and a few M are not P’, while Peterson’s basisis that ‘Almost all M are P’ is to be understood as ‘Almostall or more M are P’. Vetterlein’s study raises an interestingresearch topic: can one modify TSR’s axioms so that theyvalidate Peterson’s valid Intermediate Syllogisms and onlythem? A favorable solution would link Peterson’s Interme-diate Syllogisms to first-order logic language of sets.
6.2 Peterson’s Intermediate Syllogisms as fuzzy theoriesin Pavelka logic
In applying Peterson’s Intermediate Syllogisms in real lifesituations we proceed in the following manner. For example,in some imaginary world, the statements ‘Almost all math-ematicians are poor’ and ‘Many mathematicians are stingy’would be assumed to be true, i.e. we associate with thema truth value 1 instead of value 0. Since the statements areinstances of the two premises of PKI-III which is a validsyllogism in Peterson’s system, we may associate the truthvalue 1 (true) with the statement ‘Some of the stingy arepoor’, too. Now we may ask what happens if we are slightlyunsure of the truth of the statements; instead of value 1, weassociate, say, values 0.9 and 0.8 with them, respectively.
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Which truth value should then be associated with the con-clusion? We argue that a well-founded value would be 0.7.Next we motivate our position.
A fundamental principle of fuzzy logic is to keep its for-malism simple, understandable and easy to apply. Thereforewe present the following approach. Since Peterson dealswith intermediate categorical propositions, let us considerthem within the framework of fuzzy propositional logicinstead of fuzzy predicate logic, namely in Łukasiewicz stylefuzzy propositional logic that has been developed by Pavelka(1979). (A pure algebraic generalization, a comprehensiveoverview and many real life examples of applications ofPavelka’s approach can be found in textbook Turunen (1999),see also Hájek (1998), we do not explain all details here.) InHájek’s monograph Hájek (1998) Pavelka’s logic is calledRational Pavelka Logic and in its Chapter 8.4. Hájek has aninteresting study on what he calls the logic of ‘Many’ corre-sponding to natural language expressions like ‘Many studentspass the examination without any problem’. We observe thatsuch statements can be viewed as statements K on Peterson’slist. Hájek treats the quantifier ‘Many’ as a modality in sen-tential logic (although he does focus on syllogisms). To beexact, ‘Many’, ‘Most’ and ‘Almost all’ are not modalitiesin the linguistic meaning of the word, however, this does notpreclude Hájek to treat them mathematically in the same waythan modalities are treated in propositional logic.
This gives us a suggestion to take a similar approach wetook in Kukkurainen and Turunen (2002) where we showedthat fuzzy IF–THEN inference rules can be treated as specialaxioms of a fuzzy theory in Pavelka fuzzy propositional logic;such an approach has proven to be very useful in many realworld modelling projects, we refer to our works (Dubrovinand Jolma 2002; Ketola et al. 2004; Niittymäki and Turunen2003). We would further stress that while treating ‘Many’,‘Most’ and ‘Almost all’ as modalities we limit only to Peter-son’s Intermediate Syllogisms. Our purpose is not to intro-duce an overall logical formalism of generalized quantifiers.Let us clarify our idea by the following example.
Consider an instance of a valid syllogism PKI-III:
(A)Almost all mathematicians are poorMany mathematicians are stingySome of the stingy are poor
‘Almost all mathematicians are poor’, ‘Many mathematiciansare stingy’ and ‘Some of the stingy are poor’ are treated asatomic propositions and denoted by α, β and γ , respectively.Then the instance (A) of syllogism PKI-III corresponds to aproposition α ⇒ (β ⇒ γ ), or equivalently (α&β) ⇒ γ ; it isto be considered true at the highest degree, not by its logicalform (it is not a tautology) but on the basis of Peterson’sinvestigation. In other words, it is a special axiom in a fuzzytheory T of Peterson’s Intermediate Syllogisms in the senseof Pavelka (cf. Definition 29 in Turunen 1999). However, α
and β, the other two special axioms of T need not be true
at the highest degree; for some reason or another we mayassociate values a, b ∈ [0, 1] with α and β, respectively.Then the fuzzy theory T has the following special axioms.(i) α ⇒ (β ⇒ γ ), true at the degree 1, (ii) α, true at a degreea ∈ [0, 1] and (iii) β, true at a degree b ∈ [0, 1].
We are interested in how true the conclusion γ is. InPavelka logic we have the following answer. Consider firstthe proposition γ from a syntactic point of view. It has thefollowing (meta)proof in the fuzzy theory T :
formula degree justification(a) α ⇒ (β ⇒ γ ) 1 special axiom (i)(b) α a special axiom (ii)(c) β ⇒ γ 1 � a by Generalized Modus Ponens from (a), (b)(d) β b special axiom (iii)(e) γ a � b by Generalized Modus Ponens from (c), (d)
Thus, the provability degree of γ is at least a � b. Now con-sider the fuzzy theory T from a semantic point of view. Avaluation v such that v(α ⇒ (β ⇒ γ )) = 1, v(α) = a,v(β) = b satisfies T , therefore T is consistent and thus, acomplete fuzzy theory (cf. Definitions 23, 24, 30, Proposi-tion 94 and Theorem 25 in Turunen 1999). Moreover, sinceγ is considered as an atomic proposition, we may associatethe value a � b with it, that is v(γ ) = a � b; indeed, wereason that
hence it is sufficient to have a � b = v(α) � v(β) ≤ v(γ ),thus the truth degree of γ is a � b. Since T is complete weconclude that the provability degree of γ over T as well asthe truth degree of γ over T is a � b.
Notice that the above reasoning is applicable to any validPeterson’s Intermediate Syllogism. This leads to the follow-ing
Theorem 6 AssumeA quantity Q1 of ... are ...A quantity Q2 of ... are ...A quantity Q3 of S are P
is an instance of a valid Intermediate Syllogism on Peterson’slist. Let the degree of truth of the first and second premisebe a, b ∈ [0, 1], respectively. Then the conclusion is trueat a degree a � b, where � is the Łukasiewicz product. Weexpress such a fuzzy Peterson’s Intermediate Syllogisms inthe following way:
A quantity Q1 of ... are ... , aA quantity Q2 of ... are ... , bA quantity Q3 of S are P , a � b.
In particular, if a = 0.9 and b = 0.8, then the truth degreeof the conclusion is a � b = 0.7.
Taking another example from Murinová and Novák(2012), we have the following application of a fuzzy PPI-III.
Almost all old people are ill , 0.85Almost all old people have grey hair , 0.95Some people with grey hair are ill , 0.80
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In Ł-FTT reasoning (Murinová and Novák 2012), the con-clusion ‘Almost all old people are ill is true at a degree 0.90’,hence there is difference compared to Ł-FTT approach. Moregenerally, it is obvious that we have
Corollary 7 If Peterson’s Intermediate Syllogisms are inter-preted as special axioms of fuzzy theories in the sense ofPavelka logic, and the truth values of the premises are a andb, respectively, then the truth of the conclusion is equal toa � b, not greater than a � b.
7 Conclusion
We have studied Peterson’s Intermediate Syllogisms, infer-ence systems that extend Aristotelian syllogisms on cate-gorical propositions containing ‘All’ and ‘Some’ by accept-ing three intermediate categorical propositions containing‘Many’, ‘Most’ and ‘Almost all’. The 105 valid Peterson’sIntermediate Syllogisms, out of 4,000 possible, are ratheran empirical fact of correct reasoning than a theory of pos-sible way of reasoning. Indeed, no further justification orvalidation is needed to recognize e.g. that ‘All M are P’ and‘Many M are not P’ are two (Boolean) intermediate categori-cal propositions which are conflicting, i.e. are contradictory.Similarly, if ‘All M are P’ and ‘Many M are S’, then nec-essarily ‘Some S are P’. These are simple facts we accept.In this paper we do not define any new theory on Peterson’sIntermediate Syllogisms; we only demonstrate that, by asso-ciating certain values V, W and U on standard ŁukasiewiczMV-algebra with the first and second premise and the con-clusion, respectively, the validity of the corresponding Peter-son’s Intermediate Syllogism is determined by a simple MV-algebra (in-)equation. Indeed, all valid syllogisms (and onlythem) in Figure I and Figure II are determined by equationsof type W ⊕ U = 1, U∗ ⊕ W = 1, or W ∗ ⊕ U = 1. In Fig-ure III the inequations are V � W �= 0 and V ∗ � W �= 0. InFigure IV validity of a syllogism depends only on the orderof the values V, W and U. These observations justify the titleof this paper.
We also discuss possible extensions of Peterson’s sys-tem. We claim that, due to the empirical nature of the 105valid intermediate syllogisms including the original 24 Aris-totelian ones, a proper extension or restriction must be con-servative in a sense that validity or invalidity of any existingsyllogism must remain in this extension or restriction. Inthis respect our approach differs from Peterson’s system offractional syllogisms. Of course, a proper extension must bebased on a linguistic analysis of new quantifiers, their relationto the existing quantifiers, and pairs of contradictory interme-diate categorical propositions. In practice, this implies thatan extension of Peterson’s syllogistic system must start bya conservative extension of Peterson’s square, similarly asPeterson’s square is obtained from Aristotle’s square by a
conservative extension. We review a few (fuzzy) logic basedstudies on generalized syllogisms and suggest some interest-ing research topics.
Finally, we show how Peterson’s Intermediate Syllogismscan be viewed as fuzzy theories in Pavelka’s fuzzy propo-sitional logic; after all, intermediate syllogisms deal withintermediate categorical propositions. In other words we dealwith intermediate quantities as modalities in sentential fuzzylogic; this reminds Hájek’s study on ‘Many’ in his book(Hájek 1998). Our approach extends valid bivalent Peter-son’s Intermediate Syllogisms to cases where premises aretrue to a degree, and presents a simple method to determinethe unique degree of truth of the corresponding conclusion.Thus, we introduced a fuzzy version of Peterson’s Interme-diate Syllogisms.
Acknowledgments The author was supported by the Czech Techni-cal University in Prague under project SGS12/187/OHK3/3T/13. Theauthor is grateful for M. Navara, P. Murinová, T. Vetterlein, P. Cin-tula, C. Noguera and R. Horcík for their comments to improve the finalversion of this research.
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