Second Interlude
Modelling the Syllogism Second InterludeModelling the Syllogism People do not understand how that which is at variance with itself, agrees with itself. Heraclitus
It is impossible for the same attribute at once to belong and not to belong to the same thing and in the same relation. Aristotle
I
n this interlude, I show how relational logic can be used to model syllogistic reasoning, originally developed by Aristotle some 2,300 years ago. Aristotle was concerned with various properties of things and with what conclusions can be made about these properties from assumed premises. These properties can quite easily be represented in relational logic. However, I am not here primarily concerned with the nature of the properties of the entities under discussion. I wish to model the underlying structure of Aristotle’s thought processes. This can best be done by studying the concepts of the syllogism and their relationships to each other.
Definitions A syllogism consists of three propositions (also called statements or sentences), called the major premise, minor premise, and conclusion, respectively. Each proposition has two terms called the subject and predicate.
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The terms in a proposition are related to each other in four different ways, shown in Table 34. Class Attribute names
Syllogistic propositions Name Form A All S are P
Diagram P S
E
All S are not P P
S
Attribute values
I
O
Some S are P
S S
P
or
P
S
P
or
P
Some S are not P
S
Table 34. Types of propositions in syllogisms
Each of these propositions has a number of dualistic attributes that characterize the propositions. They are grouped together in pairs depending on whether A is paired with E, I, or O. Table 35, which is an extension of Table 34, shows these attributes. Any two of these attributes uniquely defines the proposition. So we could call them defining attributes, with the third being derived from the other two. Class Attribute names
Attribute values
Syllogistic propositions Name Universality A universal E universal I particular O particular
Positivity positive negative positive negative
Symmetry asymmetrical symmetrical symmetrical asymmetrical
Table 35. Attributes of propositions
Aristotle called the symmetrical propositions convertible because they are equivalent when the terms are interchanged. A and E are also convertible into weaker forms, I and O, respectively. Furthermore, if we assume Aristototle’s rules of logic, A and O and E and I are contradictory; they exclude each other. One other property of these propositions relates to the terms in the proposition, rather than the propositions themselves. A term is distributed if, in 304
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some sense, it refers to all entities with the particular property (called a class), otherwise it is undistributed. The subject of universal propositions and the predicate of negative propositions are distributed. How these definitions relate to syllogistic propositions is given in Table 36, attributes that can be seen quite clearly from the diagrams in Table 34. Class Attribute names
Attribute values
Syllogistic propositions Name Subject A distributed E distributed I undistributed O undistributed
Predicate undistributed distributed undistributed distributed
Table 36. Distribution of terms in syllogism
The terms of the three propositions of the syllogism are related to each other in two ways: 1. One term is common to the major and minor premises; it is called the middle term (M). 2. The predicate (P) of the conclusion is the major term of the syllogism and the subject (S) is the minor term, because they are the nonmiddle terms in the major and minor premises, respectively. As each proposition has one of four types and as there are three propositions in each syllogism, there are 43 = 64 different syllogistic forms, called moods. These are naturally called AAA, AAE, AAI, etc. In addition, the syllogism can have one of four figures, depending on whether the middle term is the subject or predicate in the major and minor premises.
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(Curiously, for some reason, Aristotle only recognized three of these figures; the fourth was discovered only in the Middle Ages.) Class Attribute names
Attribute values
Syllogistic figures Name Figure I
M P S M S P
II
P M S M S P
III
M P M S S P
IV
P M M S S P
Table 37. Syllogistic figures
There are thus 64 x 4 = 256 possible syllogisms in total. Aristotle examined each mood and figure in turn to determine whether it was valid or not. He then derived a number of common properties of these syllogisms, which can be called rules of deductions. I reverse this process here. These are the rules that Aristotle discovered: 1. Relating to premises irrespective of conclusion or figure (a) No
inference can be made from two particular premises.
(b) No
inference can be made from two negative premises.
2. Relating to propositions irrespective of figure (a) If
one premise is particular, the conclusion must be particular.
(b) If
one premise is negative, the conclusion must be negative.
3. Relating to the distribution of terms (a) The
middle term must be distributed at least once.
(b) A
predicate distributed in the conclusion must be distributed in the major premise.
(c) A
subject distributed in the conclusion must be distributed in the minor premise.
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Application of rules We can now apply these rules of inference to determine the validity of each mood in each figure. As the first rule applies irrespective of conclusion or figure, we need consider just sixteen pairs of premises. By rule 1(a), these pairs of premises are invalid: II IO OI OO By rule 1(b), these pairs of premises are invalid: EE EO OE OO I could now show the sixteen premises in the form of a relation, indicating which has an attribute value invalid by one or both of the two rules. However, it is simpler to indicate invalidity by type style, bold and italics indicating invalidity by rules 1(a) and 1(b), respectively. The nine potentially valid premises are marked in plain style: AA EA IA OA
AE EE IE OE
AI EI II OI
AO EO IO OO
Rule 2 applies to the moods of syllogisms. As there are four moods and nine potentially valid pairs of premises from rule 1, we need to consider 36 moods. By rule 2(a), these moods are invalid: xIA IxA xOA OxA xIE IxE xOE OxE By rule 2(b), these pairs of premises are invalid: xEA ExA xOA OxA xEI ExI xOI OxI Rule 2 therefore leaves 16 potentially valid moods of syllogism, the invalid ones being marked in italics and bold, as for rule 1: AAA AIA EAA EIA IAA OAA
AAE AIE EAE EIE IAE OAE
AAI AII EAI EII IAI OAI
AAO AIO EAO EIO IAO OAO
AEA AOA
AEE AOE
AEI AOI
AEO AOO
IEA
IEE
IEI
IEO
The third rule is rather more complicated as it concerns the distribution of each of the three types of term in the syllogism for each mood and figure. Draft 14 February 2004
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First considering rule 3(a) about the distribution of the middle term. The right-hand four columns in the following table show the distribution of the middle term in the two premises. The ones that are invalid by this rule are the ones that are undistibuted in both premises, indicated by italics in the table. The other four columns show the sixteen potentially valid moods that we are considering after eliminating the others by applying rules 1 and 2. AAx AEx AIx AOx EAx EEx EIx EOx IAx IEx IIx IOx OAx OEx OIx OOx
308
xxA AAA
xxE AAE AEE
xxI AAI AII
EAE
IAI
xxO AAO AEO AIO AOO EAO
I D–U D–D D–U D–D D–U
II U–U U–D U–U U–D D–U
III D–D D–D D–U D–U D–D
IV U–D U–D U–U U–U D–D
EIO
D–U
D–U
D–U
D–U
IAO IEO
U–U U–D
U–U U–D
U–D U–D
U–D U–D
OAO
U–U
D–U
U–D
D–D
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Rule 3(b) concerns the distribution of the predicate. The ones that are invalid by this rule are the ones that are distributed in the conclusion, but not distributed in the major premise, indicated by italics in the table. AxA AxE AxI AxO ExA ExE ExI ExO IxA IxE IxI IxO OxA OxE OxI OxO
xAx AAA AAE AAI AAO
xEx
xIx
xOx
AII AIO
AOO
AEE AEO
EAE EAO
IAI IAO
EIO
IEO
OAO
I U–U U–D U–U U–D
II D–U D–D D–U D–D
III U–U U–D U–U U–D
IV D–U D–D D–U D–D
D–D
D–D
D–D
D–D
D–D
D–D
D–D
D–D
U–U U–D
U–U U–D
U–U U–D
U–U U–D
D–D
U–D
D–D
U–D
Rule 3(c) is similar to 3(b) except that it applies to the subject in the minor premise. The pairs of minor premise–conclusion that are invalid are again indicated by italics in the table.
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xAA xAE xAI xAO xEA xEE xEI xEO xIA xIE xII xIO xOA xOE xOI xOO
Axx AAA AAE AAI AAO
Exx
Ixx
Oxx
IAI IAO
OAO
EAE EAO
AEE AEO
AII AIO
IEO
EIO
AOO
I D–D D–D D–U D–U
II D–D D–D D–U D–U
III U–D U–D U–U U–U
IV U–D U–D U–U U–U
D–D
D–D
D–D
D–D
D–U
D–U
D–U
D–U
U–U U–U
U–U U–U
U–U U–U
U–U U–U
U–U
U–U
D–U
D–U
Eliminating the syllogisms, both mood and figure, made invalid by rule 3 leaves just 12 moods and 25 syllogisms. One of them, AAO in figure IV, is in rather a curious position. Aristotle’s rules allow it. However, it is only conditionally valid. As this is a syllogism of figure IV, and as Aristotle did not discover this figure, it is not surprising that he did not consider it. AAO in figure IV is only valid if P is an actual subset of M and if M is an actual subset of S. However, if P, M, and S are equivalent, there are no S that are not P. So we need to eliminate it from the list for this reason, not covered by Aristotle’s rules. It is only conditionally valid. There are five syllogisms that have a universal conclusion. So there are also five corresponding syllogisms with a particular conclusion, which we can also eliminate, as they are weak forms. These are: Strong AAA I EAE I AEE II EAE II AEE IV
Weak AAI I EAO I AEO II EAO II AEO IV
This leaves us with 20 valid syllogisms, found by Aristotle and his successors: 310
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First figure AAA, EAE, AII, EIO Second figure EAE, AEE, EIO, AOO Third figure AAI, IAI, AII, EAO, OAO, EIO Fourth figure AAI, AEE, IAI, EAO, EIO Students in the Middle Ages were expected to know all these by heart. For instance, the statutes of the University of Oxford in the fourteenth century included this rule: “Batchelors and Masters of Arts who do not follow Aristotle’s philosophy are subject to a fine of 5s. for each point of divergence, as well as for infractions of the rules of the Organum”.1 Not surprising therefore that they needed a mnemonic to remember this rather arbitrary set of letters: Barbara, Celarent, Darii, Ferioque, prioris: Cesare, Camestres, Festino, Baroko, secundae: Tertia, Darapti, Disamis, Datisi, Felpaton, Bokardo, Ferison, habet: Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.2
I don’t know what Tertia (EIA in the third figure) is doing here. She was eliminated by rule 2(a). Curiously, C. W. Kilmister, whose book Language, Logic and Mathematics this mnemonic is taken from, did not point out the error. These syllogisms can be further reduced because propositions E and I are symmetrical; the terms in these propositions can be interchanged. When one or both premises is symmetrical, the mood stays unchanged, but the figure changes. When the conclusion is symmetrical, interchanging the subject and predicate means that the major and minor premises must change position, resulting in a change in both mood and figure. This means that there are just eight core syllogisms out of the 256 candidates that we started with.
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Syllogism: Mood (figure) AAA (I) AII (I) EAE (I) EIO (I) AOO (II) AAI (III) EAO (III) OAO (III)
Equivalent to AAI (IV) ≡ AAI (I) [weak form] AII (III), IAI (III), IAI (IV) EAE (II), AEE (II), AEE (IV) EIO (II), EIO (IIII), EIO (IV) — Itself EAO (IV) —
To sum up, the table on the next page shows all 256 syllogisms, which are valid, which are equivalent to a valid syllogism, and which rule first eliminated them from the list. However, all this is rather abstract and mechanical. It tells us little about the meanings of this inferences. So tables 38 on page 314 and 39 on page 320 provide a list of examples of each of the eight valid syllogisms, together with a corresponding Euler-Venn diagram. Note that because I and O can be represented in two ways in Euler-Venn diagrams, as illustrated in Table 34 on page 304, there are sometimes four or five corresponding diagrams for one particular mood and figure.
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Class of syllogisms Figure Mood AAA AAE AAI AAO AEA AEE AEI AEO AIA AIE AII AIO AOA AOE AOI AOO EAA EAE EAI EAO EEA EEE EEI EEO EIA EIE EII EIO EOA EOE EOI EOO
I Valid 3(b) Weak 3(b) 3(b) 3(b)
Valid 3(b)
3(b) Valid Weak
II
III 3(c) 3(b,c) 3(a) Valid 3(b) 2(b) Equiv 3(b) 2(b) Weak 3(b) 2(a) 2(a) Equiv 3(a) 3(b) 2(a,b) 2(a) 2(b) Valid 3(b) 2(b) Equiv 3(c) 2(b) Weak Valid
1(b)
Valid
2(a,b) 2(a) 2(b) Equiv Equiv
1(b)
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Figure IV 3(c) 3(c) Equiv Cond
Mood IAA IAE IAI IAO IEA Equiv IEE IEI Weak IEO IIA IIE III 3(a) IIO IOA IOE IOI 3(a) IOO OAA 3(c) OAE OAI Equiv OAO OEA OEE OEI OEO OIA OIE OII Equiv OIO OOA OOE OOI OOO
I
3(a) 3(a,b)
3(b)
II
III
IV
2(a) 2(a) 3(a) Equiv Equiv 3(a,b) 3(b) 3(b) 2(a,b) 2(a) 2(b) 3(b) 3(b) 3(b)
1(a)
1(a)
3(a)
2(a,b) 2(a) 2(b) 3(b) Valid
3(b)
1(b)
1(a)
1(a,b)
313
314 All M are P Some S are M Some S are P [Some S are not P] All M are P Some S are M Some S are P [All S are P]
AII (I)
AII (I)
All M are not P All S are M All S are not P
EAE (I)
Table 38. Valid syllogisms sorted by mood and figure
All Catholics are Christians Some Germans are Catholics Some Germans are Christians [Some Germans are not Christians] All mathematicians are human Some meditators are mathematicians Some meditators are human [All meditators are human]
All primates are not bears All humans are primates All humans are not bears
Class of valid syllogisms (Sheet 1 of 6) Mood (Figure) Form Example AAA (I) All M are P All primates are mammals All S are M All humans are primates All S are P All humans are mammals
S
P
S
S
M
M
S
Diagram
M
M
P
P
P
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14 February 2004 All M are not P Some S are M Some S are not P [Some S are P] All M are not P Some S are M Some S are not P [All P are S]
EIO (I)
EIO (I)
All M are P Some S are M Some S are P [All M are S] [All P are S]
AII (I)
Table 38. Valid syllogisms sorted by mood and figure
All men are not women Some mathematicians are men Some mathematicians are not women [Some mathematicians are women] All meat eaters are not elephants Some wild animals are meat eaters Some wild animals are not elephants [All elephants are wild animals]
All humans are primates Some mammals are humans Some humans are primates [All humans are mammals] [All primates are mammals]
Class of valid syllogisms (Sheet 2 of 6) Mood (Figure) Form Example AII (I) All M are P All humans are mammals Some S are M Some mammals are primates Some S are P Some primates are humans [All P are S] [All humans are primates]
M
M
M
M
Diagram
S
P
P
S
P
S
S
P
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315
316 Table 38. Valid syllogisms sorted by mood and figure
Class of valid syllogisms (Sheet 3 of 6) Mood (Figure) Form Example EIO (I) All M are not P All bears are not meditators Some S are M Some mathematicians are meditators Some S are not P Some mathematicians are not bears [All S are not P] [All mathematicians are not bears] EIO (I) All M are not P All elephants are not meat eaters Some S are M Some wild animals are elephants Some S are not P Some wild animals are not meat eaters [All M are S] [All elephants are wild animals] All humans are not bears EIO (I) All M are not P Some primates are humans Some S are M Some primates are not bears Some S are not P [All primates are not bears] [All S are not P] AOO (II) All P are M All Catholics are Christians Some S are not M Some Germans are not Christians Some S are not P Some Germans are not Catholics [Some S are P] [Some Germans are Catholics] All tigers are meat eaters AOO (II) All P are M Some wild animals are not meat eaters Some S are not M Some wild animals are not tigers Some S are not P [All tigers are wild animals] [All P are S] S
S
S
S
M
M
M
P
Diagram
M
P
P
S
M
P
P
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14 February 2004 All tigers are wild animals All tigers are meat eaters Some meat eaters are wild animals [Some wild animals are not meat eaters] [Some meat eaters are not wild animals] All humans are primates All humans are mammals Some mammals are primates [All primates are mammals]
All humans are primates Some mammals are primates Some mammals are human [All primates are mammals] [All humans are mammals]
Table 38. Valid syllogisms sorted by mood and figure
All M are P All M are S Some S are P [Some P are not S] [Some S are not P] All M are P All M are S Some S are P [All P are S]
AAI (III)
AAI (III)
All P are M Some S are not M Some S are not P [All M are S] [All P are S]
AOO (II)
Class of valid syllogisms (Sheet 4 of 6) Mood (Figure) Form Example AOO (II) All P are M All elephants are wild animals Some S are not M Some meat eaters are not wild animals Some S are not P Some meat eaters are not elephants [All S are not P] [All elephants are not meat eaters]
S
S
M
M
P
Diagram
P
M
S
P
S
P
M
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317
318 Table 38. Valid syllogisms sorted by mood and figure
Class of valid syllogisms (Sheet 5 of 6) Mood (Figure) Form Example EAO (III) All M are not P All elephants are not meat eaters All M are S All elephants are wild animals Some S are not P Some wild animals are not meat eaters [Some S are P] [Some wild animals are meat eaters] EAO (III) All M are not P All men are not women All M are S All men are human Some S are not P Some humans are not women [All P are S] [All women are human] All primates are not bears EAO (III) All M are not P All primates are humans All M are S Some humans are not bears Some S are not P [All humans are not bears] [All S are not P] OAO (III) Some M are not P Some Germans are not Christians All M are S All Catholics are Christians Some S are not P Some Christians are not Catholics [Some P are not S] [Some Christians are Catholics] Some meditators are not mathematicians OAO (III) Some M are not P All meditators are human All M are S Some humans are not mathematicians Some S are not P [All mathematicians are humans] [Some P are not M] S
M
M
S
S
M
M
S
M
P
S
P
Diagram
P
P
P
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14 February 2004 Table 38. Valid syllogisms sorted by mood and figure
Class of valid syllogisms (Sheet 6 of 6) Mood (Figure) Form Example OAO (III) Some M are not P Some primates are not humans All M are S All primates are mammals Some S are not P Some mammals are not humans [All P are M] [All humans are primates] [All P are S] [All humans are mammals] P
Diagram
M
S
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319
320
P
M
M
M
S
P
S
P
S
All M are P Some S are M Some S are P [All M are S] [All P are S] All P are M Some S are not M Some S are not P [All M are S] [All P are S]
All humans are primates Some mammals are humans Some humans are primates [All humans are mammals] [All primates are mammals] All humans are primates Some mammals are primates Some mammals are human [All primates are mammals] [All humans are mammals]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
All M are P Some S are M Some S are P [All P are S]
Form All M are P All S are M All S are P
All humans are mammals Some mammals are primates Some primates are humans [All humans are primates]
Class of valid syllogisms (Sheet 1 of 5) Diagram Example All primates are mammals P All humans are primates M All humans are mammals S
AOO (II)
AII (I)
AII (I)
Mood (Figure) AAA (I)
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S
M
M
S
S
M
P
M
S
P
P
P
All M are not P All S are M All S are not P All M are not P Some S are M Some S are not P [All S are not P] All M are not P All M are S Some S are not P [All S are not P]
All primates are not bears All humans are primates All humans are not bears All humans are not bears Some primates are humans Some primates are not bears [All primates are not bears] All primates are not bears All primates are humans Some humans are not bears [All humans are not bears]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
OAO (III)
Some M are not P All M are S Some S are not P [All P are M] [All P are S]
Some primates are not humans All primates are mammals Some mammals are not humans [All humans are primates] [All humans are mammals]
EAO (III)
EIO (I)
EAE (I)
Mood (Figure) AAI (III)
Form All M are P All M are S Some S are P [All P are S]
Class of valid syllogisms (Sheet 2 of 5) Diagram Example All humans are primates S All humans are mammals P Some mammals are primates M [All primates are mammals]
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321
322 Form All M are P Some S are M Some S are P [Some S are not P] All P are M Some S are not M Some S are not P [Some S are P] Some M are not P All M are S Some S are not P [Some P are not S] All M are P Some S are M Some S are P [All S are P] Some M are not P All M are S Some S are not P [Some P are not M]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
Class of valid syllogisms (Sheet 3 of 5) Diagram Example All Catholics are Christians Some Germans are Catholics S M P Some Germans are Christians [Some Germans are not Christians] All Catholics are Christians Some Germans are not Christians S P M Some Germans are not Catholics [Some Germans are Catholics] Some Germans are not Christians All Catholics are Christians S M P Some Christians are not Catholics [Some Christians are Catholics] All mathematicians are human P Some meditators are mathematicians S M Some meditators are human [All meditators are human] Some meditators are not mathematicians S All meditators are human M P Some humans are not mathematicians [All mathematicians are humans] OAO (III)
AII (I)
OAO (III)
AOO (II)
Mood (Figure) AII (I)
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14 February 2004 Form All M are not P Some S are M Some S are not P [Some S are P] All M are not P Some S are M Some S are not P [All P are S] All M are not P Some S are M Some S are not P [All M are S] All P are M Some S are not M Some S are not P [All S are not P] All M are not P All M are S Some S are not P [Some S are P]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
Class of valid syllogisms (Sheet 4 of 5) Diagram Example All men are not women Some mathematicians are men M S P Some mathematicians are not women [Some mathematicians are women] All meat eaters are not elephants S Some wild animals are meat eaters M P Some wild animals are not elephants [All elephants are wild animals] All elephants are not meat eaters S Some wild animals are elephants M P Some wild animals are not meat eaters [All elephants are wild animals] All elephants are wild animals M Some meat eaters are not wild animals S P Some meat eaters are not elephants [All elephants are not meat eaters] All elephants are not meat eaters S All elephants are wild animals M P Some wild animals are not meat eaters [Some wild animals are meat eaters] EAO (III)
AOO (II
EIO (I)
EIO (I)
Mood (Figure) EIO (I)
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323
324 Form All M are not P Some S are M Some S are not P [All S are not P] All P are M Some S are not M Some S are not P [All P are S] All M are P All M are S Some S are P [Some P are not S] [Some S are not P] All M are not P All M are S Some S are not P [All P are S]
Table 39. Valid syllogisms sorted by type of Euler-Venn diagram
Class of valid syllogisms (Sheet 5 of 5) Diagram Example All bears are not meditators Some mathematicians are meditators M S P Some mathematicians are not bears [All mathematicians are not bears] All tigers are meat eaters Some wild animals are not meat eaters S P M Some wild animals are not tigers [All tigers are wild animals] All tigers are wild animals All tigers are meat eaters S M P Some meat eaters are wild animals [Some wild animals are not meat eaters] [Some meat eaters are not wild animals] All men are not women S All men are human M P Some humans are not women [All women are human] EAO (III)
AAI (III)
AOO (II)
Mood (Figure) EIO (I)
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