modelling the syllogismimed/courses/seminar/modellingsyllogism.pdf · modelling the syllogism ......

Draft 14 February 2004 303

Second Interlude

Modelling the Syllogism

Second InterludeModelling the Syllogism

People do not understand how thatwhich 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

n this interlude, I show how relational logic can be used to model syllogisticreasoning, originally developed by Aristotle some

2

,

300

years ago. Aristotlewas concerned with various properties of things and with what conclusions

can be made about these properties from assumed premises. These propertiescan quite easily be represented in relational logic.

However, I am not here primarily concerned with the nature of the propertiesof the entities under discussion. I wish to model the underlying structure ofAristotle’s thought processes. This can best be done by studying the concepts ofthe syllogism and their relationships to each other.

Definitions

A syllogism consists of three propositions (also called statements orsentences), called the major premise, minor premise, and conclusion,respectively. Each proposition has two terms called the subject and predicate.

I

Second Interlude


304 Draft 14 February 2004

The terms in a proposition are related to each other in four different ways,shown in Table

34

.

Each of these propositions has a number of dualistic attributes thatcharacterize the propositions. They are grouped together in pairs depending onwhether 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 theproposition. So we could call them defining attributes, with the third beingderived from the other two.

Aristotle called the symmetrical propositions convertible because they areequivalent when the terms are interchanged. A and E are also convertible intoweaker forms, I and O, respectively. Furthermore, if we assume Aristototle’s rulesof 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 theproposition, rather than the propositions themselves. A term is distributed if, in

Class

Syllogistic propositions

Attribute names

Name Form Diagram

Attribute values

A All S are P

E All S are not P

I Some S are P

or

O Some S are not P

or

Table 34.

Types of propositions in syllogisms

Class


Attribute names

Name Universality Positivity Symmetry

Attribute values

A universal positive asymmetrical

E universal negative symmetrical

I particular positive symmetrical

O particular negative asymmetrical

Table 35.

Attributes of propositions

S

P

PS

S P

S

P

S P

S

P

Second Interlude


Draft 14 February 2004

305

some sense, it refers to all entities with the particular property (called a class),otherwise it is undistributed. The subject of universal propositions and thepredicate of negative propositions are distributed. How these definitions relateto syllogistic propositions is given in Table

36

, attributes that can be seen quiteclearly from the diagrams in Table

34

.

The terms of the three propositions of the syllogism are related to each otherin two ways:

1.

One term is common to the major and minor premises; it is called themiddle term (M).

2.

The predicate (P) of the conclusion is the major term of the syllogism andthe subject (S) is the minor term, because they are the nonmiddle terms inthe major and minor premises, respectively.

As each proposition has one of four types and as there are three propositionsin each syllogism, there are

4

3

=

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 whetherthe middle term is the subject or predicate in the major and minor premises.

Class


Attribute names

Name Subject Predicate

Attribute values

A distributed undistributed

E distributed distributed

I undistributed undistributed

O undistributed distributed

Table 36.

Distribution of terms in syllogism

Second Interlude



(Curiously, for some reason, Aristotle only recognized three of these figures; thefourth was discovered only in the Middle Ages.)

There are thus

64

x

4

=

256

possible syllogisms in total.

Aristotle examined each mood and figure in turn to determine whether it wasvalid or not. He then derived a number of common properties of thesesyllogisms, 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 inthe major premise.

(c)

A subject distributed in the conclusion must be distributed in theminor premise.

Class

Syllogistic figures

Attribute names

Name Figure

Attribute values

I

II

III

IV

Table 37.

Syllogistic figures

M PS MS P

P MS MS P

M PM SS P

P MM SS P

Second Interlude



307

Application of rules

We can now apply these rules of inference to determine the validity of eachmood in each figure.

As the first rule applies irrespective of conclusion or figure, we need considerjust 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, indicatingwhich has an attribute value invalid by one or both of the two rules. However, itis simpler to indicate invalidity by type style, bold and italics indicatinginvalidity by rules

1

(a) and

1

(b), respectively. The nine potentially valid premisesare marked in plain style:

Rule

2

applies to the moods of syllogisms. As there are four moods and ninepotentially valid pairs of premises from rule

1

, we need to consider

36

moods.By rule

2

(a), these moods are invalid:xIA IxA xOA OxAxIE IxE xOE OxE

By rule

2

(b), these pairs of premises are invalid:xEA ExA xOA OxAxEI ExI xOI OxI

Rule

2

therefore leaves

16

potentially valid moods of syllogism, the invalidones being marked in italics and bold, as for rule

1

:

The third rule is rather more complicated as it concerns the distribution ofeach of the three types of term in the syllogism for each mood and figure.

AA AE AI AO

EA

EE

EI

EO

IA IE

II IO

OA

OE

OI

OO

AAA AAE AAI AAO

AEA

AEE

AEI

AEO

AIA AIE

AII AIO

AOA

AOE

AOI

AOO

EAA

EAE

EAI

EAO

EIA

EIE

EII

EIO

IAA IAE

IAI IAO

IEA

IEE

IEI

IEO

OAA

OAE

OAI

OAO

Second Interlude



First considering rule

3

(a) about the distribution of the middle term. Theright-hand four columns in the following table show the distribution of themiddle term in the two premises. The ones that are invalid by this rule are theones that are undistibuted in both premises, indicated by italics in the table. Theother four columns show the sixteen potentially valid moods that we areconsidering after eliminating the others by applying rules

1

and

2

.

xxA xxE xxI xxO I II III IV

AAx AAA AAE AAI AAO D–U

U–U

D–D U–D

AEx AEE AEO D–D U–D D–D U–D

AIx AII AIO D–U

U–U

D–U

U–U

AOx AOO D–D U–D D–U

U–U

EAx EAE EAO D–U D–U D–D D–D

EEx

EIx EIO D–U D–U D–U D–U

EOx

IAx IAI IAO

U–U U–U

U–D U–D

IEx IEO U–D U–D U–D U–D

IIx

IOx

OAx OAO

U–U

D–U U–D D–D

OEx

OIx

OOx

Second Interlude



309

Rule

3

(b) concerns the distribution of the predicate. The ones that are invalidby this rule are the ones that are distributed in the conclusion, but notdistributed in the major premise, indicated by italics in the table.

Rule

3(c) is similar to 3(b) except that it applies to the subject in the minorpremise. The pairs of minor premise–conclusion that are invalid are againindicated by italics in the table.

xAx xEx xIx xOx I II III IV

AxA AAA U–U D–U U–U D–U

AxE AAE AEE U–D D–D U–D D–D

AxI AAI AII U–U D–U U–U D–U

AxO AAO AEO AIO AOO U–D D–D U–D D–D

ExA

ExE EAE D–D D–D D–D D–D

ExI

ExO EAO EIO D–D D–D D–D D–D

IxA

IxE

IxI IAI U–U U–U U–U U–U

IxO IAO IEO U–D U–D U–D U–D

OxA

OxE

OxI

OxO OAO D–D U–D D–D U–D



Eliminating the syllogisms, both mood and figure, made invalid by rule 3leaves just 12 moods and 25 syllogisms. One of them, AAO in figure IV, is inrather a curious position. Aristotle’s rules allow it. However, it is onlyconditionally valid. As this is a syllogism of figure IV, and as Aristotle did notdiscover 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 anactual subset of S. However, if P, M, and S are equivalent, there are no S that arenot P. So we need to eliminate it from the list for this reason, not covered byAristotle’s rules. It is only conditionally valid.

There are five syllogisms that have a universal conclusion. So there are alsofive corresponding syllogisms with a particular conclusion, which we can alsoeliminate, as they are weak forms. These are:

This leaves us with 20 valid syllogisms, found by Aristotle and his successors:

Axx Exx Ixx Oxx I II III IV

xAA AAA D–D D–D U–D U–D

xAE AAE EAE D–D D–D U–D U–D

xAI AAI IAI D–U D–U U–U U–U

xAO AAO EAO IAO OAO D–U D–U U–U U–U

xEA

xEE AEE D–D D–D D–D D–D

xEI

xEO AEO IEO D–U D–U D–U D–U

xIA

xIE

xII AII U–U U–U U–U U–U

xIO AIO EIO U–U U–U U–U U–U

xOA

xOE

xOI

xOO AOO U–U U–U D–U D–U

Strong Weak

AAA I AAI I

EAE I EAO I

AEE II AEO II

EAE II EAO II

AEE IV AEO IV



First figure AAA, EAE, AII, EIOSecond figure EAE, AEE, EIO, AOOThird figure AAI, IAI, AII, EAO, OAO, EIOFourth figure AAI, AEE, IAI, EAO, EIO

Students in the Middle Ages were expected to know all these by heart. Forinstance, the statutes of the University of Oxford in the fourteenth centuryincluded this rule: “Batchelors and Masters of Arts who do not follow Aristotle’sphilosophy are subject to a fine of 5s. for each point of divergence, as well as forinfractions of the rules of the Organum”.1

Not surprising therefore that they needed a mnemonic to remember thisrather 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 waseliminated by rule 2(a). Curiously, C. W. Kilmister, whose book Language, Logicand Mathematics this mnemonic is taken from, did not point out the error.

These syllogisms can be further reduced because propositions E and I aresymmetrical; the terms in these propositions can be interchanged. When one orboth premises is symmetrical, the mood stays unchanged, but the figure changes.When the conclusion is symmetrical, interchanging the subject and predicatemeans that the major and minor premises must change position, resulting in achange in both mood and figure.

This means that there are just eight core syllogisms out of the 256 candidatesthat we started with.



To sum up, the table on the next page shows all 256 syllogisms, which arevalid, which are equivalent to a valid syllogism, and which rule first eliminatedthem from the list.

However, all this is rather abstract and mechanical. It tells us little about themeanings of this inferences. So tables 38 on page 314 and 39 on page 320provide a list of examples of each of the eight valid syllogisms, together with acorresponding Euler-Venn diagram. Note that because I and O can berepresented in two ways in Euler-Venn diagrams, as illustrated in Table 34 onpage 304, there are sometimes four or five corresponding diagrams for oneparticular mood and figure.

Syllogism: Mood (figure) Equivalent to

AAA (I) AAI (IV) ≡ AAI (I) [weak form]

AII (I) AII (III), IAI (III), IAI (IV)

EAE (I) EAE (II), AEE (II), AEE (IV)

EIO (I) EIO (II), EIO (IIII), EIO (IV)

AOO (II) —

AAI (III) Itself

EAO (III) EAO (IV)

OAO (III) —



Class of syllogisms

Figure Figure

Mood I II III IV Mood I II III IV

AAA Valid

3(a)

3(c) 3(c) IAA 2(a)

AAE 3(b) 3(b,c) 3(c) IAE 2(a)

AAI Weak Valid Equiv IAI 3(a) 3(a) Equiv Equiv

AAO 3(b) 3(b) Cond IAO 3(a,b) 3(a,b) 3(b) 3(b)

AEA 2(b) IEA 2(a,b)

AEE 3(b) Equiv 3(b) Equiv IEE 2(a)

AEI 2(b) IEI 2(b)

AEO 3(b) Weak 3(b) Weak IEO 3(b) 3(b) 3(b) 3(b)

AIA 2(a) IIA

1(a)AIE 2(a) IIE

AII Valid3(a)

Equiv3(a)

III

AIO 3(b) 3(b) IIO

AOA 2(a,b) IOA

1(a)AOE 2(a) IOE

AOI 2(b) IOI

AOO 3(b) Valid 3(b) 3(a) IOO

EAA 2(b) OAA 2(a,b)

EAE Valid Equiv 3(c) 3(c) OAE 2(a)

EAI 2(b) OAI 2(b)

EAO Weak Weak Valid Equiv OAO 3(a) 3(b) Valid 3(b)

EEA

1(b)

OEA

1(b)EEE OEE

EEI OEI

EEO OEO

EIA 2(a,b) OIA

1(a)EIE 2(a) OIE

EII 2(b) OII

EIO Valid Equiv Equiv Equiv OIO

EOA

1(b)

OOA

1(a,b)EOE OOE

EOI OOI

EOO OOO


314 14 February 2004

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 1

of 6

)

Moo

d (F

igur

e)Fo

rmEx

ampl

eD

iagr

am

AA

A (

I)A

ll M

are

PA

ll S

are

MA

ll S

are

P

All

prim

ates

are

mam

mal

sA

ll hu

man

s ar

e pr

imat

esA

ll hu

man

s ar

e m

amm

als

EA

E (

I)A

ll M

are

not

PA

ll S

are

MA

ll S

are

not P

All

prim

ates

are

not

bea

rsA

ll hu

man

s ar

e pr

imat

esA

ll hu

man

s ar

e no

t bea

rs

AII

(I)

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[Som

e S

are

not P

]

All

Cat

holic

s ar

e C

hris

tian

sSo

me

Ger

man

s ar

e C

atho

lics

Som

e G

erm

ans

are

Chr

isti

ans

[Som

e G

erm

ans a

re n

ot C

hrist

ians

]

AII

(I)

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[All

S ar

e P]

All

mat

hem

atic

ians

are

hum

anSo

me

med

itat

ors

are

mat

hem

atic

ians

Som

e m

edit

ator

s ar

e hu

man

[All

med

itato

rs a

re h

uman

]

Tabl

e 38

. Val

id s

yllo

gism

s so

rted

by

moo

d an

d fig

ure

P

M

S

P

M

S

SM

P

P

SM


14 February 2004 315

AII

(I)

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[All

P ar

e S]

All

hum

ans

are

mam

mal

sSo

me

mam

mal

s ar

e pr

imat

esSo

me

prim

ates

are

hum

ans

[All

hum

ans a

re p

rim

ates

]

AII

(I)

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[All

M a

re S

][A

ll P

are

S]

All

hum

ans

are

prim

ates

Som

e m

amm

als

are

hum

ans

Som

e hu

man

s ar

e pr

imat

es[A

ll hu

man

s are

mam

mal

s][A

ll pr

imat

es a

re m

amm

als]

EIO

(I)

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[S

ome

S ar

e P]

All

men

are

not

wom

enSo

me

mat

hem

atic

ians

are

men

Som

e m

athe

mat

icia

ns a

re n

ot w

omen

[Som

e m

athe

mat

icia

ns a

re w

omen

]

EIO

(I)

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll P

are

S]

All

mea

t eat

ers

are

not e

leph

ants

Som

e w

ild a

nim

als

are

mea

t eat

ers

Som

e w

ild a

nim

als

are

not e

leph

ants

[All

elep

hant

s are

wild

ani

mal

s]

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 2

of 6

)

Moo

d (F

igur

e)Fo

rmEx

ampl

eD

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am

Tabl

e 38

. Val

id s

yllo

gism

s so

rted

by

moo

d an

d fig

ure

S

P

M

P

S

M

SM

P

SM

P



EIO

(I)

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll S

are

not P

]

All

bear

s ar

e no

t med

itat

ors

Som

e m

athe

mat

icia

ns a

re m

edit

ator

sSo

me

mat

hem

atic

ians

are

not

bea

rs[A

ll m

athe

mat

icia

ns a

re n

ot b

ears

]

EIO

(I)

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll M

are

S]

All

elep

hant

s ar

e no

t mea

t eat

ers

Som

e w

ild a

nim

als

are

elep

hant

sSo

me

wild

ani

mal

s ar

e no

t mea

t eat

ers

[All

elep

hant

s are

wild

ani

mal

s]

EIO

(I)

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll S

are

not P

]

All

hum

ans

are

not b

ears

Som

e pr

imat

es a

re h

uman

sSo

me

prim

ates

are

not

bea

rs[A

ll pr

imat

es a

re n

ot b

ears

]

AO

O (

II)

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[Som

e S

are

P]

All

Cat

holic

s ar

e C

hris

tian

sSo

me

Ger

man

s ar

e no

t Chr

isti

ans

Som

e G

erm

ans

are

not C

atho

lics

[Som

e G

erm

ans a

re C

atho

lics]

AO

O (

II)

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[All

P ar

e S]

All

tige

rs a

re m

eat e

ater

sSo

me

wild

ani

mal

s ar

e no

t mea

t eat

ers

Som

e w

ild a

nim

als

are

not t

iger

s[A

ll tig

ers a

re w

ild a

nim

als]

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 3

of 6

)

Moo

d (F

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ampl

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am

Tabl

e 38

. Val

id s

yllo

gism

s so

rted

by

moo

d an

d fig

ure

PS

M

S

MP

PM

S

SP

M

SM

P



AO

O (

II)

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[All

S ar

e no

t P]

All

elep

hant

s ar

e w

ild a

nim

als

Som

e m

eat e

ater

s ar

e no

t wild

ani

mal

sSo

me

mea

t eat

ers

are

not e

leph

ants

[All

elep

hant

s are

not

mea

t eat

ers]

AO

O (

II)

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[All

M a

re S

][A

ll P

are

S]

All

hum

ans

are

prim

ates

Som

e m

amm

als

are

prim

ates

Som

e m

amm

als

are

hum

an[A

ll pr

imat

es a

re m

amm

als]

[All

hum

ans a

re m

amm

als]

AA

I (I

II)

All

M a

re P

All

M a

re S

Som

e S

are

P[S

ome

P ar

e no

t S]

[Som

e S

are

not P

]

All

tige

rs a

re w

ild a

nim

als

All

tige

rs a

re m

eat e

ater

sSo

me

mea

t eat

ers

are

wild

ani

mal

s[S

ome

wild

ani

mal

s are

not

mea

t eat

ers]

[Som

e m

eat e

ater

s are

not

wild

ani

mal

s]

AA

I (I

II)

All

M a

re P

All

M a

re S

Som

e S

are

P[A

ll P

are

S]

All

hum

ans

are

prim

ates

All

hum

ans

are

mam

mal

sSo

me

mam

mal

s ar

e pr

imat

es[A

ll pr

imat

es a

re m

amm

als]

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 4

of 6

)

Moo

d (F

igur

e)Fo

rmEx

ampl

eD

iagr

am

Tabl

e 38

. Val

id s

yllo

gism

s so

rted

by

moo

d an

d fig

ure

S

M

P S

M

P

SP

M

S

P

M



EA

O (

III)

All

M a

re n

ot P

All

M a

re S

Som

e S

are

not P

[Som

e S

are

P]

All

elep

hant

s ar

e no

t mea

t eat

ers

All

elep

hant

s ar

e w

ild a

nim

als

Som

e w

ild a

nim

als

are

not m

eat e

ater

s[S

ome

wild

ani

mal

s are

mea

t eat

ers]

EA

O (

III)

All

M a

re n

ot P

All

M a

re S

Som

e S

are

not P

[All

P ar

e S]

All

men

are

not

wom

enA

ll m

en a

re h

uman

Som

e hu

man

s ar

e no

t wom

en[A

ll w

omen

are

hum

an]

EA

O (

III)

All

M a

re n

ot P

All

M a

re S

Som

e S

are

not P

[All

S ar

e no

t P]

All

prim

ates

are

not

bea

rsA

ll pr

imat

es a

re h

uman

sSo

me

hum

ans

are

not b

ears

[All

hum

ans a

re n

ot b

ears

]

OA

O (

III)

Som

e M

are

not

PA

ll M

are

SSo

me

S ar

e no

t P[S

ome

P ar

e no

t S]

Som

e G

erm

ans

are

not C

hris

tian

sA

ll C

atho

lics

are

Chr

isti

ans

Som

e C

hris

tian

s ar

e no

t Cat

holic

s[S

ome

Chr

istia

ns a

re C

atho

lics]

OA

O (

III)

Som

e M

are

not

PA

ll M

are

SSo

me

S ar

e no

t P[S

ome

P ar

e no

t M]

Som

e m

edit

ator

s ar

e no

t mat

hem

atic

ians

All

med

itat

ors

are

hum

anSo

me

hum

ans

are

not m

athe

mat

icia

ns[A

ll m

athe

mat

icia

ns a

re h

uman

s]

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 5

of 6

)

Moo

d (F

igur

e)Fo

rmEx

ampl

eD

iagr

am

Tabl

e 38

. Val

id s

yllo

gism

s so

rted

by

moo

d an

d fig

ure

S

MP

S

PM

M

S

P

SM

P

S

MP



OA

O (

III)

Som

e M

are

not

PA

ll M

are

SSo

me

S ar

e no

t P[A

ll P

are

M]

[All

P ar

e S]

Som

e pr

imat

es a

re n

ot h

uman

sA

ll pr

imat

es a

re m

amm

als

Som

e m

amm

als

are

not h

uman

s[A

ll hu

man

s are

pri

mat

es]

[All

hum

ans a

re m

amm

als]

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 6

of 6

)

Moo

d (F

igur

e)Fo

rmEx

ampl

eD

iagr

am

Tabl

e 38

. Val

id s

yllo

gism

s so

rted

by

moo

d an

d fig

ure

S

M

P



Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 1

of 5

)

Dia

gram

Exam

ple

Form

Moo

d (F

igur

e)

All

prim

ates

are

mam

mal

sA

ll hu

man

s ar

e pr

imat

esA

ll hu

man

s ar

e m

amm

als

All

M a

re P

All

S ar

e M

All

S ar

e P

AA

A (

I)

All

hum

ans

are

mam

mal

sSo

me

mam

mal

s ar

e pr

imat

esSo

me

prim

ates

are

hum

ans

[All

hum

ans a

re p

rim

ates

]

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[All

P ar

e S]

AII

(I)

All

hum

ans

are

prim

ates

Som

e m

amm

als

are

hum

ans

Som

e hu

man

s ar

e pr

imat

es[A

ll hu

man

s are

mam

mal

s][A

ll pr

imat

es a

re m

amm

als]

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[All

M a

re S

][A

ll P

are

S]

AII

(I)

All

hum

ans

are

prim

ates

Som

e m

amm

als

are

prim

ates

Som

e m

amm

als

are

hum

an[A

ll pr

imat

es a

re m

amm

als]

[All

hum

ans a

re m

amm

als]

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[All

M a

re S

][A

ll P

are

S]

AO

O (

II)

Tabl

e 39

. Val

id s

yllo

gism

s so

rted

by

type

of E

uler

-Ven

n di

agra

m

P

M

S

S

P

M

P

S

M

S

M

P



All

hum

ans

are

prim

ates

All

hum

ans

are

mam

mal

sSo

me

mam

mal

s ar

e pr

imat

es[A

ll pr

imat

es a

re m

amm

als]

All

M a

re P

All

M a

re S

Som

e S

are

P[A

ll P

are

S]

AA

I (I

II)

Som

e pr

imat

es a

re n

ot h

uman

sA

ll pr

imat

es a

re m

amm

als

Som

e m

amm

als

are

not h

uman

s[A

ll hu

man

s are

pri

mat

es]

[All

hum

ans a

re m

amm

als]

Som

e M

are

not

PA

ll M

are

SSo

me

S ar

e no

t P[A

ll P

are

M]

[All

P ar

e S]

OA

O (

III)

All

prim

ates

are

not

bea

rsA

ll hu

man

s ar

e pr

imat

esA

ll hu

man

s ar

e no

t bea

rs

All

M a

re n

ot P

All

S ar

e M

All

S ar

e no

t P

EA

E (

I)

All

hum

ans

are

not b

ears

Som

e pr

imat

es a

re h

uman

sSo

me

prim

ates

are

not

bea

rs[A

ll pr

imat

es a

re n

ot b

ears

]

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll S

are

not P

]

EIO

(I)

All

prim

ates

are

not

bea

rsA

ll pr

imat

es a

re h

uman

sSo

me

hum

ans

are

not b

ears

[All

hum

ans a

re n

ot b

ears

]

All

M a

re n

ot P

All

M a

re S

Som

e S

are

not P

[All

S ar

e no

t P]

EA

O (

III)

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 2

of 5

)

Dia

gram

Exam

ple

Form

Moo

d (F

igur

e)

Tabl

e 39

. Val

id s

yllo

gism

s so

rted

by

type

of E

uler

-Ven

n di

agra

m

S

P

M

S

M

P

P

M

S

PM

S

M

S

P



All

Cat

holic

s ar

e C

hris

tian

sSo

me

Ger

man

s ar

e C

atho

lics

Som

e G

erm

ans

are

Chr

isti

ans

[Som

e G

erm

ans a

re n

ot C

hrist

ians

]

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[Som

e S

are

not P

]

AII

(I)

All

Cat

holic

s ar

e C

hris

tian

sSo

me

Ger

man

s ar

e no

t Chr

isti

ans

Som

e G

erm

ans

are

not C

atho

lics

[Som

e G

erm

ans a

re C

atho

lics]

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[Som

e S

are

P]

AO

O (

II)

Som

e G

erm

ans

are

not C

hris

tian

sA

ll C

atho

lics

are

Chr

isti

ans

Som

e C

hris

tian

s ar

e no

t Cat

holic

s[S

ome

Chr

istia

ns a

re C

atho

lics]

Som

e M

are

not

PA

ll M

are

SSo

me

S ar

e no

t P[S

ome

P ar

e no

t S]

OA

O (

III)

All

mat

hem

atic

ians

are

hum

anSo

me

med

itat

ors

are

mat

hem

atic

ians

Som

e m

edit

ator

s ar

e hu

man

[All

med

itato

rs a

re h

uman

]

All

M a

re P

Som

e S

are

MSo

me

S ar

e P

[All

S ar

e P]

AII

(I)

Som

e m

edit

ator

s ar

e no

t mat

hem

atic

ians

All

med

itat

ors

are

hum

anSo

me

hum

ans

are

not m

athe

mat

icia

ns[A

ll m

athe

mat

icia

ns a

re h

uman

s]

Som

e M

are

not

PA

ll M

are

SSo

me

S ar

e no

t P[S

ome

P ar

e no

t M]

OA

O (

III)

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 3

of 5

)

Dia

gram

Exam

ple

Form

Moo

d (F

igur

e)

Tabl

e 39

. Val

id s

yllo

gism

s so

rted

by

type

of E

uler

-Ven

n di

agra

m

SM

P

SP

M

SM

P

P

SM

S

MP



All

men

are

not

wom

enSo

me

mat

hem

atic

ians

are

men

Som

e m

athe

mat

icia

ns a

re n

ot w

omen

[Som

e m

athe

mat

icia

ns a

re w

omen

]

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[S

ome

S ar

e P]

EIO

(I)

All

mea

t eat

ers

are

not e

leph

ants

Som

e w

ild a

nim

als

are

mea

t eat

ers

Som

e w

ild a

nim

als

are

not e

leph

ants

[All

elep

hant

s are

wild

ani

mal

s]

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll P

are

S]

EIO

(I)

All

elep

hant

s ar

e no

t mea

t eat

ers

Som

e w

ild a

nim

als

are

elep

hant

sSo

me

wild

ani

mal

s ar

e no

t mea

t eat

ers

[All

elep

hant

s are

wild

ani

mal

s]

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll M

are

S]

EIO

(I)

All

elep

hant

s ar

e w

ild a

nim

als

Som

e m

eat e

ater

s ar

e no

t wild

ani

mal

sSo

me

mea

t eat

ers

are

not e

leph

ants

[All

elep

hant

s are

not

mea

t eat

ers]

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[All

S ar

e no

t P]

AO

O (

II

All

elep

hant

s ar

e no

t mea

t eat

ers

All

elep

hant

s ar

e w

ild a

nim

als

Som

e w

ild a

nim

als

are

not m

eat e

ater

s[S

ome

wild

ani

mal

s are

mea

t eat

ers]

All

M a

re n

ot P

All

M a

re S

Som

e S

are

not P

[Som

e S

are

P]

EA

O (

III)

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 4

of 5

)

Dia

gram

Exam

ple

Form

Moo

d (F

igur

e)

Tabl

e 39

. Val

id s

yllo

gism

s so

rted

by

type

of E

uler

-Ven

n di

agra

m

SM

P

SM

P

S

MP

S

M

P

S

MP



All

bear

s ar

e no

t med

itat

ors

Som

e m

athe

mat

icia

ns a

re m

edit

ator

sSo

me

mat

hem

atic

ians

are

not

bea

rs[A

ll m

athe

mat

icia

ns a

re n

ot b

ears

]

All

M a

re n

ot P

Som

e S

are

MSo

me

S ar

e no

t P[A

ll S

are

not P

]

EIO

(I)

All

tige

rs a

re m

eat e

ater

sSo

me

wild

ani

mal

s ar

e no

t mea

t eat

ers

Som

e w

ild a

nim

als

are

not t

iger

s[A

ll tig

ers a

re w

ild a

nim

als]

All

P ar

e M

Som

e S

are

not M

Som

e S

are

not P

[All

P ar

e S]

AO

O (

II)

All

tige

rs a

re w

ild a

nim

als

All

tige

rs a

re m

eat e

ater

sSo

me

mea

t eat

ers

are

wild

ani

mal

s[S

ome

wild

ani

mal

s are

not

mea

t eat

ers]

[Som

e m

eat e

ater

s are

not

wild

ani

mal

s]

All

M a

re P

All

M a

re S

Som

e S

are

P[S

ome

P ar

e no

t S]

[Som

e S

are

not P

]

AA

I (I

II)

All

men

are

not

wom

enA

ll m

en a

re h

uman

Som

e hu

man

s ar

e no

t wom

en[A

ll w

omen

are

hum

an]

All

M a

re n

ot P

All

M a

re S

Som

e S

are

not P

[All

P ar

e S]

EA

O (

III)

Cla

ss o

f va

lid s

yllo

gism

s (

Shee

t 5

of 5

)

Dia

gram

Exam

ple

Form

Moo

d (F

igur

e)

Tabl

e 39

. Val

id s

yllo

gism

s so

rted

by

type

of E

uler

-Ven

n di

agra

m

PS

M

SM

P

SP

M

S

PM

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