Aristotelian Syllogisms

Euler diagrams

Aristotle 384-322 BC

INF1Afrom

Aristotle to Euler

Google ngram

inf1a-cl September 2020

Michael Fourman

Euler 1707-1783

a b c a b c

Socrates is a man All men are mortal ∴ Socrates is mortal

A deduction is speech (logos) in which, certain things having been supposed, something, different from those supposed, results, of necessity, because of their being so.

AristotleAristotle

384-322 BC

INF1Adeduction

a b c a b c

Socrates is a man All men are mortal ∴ Socrates is mortal

INF1Asyllogism

{x | x = Socrates} ✓ {x | isaMan x}{x | isaMan x} ✓ {x | isaMortal x}

))) {x | x = Socrates} ✓ {x | isaMortal x}

isSocrates x =def x = Socrates

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

isSocrates ✏ isaMan isaMan ✏ isaMortalisSocrates ✏ isaMortal

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

a ✏ b b ✏ ca ✏ c

<latexit sha1_base64

="ZEAlIupjGXdIB4yucVUk2j7V5Z0=">AAAEOXic

bVPJbhNBEO3ELGHYEjhyaREFcYgsGyUKHCxFgFAO

HIIgi5Sxop6eGrvlXobuGi8azafwJRy5wgdw5IZ

y5QeosZ0oTtLSSKX3qqte15tKcq0Ctlq/l5Ybt27

fubtyL7r/4OGjx6trTw6DK7yEA+m088eJCKCVhQ

NUqOE49yBMouEoGbyr+aMh+KCc/YKTHLpG9KzKlB

RI0OnqTqxsBr4U8fC9CH0uq4sw4S/ou8DjOIrzvr

DoTCmq09X1VrM1Pfx60J4H62x+9k/Xls/i1MnCg

EWpRQgn7VaO3VJ4VFJDFcVFgFzIgejBCYVWGAjdc

vrCim8QkvLMefos8il6+UYpTAgTk1CmEdgPV7kav

Ik7KTB73S2VzQsEK2eNskJzdLweF0+VB4l6QoGQ

XpFWLvvCC4k01IUu45nUBSz3zmWLUEAj/MSnV9CB

GBBiYSSdMcKmZZy7EbUArMrYCxUgceOy2d6Oz90

OA5UTp4XvQQw2FB7q55XxKK8qqoUwxpFKsd/Zkma

hWZF/LRzCZp1B7fJFKfVPp2wvEKoDkiKwQ+WdrY0

jO1KyqqwZklZWGxHnsQy1WZyw3c+AH5QP+FEkoE

lVOuPqK5ezxBBuTjsvTGW1sL2CBHX2RBiA1gRt8r

n0DvoCpsAAIA8kHcJlbDJyPg040dCJp7aUiS6ofM

3S/JScc4iZMEpPpgQlFsaGDtmvMw1jRTs0JULfj

QL6eiizTpnQYUaNNWS1oT1lO63mNpioiqKI03a0r

+7C9eDwVbO91XzzaWt99+18T1bYM/acvWRttsN22

R7bZwdMsm/sB/vJfjW+N/40/jbOZqnLS/M7T9nC

afz7DyJTbcs=</latexit>

Euler diagram

a b c

Aristotelian Syllogisms

Venn diagrams

Aristotle 384-322 BC

John Venn 1834–1923

INF1AAristotle via Venn

Google ngram

inf1a-cl September 2020

Michael Fourman

Venn 1834—1923

every red triangle is small

every small triangle is red

some big triangle is amber

some small disc is red

no red thing is blue

✔

❌

❓

❓

❓

redtriangle

small

Nothing here

INF1A-CLuniversal

affirmative

INF1Afrom Venn to Euler

Euler 1707-1783

John Venn 1834–1923

¿?¿?

INF1Afrom Venn to Euler

Euler 1707-1783

a b c a b c

John Venn 1834–1923 every a is b no a is c

a ✏ b

<latexit s

ha1_base64="AzzFr1Z/

xYhoyotfOknWSTRMNqM

=">AAAJU3icbVZRb9xEE

HZLgeSgtIVHXlzSINRc

rndR2ooqriIoqKAKhdK0

lbIhrNfj8+p2vc7uXi6

nrfsveIVfxQO/hRdmbSc

5n29POo+/+XZmPDszdl

wIbuxw+O+16x/c+PCjj9

fWe598evOzW7fvfP7aq

KlmcMiUUPptTA0InsOh5

VbA20IDlbGAN/Hke69/

cwbacJW/svMCjiUd5zzl

jFqEjujX5OwZNVkYhye3

N4aDYbXCrjBqhI2gWQc

nd9Z2SKLYVEJumaDGHKH

nQvPc9unUZkrPgepjB8

JQbTkTUPbI1EBB2YSOwV

FpJLVZB7SZ7GBmLuMOm

Krcmjaa4RNqDekKNKe2j

XrncaxEGW6GqdJhwsfc

mhYljmUZVqumUFFkNAbb

IhkrqZ7rpGtdm3QpvkI

rtSq4TsiJ4m0A04fHuEw

TdK6mPgmL6NEcl5RJ0s

8yibk7dgm1YLn0B6Ahhx

lTUtI8ccQr/D5NrdKlG

7QNOV9/PB/7hxDGCkUTQ

fPxFFXGPadmAkLUKrQJ

+RnXKvfF4JhK0JfzGgPW

lZu9MCTM5FRCiNj+b2B

/5NrYF5hMQSBPap3fssi

iZ7CadmEYzV4EFGV1PH

3EwDBaAMuojsimB7BBOD

N2LiAiWHRCEGtTKrmYe

20q4Jxjv2AzTWVuopQKA

802mPHEZpEbDh6C7A8H

u3gpvU5wC5g1iNxW6dy9

rXtlOQrdAy8/qOX7Xr5

fy5GXIy/jVvfU3z2tNXt

e3qtlQvwNEdi9Cb0gE1

LBxD/OHyg310a7Xdkimo

8zS7VWM9y2gx4qh2QMp/

7eE/e2a9uQXvE8vlWHT

+6Su1uX3KjhntbWotrJy

7aTKrhwUKkY16zmIkLC

FlY9sbeAf1VsVXD+v9a9

q9iSJ1US2qvnCzKfyhj

0jNuM5w5Op9XkKp0BVgl

IgJnNQGmQzg8P96q+aW

sEAkeoPnYvAKu/ViaQ8p

x7Mw5FNPrsEugQtJyU7

iVgs086OhiX7odzKotqy

KGuanRXpKU7qFu+19sM

f1G2it1UlMsmxI6hXJjS

kWpogHALqS7LNhlHbMr

PG26MCfEIY1Mt5nC6gsy

Qs8wuO3Pg/aLzQsbunS

8I/Bkuyy49WrAYddX4Bm

gIqnDEwjm+l9xP+QpDT

xb8Pumqf0d1Rq17v/xge

wsB7C0rsaIxp2aKp46U

K2Jd5BgFTnOmcnwDtvdp

Xh6NjvFKuYFYnbvt4eD

Ro8dYN25jtOyEiZrsjae

aTtzPq1gp/i+kYgbJGJ

rMNvLyDsvTdPG8+lX+nE

dJv0O+Mr9I9ugKsp/wH

S6CXWqiZr5o/KUqQh/wE

gVOsfz9XLosmzA6qQ/b

t1JZ4hQoffd6pB6iD5ns

+a+M0fI3RVd4vTMY7Q6+

/XV3Y/+75ntjLfgy+Cr

4JhgFj4P94HlwEBwGLFD

Bn8Ffwd9r/6z9t359/U

ZNvX6t2fNF0FrrN/8HZs

hb4A==</latexit>

a ✏ ¬c

<latexit sha1_base64

="5QX776EKt8dolNhTmidJx5nljJc=">AAAJWXi

cbVZtb9s2EFa6t8R7abp+3Bd1aYahcVw7SFusiI

pg64ZuKIaua9oCYZZR1MkiTIoOScUxWPWP7Ov2o

4b9mR0lJbEs04B1eu7h3el4d1I8FdzY4fDftRsff

PjRx5+sb/Q+/ezzL25u3vrytVGFZnDElFD6bUwN

CJ7DkeVWwNupBipjAW/iyQ9e/+YctOEqf2XnUzi

RdJzzlDNqETrdvEm/IedPqclCksM4ZKebW8PBsF

phVxg1wlbQrBent9b3SKJYISG3TFBjjtH/VPPc9

mlhM6XnQPWJA2GotpwJKHukMDClbELH4Kg0ktqs

A9pMdjAzl3EHTFVuTRvN8Dm1hnQFmlPbRr3zOFa

iDLfDVOkw4WNuTYsSx7IMq1VTqJhmNAbbIhkrqZ7

rpGtdm3QpvqlWalVwnZATxdsApg8Pc5km6FwVPg

mL6PEcl5RJ0s8yibk7cQm1YLn0B6AhhxlTUtI8c

cQr/D5NrdKlG7QNOV+FPB/7hxDGCkUTQfNxgSrj

nlEzASFqFdqE/JxrlfticEwl6Mt5jQHryu1eGBJ

mciohROzwd7A/cW3sc0ymIJAntc5vWWTRc1hNuz

SMZi8DirI6nj5iYBidAsuojsi2B7BNODN2LiAiW

HRCEGtTKrmYe20q4IJj12BLFTI3UUqFgWYbzHhis

8gNBw9A9oeDfbyUXie4BcwaRG6ndO7uzt2yHIXu

vpfv1/I9L9+r5cjLkZdxq3vi757UmgMvH9QyIf6

GCOzhhF6SCalg4h/nT5Sba6PdrWwRzceZpVqrGW

7bQw+VQzKGM3/viQe7tW1Ir3ke36nDJ3fInZ0rb

tRwz2prUe3kZdtJFVw4qFSMa1ZzESFhC6ue2FvA

vyq2Kjj/X+veVWzJkyoJ7dXzBZkXMgY94zbjuYO

zoppfpTPAKgEJMLMZKA3S+eHhXtU3bY1A4BjVJ+

45YPXXygRSnnNvxqGIRp9eAR2ClpPSvQRs9klHB+

PS/XhB5bQacqirGt1N09K9qFu+19sOf1W2it1Ul

KsmxI6hXJjSkWpogHALqS7LNhlHbMovGm6MCfEI

Y4UWczhbQWbIWWaXnTnwftH5VMbunS8I/Bkuyy4

9WrAYddX4BmgIauqIhQt8O7mf8xWGHi/4fdxV/4

HqjFr3fvnBDhYCOFhWYkVjTk2Bp46Ua2Jd5BgFT

nOmcnwPtvdpXh6PTvBKuYFYXbjd4eDhw0dYN25r

tOyEiZrsjaeaTtwvq1gp/i+kYgbJGJrMNvLyDsvT

dPG8+lX+nEdJv0O+Nr9I9ugKsp/wHS6CXWqiZr5

o/KUqQh/wEgXOsPz9XLoqmzA6rQ/bt1JZ4hQoff

d6pB6iD5jshfiVMVr+pugKr/cGo/3Bd7/tbx1+3

3xvrAdfBV8H3waj4FFwGDwLXgRHAQuK4K/g7+Cf

9f821jbwK6mm3lhr9twOWmvj9v+F+FzR</latex

it>

INF1Afrom Euler

to Venna b c a b c

John Venn 1834–1923

¿?¿?

INF1Afrom Euler

to Venna b c a b c

John Venn 1834–1923

a ^ 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

a ^ ¬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

¬a ^ ¬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

¬a ^ 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

not ¬, and ^, or _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

INF1Acombinations

a Venn diagram represents all imaginable combinations

for two predicates there are four combinations

John Venn 1834–1923

000

0

0011

010

2

011

3

1004

101

5

110

6 111

7

a b c

INF1Acombinations

 rotationally symmetric Venn diagrams for n predicates exist if and only if n is a prime number.

000

0

0011

010

2

011

3

1004

101

5

110

6 111

7

bINF1A

Venn diagram

every a is b no a is ca ✏ b

<latexit sha1_base64="AzzFr1Z/xYhoyotfOk

nWSTRMNqM=">AAAJU3icbVZRb9xEEHZLgeSgtIVHXlzSINRcrndR2ooqriIoqKAKhdK0lbIhrNfj8+

p2vc7uXi6nrfsveIVfxQO/hRdmbSc5n29POo+/+XZmPDszdlwIbuxw+O+16x/c+PCjj9fWe598evOz

W7fvfP7aqKlmcMiUUPptTA0InsOh5VbA20IDlbGAN/Hke69/cwbacJW/svMCjiUd5zzljFqEjujX5O

wZNVkYhye3N4aDYbXCrjBqhI2gWQcnd9Z2SKLYVEJumaDGHKHnQvPc9unUZkrPgepjB8JQbTkTUPbI

1EBB2YSOwVFpJLVZB7SZ7GBmLuMOmKrcmjaa4RNqDekKNKe2jXrncaxEGW6GqdJhwsfcmhYljmUZVq

umUFFkNAbbIhkrqZ7rpGtdm3QpvkIrtSq4TsiJ4m0A04fHuEwTdK6mPgmL6NEcl5RJ0s8yibk7dgm1

YLn0B6AhhxlTUtI8ccQr/D5NrdKlG7QNOV9/PB/7hxDGCkUTQfPxFFXGPadmAkLUKrQJ+RnXKvfF4J

hK0JfzGgPWlZu9MCTM5FRCiNj+b2B/5NrYF5hMQSBPap3fssiiZ7CadmEYzV4EFGV1PH3EwDBaAMuo

jsimB7BBODN2LiAiWHRCEGtTKrmYe20q4Jxjv2AzTWVuopQKA802mPHEZpEbDh6C7A8Hu3gpvU5wC5

g1iNxW6dy9rXtlOQrdAy8/qOX7Xr5fy5GXIy/jVvfU3z2tNXte3qtlQvwNEdi9Cb0gE1LBxD/OHyg3

10a7Xdkimo8zS7VWM9y2gx4qh2QMp/7eE/e2a9uQXvE8vlWHT+6Su1uX3KjhntbWotrJy7aTKrhwUK

kY16zmIkLCFlY9sbeAf1VsVXD+v9a9q9iSJ1US2qvnCzKfyhj0jNuM5w5Op9XkKp0BVglIgJnNQGmQ

zg8P96q+aWsEAkeoPnYvAKu/ViaQ8px7Mw5FNPrsEugQtJyU7iVgs086OhiX7odzKotqyKGuanRXpK

U7qFu+19sMf1G2it1UlMsmxI6hXJjSkWpogHALqS7LNhlHbMrPG26MCfEIY1Mt5nC6gsyQs8wuO3Pg/

aLzQsbunS8I/Bkuyy49WrAYddX4BmgIqnDEwjm+l9xP+QpDTxb8Pumqf0d1Rq17v/xgewsB7C0rsaI

xp2aKp46UK2Jd5BgFTnOmcnwDtvdpXh6NjvFKuYFYnbvt4eDRo8dYN25jtOyEiZrsjaeaTtzPq1gp/

i+kYgbJGJrMNvLyDsvTdPG8+lX+nEdJv0O+Mr9I9ugKsp/wHS6CXWqiZr5o/KUqQh/wEgVOsfz9XLo

smzA6qQ/bt1JZ4hQoffd6pB6iD5ns+a+M0fI3RVd4vTMY7Q6+/XV3Y/+75ntjLfgy+Cr4JhgFj4P94

HlwEBwGLFDBn8Ffwd9r/6z9t359/UZNvX6t2fNF0FrrN/8HZshb4A==</latexit>

a b c

a c

a ✏ ¬c

<latexit sha1_base64="5QX776EKt8dolNhTmi

dJx5nljJc=">AAAJWXicbVZtb9s2EFa6t8R7abp+3Bd1aYahcVw7SFusiIpg64ZuKIaua9oCYZZR1M

kiTIoOScUxWPWP7Ov2o4b9mR0lJbEs04B1eu7h3el4d1I8FdzY4fDftRsffPjRx5+sb/Q+/ezzL25u

3vrytVGFZnDElFD6bUwNCJ7DkeVWwNupBipjAW/iyQ9e/+YctOEqf2XnUziRdJzzlDNqETrdvEm/Ie

dPqclCksM4ZKebW8PBsFphVxg1wlbQrBent9b3SKJYISG3TFBjjtH/VPPc9mlhM6XnQPWJA2GotpwJK

HukMDClbELH4Kg0ktqsA9pMdjAzl3EHTFVuTRvN8Dm1hnQFmlPbRr3zOFaiDLfDVOkw4WNuTYsSx7I

Mq1VTqJhmNAbbIhkrqZ7rpGtdm3QpvqlWalVwnZATxdsApg8Pc5km6FwVPgmL6PEcl5RJ0s8yibk7c

Qm1YLn0B6AhhxlTUtI8ccQr/D5NrdKlG7QNOV+FPB/7hxDGCkUTQfNxgSrjnlEzASFqFdqE/Jxrlfti

cEwl6Mt5jQHryu1eGBJmciohROzwd7A/cW3sc0ymIJAntc5vWWTRc1hNuzSMZi8DirI6nj5iYBidAs

uojsi2B7BNODN2LiAiWHRCEGtTKrmYe20q4IJj12BLFTI3UUqFgWYbzHhis8gNBw9A9oeDfbyUXie4

BcwaRG6ndO7uzt2yHIXuvpfv1/I9L9+r5cjLkZdxq3vi757UmgMvH9QyIf6GCOzhhF6SCalg4h/nT5

Sba6PdrWwRzceZpVqrGW7bQw+VQzKGM3/viQe7tW1Ir3ke36nDJ3fInZ0rbtRwz2prUe3kZdtJFVw4q

FSMa1ZzESFhC6ue2FvAvyq2Kjj/X+veVWzJkyoJ7dXzBZkXMgY94zbjuYOzoppfpTPAKgEJMLMZKA3

S+eHhXtU3bY1A4BjVJ+45YPXXygRSnnNvxqGIRp9eAR2ClpPSvQRs9klHB+PS/XhB5bQacqirGt1N0

9K9qFu+19sOf1W2it1UlKsmxI6hXJjSkWpogHALqS7LNhlHbMovGm6MCfEIY4UWczhbQWbIWWaXnTn

wftH5VMbunS8I/Bkuyy49WrAYddX4BmgIauqIhQt8O7mf8xWGHi/4fdxV/4HqjFr3fvnBDhYCOFhWYk

VjTk2Bp46Ua2Jd5BgFTnOmcnwPtvdpXh6PTvBKuYFYXbjd4eDhw0dYN25rtOyEiZrsjaeaTtwvq1gp

/i+kYgbJGJrMNvLyDsvTdPG8+lX+nEdJv0O+Nr9I9ugKsp/wHS6CXWqiZr5o/KUqQh/wEgXOsPz9XL

oqmzA6rQ/bt1JZ4hQoffd6pB6iD5jshfiVMVr+pugKr/cGo/3Bd7/tbx1+33xvrAdfBV8H3waj4FFwG

DwLXgRHAQuK4K/g7+Cf9f821jbwK6mm3lhr9twOWmvj9v+F+FzR</latexit>