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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
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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
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hb4A==</latexit>
a ✏ ¬c
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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 ^ ¬bAAAELHicbVNNb9NAEN02fBTz1QI3LiuqShyqKCmt4BKpAgn1wKEI+iHVUbVej51V1rtmd9zEWP4vXOHAr+GCEFd+B2MnlZq0K0Uavzc783ZeJsq18tjr/V5Z7dy6fefu2r3g/oOHjx6vbzw59rZwEo6k1dadRsKDVgaOUKGG09yByCINJ9H4XcOfXIDzyprPWOYwzERqVKKkQILO15+FBlIuwgnEKfD2IwrO1zd73V57+PWgPw822fwcnm90WBhbWWRgUGrh/Vm/l+OwEg6V1FAHYeEhF3IsUjij0IgM/LBq5dd8i5CYJ9bRzyBv0as3KpF5X2YRZWYCR36Za8AbuViJ1FHCUn9M3gwrZfICwchZ+6TQHC1vJsRj5UCiLikQ0il6AZcj4YREmuNC/ensAQtY7qxNFiGPmXCli5fQsRgTYmAibZYJE1dhbifUArCuQieUh8hOq25/L7w02I9VTpwWLoUQjC8cNI+uwkle11QLYYoTFeNosCuzhWZF/qWwCNtNBrXLF6U0/zNl0mZO2iMpAnOhnDWNnWRSTAZWDUPSqnor4DyUvrGQE7b/CfC9ch4/iAg0qYpnXHPlapa4gJvTLgtTWS1MWpCgwYHwY9CaoG0+lz5AV0ALjAFyT9LBX8XKiXWxx1LDIGxtqSJdUPmGpfkpOecQE5EpXbYEJRaZ8QOyXycaporWpiX8yE48umYos06J0H5GTTUkjaGpMoNedw+yoA6CgNPO9Jc35HpwvNPtv+rufNzd3H8735419py9YC9Zn71m++yAHbIjJtlX9o19Zz86Pzu/On86f2epqyvzO0/Zwun8+w9eX2d0
¬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
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a b c
a c
a ✏ ¬c
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