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A little categorical logic
11
Week 12: A little categorical logic
Venn diagrams
- restricted quantity of the relative basis sets
- easy to use for arguments that involve 2 or 3 sets
- 4 sets involve the usage of ellipses!
categorical syllogisms
second characteristic attribute of a valid deduction:
The informative or factual content of the conclusion is already completely, at least implicitly, made up by the premises.
logical basis for validity of diagrams
1. All logicians are mathematicians.
2. Some philosophers are not mathematicians.
:. Some philosophers are not logicians. :.
1. All spiders are octopods. S A M
2. No insects are octopods. P E M
:. No spider is an insect. :. S E P
logicians
x
philosophers
never put an "x" in a cross-hatched area because they are always empty sets!
mathematicians
display the first premise in a diagram:
"All spiders are octopods." = All S are O.
valid
conclusion
I
S
spiders
O
octopods
display the second premise in a diagram:
"No insects are octopods." = No I is an O.
step
I
S
insects
arguments which consist of 3 categorical statements with 3 different expressions:
O
octopods
Conclusion: :. No spider is an insect.
= No S is P.
spiders
I
S
insect
area where I set and S set
overlap is shaded (empty set)
step
O
octopods
syllogism is valid!
basic diagram
basic diagrams
DIAGRAMS
add additional information about the
INCLUSION, EXCLUSION or OVERLAPPING of sets
"All whales are mammals."
cross-hatching (or shading)
for expressing an area of a set with no elements
(categorical) syllogisms
standard diagram for E statements
S
I
arguments consisting of categorical statements
empty set
standard diagram for O statements
"Some animals are not predators."
standard diagram for I statements
A
"Some diamonds are expensive."
P
x
D
some
E
x
syllogism:
premise 1 (major premise)
premise 2 (minor premise)
conclusion
subject and predicate statements: statements about sets
A-Form
x represents at least one item of a category falls in that zone ('some')
generally affirmative
categorical statements:
expressions about the relation between sets of items
E-Form
generally negative
I-Form
O-Form
particularly affirmative
particularly negative
"Some diamonds are expensive."
Dr. N.E.J.A. Bowen
"No spider is an insect."
step
"Some animals are not predators."
standard diagram for A statements
W
M