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| Lausanne: hotelroomsearchdotnet: Forward, January 2018 |
LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York.
The Langer philosophy text review, continues.
Some key symbols from the textbook:
≡df = Equivalence by definition
: = Equal (s)
ε = Epsilon and means is
⊃ = Is the same as
⊨ is Entails
˜ = Not
∃ = There exists
∃! = There exists
∴ = Therefore
· = Therefore
< = Is included
v = a logical inclusive disjunction (disjunction is the relationship between two distinct alternatives).
x = variable
· = Conjunction meaning And
To combine various classes, that are not the same, through the use disjunction or conjunction, positive statements are required to connect membership. (145-146). Langer's example:
C = Cats
D = Dogs
(x) : x ε C ⊃ (x ε -D) (146).
Variable equals variable is Cats, is the same as variable is not Dogs.
Also
C < -D
Cats is included in not Dogs.
D < -C
Dogs is included in not Cats.
Langer explains that this principle of dichotomy provides liberty to logical expression. (146). Negative statements can be turned into positive statements, and positive statements can be turned into negative ones. This approach assumes that if something is not A or B it is -A x -B. (146).
Everything that is not in the class of Cats or Dogs, is not Cats or Dogs.
H = Horses
H ` C < H ` D
Horses are not cats is included with horses are not dogs.
(x) : (x ε H) ⊃ (x ε -C) ⊃ (x ε -D)
Variable equals variable is Horses, means variable is not the same as Cats, means variable is not the same as Dogs.
Horses are not the same as cats and horses are not the same as dogs.
(x) : (x ε H) ⊨ (x ε -C) ⊨ (x ε -D)
Variable equals variable is Horses, entails variable is not Cats, entails variable is not Dogs.
Horses are not the same as cats and horses are not the same as dogs.
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