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| Germany: Colourized |
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
0= Null class
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Null class entails the class of cats and the class of not-cats. The null class 0 is included in each complementary pair (147). Every class has a complement and therefore every class includes null class 0. (147).
Langer continues:
As every class in the universe of discourse has a complement, the sub-classes of any class, also has a complement. (147). If two classes overlap we have like, the following example from the author:
Class A: Soldiers
Class B: Brave men (Note this was written in 1953 and revised in 1967)
Class C: Brave soldiers (148).
Class C: Brave soldiers is a sub-class of Class A and Class B. (148).
C = A x B (148). (Note it is A x B, not A + B; these groups overlap, they are not combined. Not all soldiers are brave, and some brave men are not soldiers and may be police officers, for example.)
The negative complement to Class C, is Class -C. This class is also known as -(A x B). (148). Langer states this everything that is not both a soldier and brave. (148).
This is Class A which is not Class B.
This is Class B which is not Class A.
This is everything that is neither Class A or Class B. (148).
Langer reasons that the sum (now +) of Class A and Class B is Class D.
This is all soldiers, brave or not, and all brave men, soldiers or not. (148).
D= A+B (148). (Not x/times. Here they are combined).
The complement of D is -D, which is -(A + B).
This is the class that is neither soldiers or brave men. (149).
Is this technical and somewhat tricky presentation from Langer, I can again philosophically appreciate how the idea of Class (Which Pirie also mentions in his philosophy text) allows one to reason and avoid contradiction and the illogical.
I also appreciate that in philosophy there is both positive and negative class and premises. Christian theology both emphasizes the positive and the negative counterpart.
Once again, this demonstrates within a theistic model, that all truth is God's truth.
C x T = Y
Christians x theologians equals Christian theologians.
C + T = Z
Christians + theologians equals Christians and all theologians, Christian or not.
Christian theologians is not the same as Christians and theologians in reasoning.
Y ˜ ⊃ Z
