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
Fundamentum Divisionis
Whenever there is a class formed within any universe of discourse, then every individual in that universe must either belong to the class, or not belong to it. (142). A class of two-storeyed houses has every house in it that is two-storeyed, or the house is not in that class. (142).
In the universe of creatures there is a sub-class of cats, then every creature is a cat or non-cat. (142).
With Aristotle's class of fundamentum divisionis, there is what is A and what is not A. (142). For every x, either x is an A or x is not an A. (142).
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| In this example, the variable, x (creatures). |
Therefore x ε A, or there are creatures that are cats is one defining form, there is also the defining form ˜(x ε A) that there are creatures that are not cats. (142). Everything in this universe of discourse belongs to A or ˜A. (142).
A creature is either a cat or is not. Therefore (x ε A) V ˜(x ε A). There are creatures that are cats, therefore it is true, there are creatures that are not cats.
The universe class of all creatures is defined by Langer as I, therefore, I = A + -A (142-143).
All creatures equals cats and non-cats.
I = H + -H
All creatures equals human beings and non-human beings.
(x ε H) V ˜(x ε H)
There are creatures that are human beings, therefore it is true, there are creatures that are not human beings.
