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| Vancouver: This 6:30 am |
LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York.
Key symbols
≡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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Previously
The Universe of Classes
We documented a system of houses (157), as example of K=interpreted as houses and a dyadic. (157) 'When a relation-symbol stands in a construct, the number of terms grouped with it reveals the degree of the relation. But when it is not actually used, but merely spoken of, it is sometimes convenient to have some way of denoting its degree. This may be done by adding a numerical subscript; for example, "kd2" means that "killing" is dyadic (a pair), "bt3" that "between" is triadic.' (55).
In such a dyadic system, all the elements have to expressible in terms of two elements. (157). There is a fixed element that relates to an element on the other end of the pole, so to speak. (157). Every class is therefore relative to some given element. (157). The defining form of the class must be in dyad (157) such as with Langer's example of K nt2. These are the houses north of a stated element. Langer explains that if the elements had begun with a triadic (group of three not two as in dyadic relation), such as using the term 'between', then two fixed elements would have been used to generate a class. (157). The class between a and b or the class of terms not between a and b.
Using dyadic: K=interpreted as houses nt=interpreted as north of... K= (a, b, c... =nt2 ) a, b, c... are houses north of x within this deductive system and universe of discourse.
September 25, 2017
Langer
'Ordinarily, however, we do not invoke relationship to a given term in order to form a class; we form such classes as 'white houses.' two-storeyed houses,' etc., without reference to any given term. What sort of formal context does this requires? What relation functions among the elements of our universe to generate a class of 'white houses'? (157).
'If the is no relation among the elements of the universe...how can we have any elementary structure of terms.' (157). As in the propositional forms in any given context, to define classes of elements. (157).
The answer Langer provides is:
Predication
For example, the term 'being white' (Reader again, please be aware of the context and time of this textbook being 1953 and 1967. The text is not playing philosophical or political games with any modern context.) If there is no second term, such as 'white house', it cannot be stated that x has any relation to any other term. (158). This relation of 'nomadic degree', (158), is called a predicate. (158).
To add some unfortunate confusion, Langer then states that within the philosophical community (of that day) there is 'considerable disagreement' on whether or not it is fair to call a predicate a nomadic degree. (158).
To be blunt, this is another example of how philosophy often offers shades of gray/grey...
Langer example:
wt= Is white
wt x=A definition of the class of things white, without relating what is white to any other term. (Based on 159).
Therefore the term wt needs to be connected with other terms.
(∃! x) = wt · x (Based on 159).
There exists x equals white equals and means x. But it connects to nothing else.
However:
K=int 'houses'
(∃! wt) . (K)
There exists white and houses.
There exists white houses.
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