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
Previous article of the Langer review text
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.
October 23, 2017
Further:
The symbol wt2 can be used as the symbol for the universe where there are white houses. (161). Such as our present universe.
This means:
wt x
There are white houses. (161).
˜ (wt x)
There are not white houses. (161).
(∃a) : wt a (161).
There exists (a=houses) equals there exists white houses,
(∃!a) ⊨ wt a
There exists (a=houses) entails there exists white houses.
(∃a) : ˜ wt a (161).
There exists (a=houses) equals there is not white houses.
Relevance?
With one element to quantify, Langer provides four possible general propositions. (161).
(a) : wt a
(A) equals white houses.
(a) : ˜ wt a
(A) equals not white houses.
(∃a) : wt a
There exists (a) equals white houses exist.
(∃a) : ˜ wt a
There exists (a) equals white houses do not exist.
Langer writes that the quantifiers can be manipulated (161) so there is just one element which is white or there can be two, three, etcetera...wt2, wt3...(161)
I suppose practically, philosophically white houses can be separated from the class of non-white houses that are a sub class of houses. One could also separate houses based on actual colours as in green houses, red houses, white houses, etcetera.
Classes have sub-classes (117).
Class: Sheep
Sub-class: White sheep
Sub-class: Black sheep (117).
Class: Christians
Sub-class: Roman Catholic
Sub-class: Protestant
Sub-class: Orthodox
Sub-class: Non-denominational
Each sub-class would also not (˜ ) equal the other sub-class.
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