Logical operation

British Columbia: trekearth

LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy)

The continuation of text review:

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
cls = Class
int = Interpretation
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Page 193 presents a new section Relations and Operations.

Langer explains that

(a) (∃ -a) (193)

In this deductive system, every element generates its complement, as in (a) would generate the complement (-a). Therefore, as there is a positive (a) this means negative (-a) also exists. (192).

(a) = (∃! -a)

Langer explains that for every (a) there is at least one (-a). (192). The proposition which defines the nature (-a) is stated to describe the operation (-), upon the element (a). (192). This operation generates the complement (-a).

She further goes on to explain that the element might of course (as it is symbolic logic) have any name we like. The symbols (b) or (c) could be used. Wherever there is (a), there is the operation (b), or wherever there is (a), there is the operation (c). (192). But she explains that this type of symbolism would be correct but not helpful. (193).

Practically, a deductive system needs to make sense to the reader, and the presenter! Regardless of symbols used. As example, for a white house that exists in a deductive system, there exists a complement, a non-white house in that same deductive system. This presentation from Langer is not so much straightforward, but is it logical.

And logic is required for good reasoning, and the reasoning presented by Langer in this textbook may often be more useful than the symbols provided.

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