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The review continues:
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
∧ = Logical conjunction
Inclusion
Philosopher Langer introduces a 'new relation' called inclusion. (Note I have had this symbol in my key, as she introduced it earlier in the textbox). < means that 'a' is included in 'b' (226). This means the same thing as the sum of 'a' and 'b' is 'b' itself. (226). In mathematics < is used for 'less than' (134), but is defined as is included and inclusion within symbolic logic (135).
She explains this importance...
< in this system means a + b = b, and this is the equivalent of ab = a; a proposition in terms of + (plus, addition) and x (times, multiplication), as in a x b = b or a x b = a. (227). This regarding classes does not demonstrate metaphysical nature (238), or reality or nature (238), but 'merely shows the pattern of class-relationships which we may call disjunction and conjunction, or total and partial inclusions, or overlapping, or anything else that fits the formal conditions.' (238).
(Disjunction is the relationship between two distinct alternatives).
Symbolic logic
A conjunction is a compound statement formed by combining two statements using the word and. In symbolic logic, the conjunction of p and q is written p∧q . A conjunction is true only if both the statements in it are true.
Pragmatically
It seems to me, symbolic logic can provide a system, with logical formal conditions for presenting propositions. I have noted that the pragmatic benefits of using symbolic logic for reasoning for me, are questionable, in that symbolic logic is difficult for the average reader to understand.
Propositions are statements, and I was required to present propositions for my British MPhil and Doctoral survey questionnaires. Interestingly, working through this section of Langer, I was not considering it as significantly philosophically beneficial, however, it became crystal clear while researching this article that a conclusion is considered a proposition in contrast to a premise or premises which support (s) the conclusion. This was not often clearly stated in much of the philosophical reading I have done over the years. Indeed in my British academic survey questionnaires, I was formally presenting:
Proposition = Conclusion
or
Statement = Proposition = Conclusion
One key issue in writing my Doctoral thesis's main body was to avoid presenting an assertion without an argument. As an assertion = statement = proposition = conclusion. This was left for the questionnaire surveys.
As noted in an earlier review with this text; an implied proposition is true if all the premises are true. (188). The implied proposition could also be defined as the conclusion. If the premises are false, she opines that the proposition may or may not be true. (188).
This section from Langer ended up being beneficial research.
Proposition < Conclusion
A proposition is inclusive to a conclusion. < means that 'a' is included in 'b' (226).
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