Logically impossible: Fatal condition

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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Langer explains that a proposition can only be known via another proposition. (183). Implication is a relation that only holds among propositions. (183). Propositions are regarded as postulates. (185).

A postulate needs to belong to the system, in the language of that system.
A postulate should imply further propositions of that system.
A postulate should not contradict any other accepted postulate, or any other proposition implied by another postulate. (185).

In other words, symbolic logic requires non-contradiction within its system in a universe of discourse.

Requirements

Coherence: Every proposition in the system must cohere to the established conceptual structure. (185). It must be in coherence with the rest.

Contributiveness : A postulate should contribute and have implication. (185-186).

Consistency: Most important states Langer (186). Two contradictory propositions (or premises) cannot contradict each other in a system. (186). The inconsistent is logically impossible. It is a fatal condition. (186). It is not logic at all. (186).

Independence: Postulates should be independent from each other. (186). If a proposition is deductible from a postulate already provided, then it is a theorem, a necessary fact, not another assumption. (186). Something provable in a theorem would be error to include as a postulate. (186).

I would reason that within philosophy there would be plenty of debate on what is a proposition/premise within systems and what would be a theorem.

Langer explains that when a theorem needs elucidation, any proposition  implied by another proposition as granted and proved within a system is a theorem. (186-187).

Within a biblical, system and universe of discourse...

Gd = God

(∃! Gd)

God exists. Would be viewed as necessary and a theorem.

Within an atheistic, system and universe of discourse...

˜ (∃! Gd)

God does not exist. Would be viewed as necessary and a theorem.

Noted: Some atheists would state they do not know if God exists, and not definitely that God does not exist. But my example stands as valid. Some atheists do fit within my proposition.

Further

Sc = Scripture

(∃! Sc) ⊨ (∃! Gd)

Scripture exists entails God exists.

From a Christian worldview, revealed, supernaturally inspired Scripture entails that God exists.