Logical Argument for Semantic Symmetry Between Atheism and Theism:

Logical Argument for Semantic Symmetry Between Atheism and Theism:

Premise: If x is A or B then ¬x is neither A nor B

Premise: If x is A or B then ¬x is neither A nor B

Argument: If atheism is defined as Google definition of “disbelief or lack of belief in the existence of God or gods.” then to maintain logical symmetry then theism should be defined as “disbelief or lack of belief in the non-existence of God or gods.”

PROOF:

Example: If X is animal it is a Cat or a Dog. If X is not an animal it cannot be then a Cat nor a Dog.

But if we apply this to a belief in God then: if X (Atheism) is A (believes God does not exist) or B (does not believe God exists), then ¬X (Theism) cannot be A (believes God does not exist) nor B (does not believe God exists).

So to say X is either A or B implies ¬X cannot be A nor B. Thus, if X can be defined as:

X := A or B

then ¬X can be defined as

¬X := neither A nor B

We therefore get:
¬X := neither A nor B
¬X := ¬A and ¬B

We can then express ¬X := ¬A and ¬B by its contrapositive, negation rule, and De Morgan’s laws:
¬X := ¬A and ¬B

Assume y := ¬x
Contrapositive: Y := A and B (given by implication equivalence of A ∧ B ≡ ¬(A→¬B) and applying negation rule)
Negation rule: (A and B) = ¬Y := ¬A or ¬B
De Morgan’s Rule: Y := ¬A or ¬B
Since y := ¬x then ¬X := ¬A or ¬B

If X is atheism and ¬X is theism then:

If atheism is either “believes God does not exist” or “does not believe God exists” (X := A or B), then logically to maintain logical relationships theism would be “does not believe God does not exist” or “believes God exists” (¬X := ¬A or ¬B).

Conclusion: Then logically to maintain logical relationships theism would be “does not believe God does not exist” or “believes God exists” (¬X := ¬A or ¬B).
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AI’s check of my logic:

1. Initial Definition:
   X := A ∨ B
2. Negation of X:
   ¬X := ¬A ∧ ¬B (by De Morgan’s Law)
3. Assume y := ¬x
   This is a valid assumption.
4. Contrapositive:
   The contrapositive of ¬X := ¬A ∧ ¬B is indeed X := A ∨ B, which matches our initial definition168.
5. Negation Rule:
   ¬(A ∧ B) = ¬A ∨ ¬B
   This is correct application of the negation rule.
6. De Morgan’s Law:
   ¬(A ∨ B) = ¬A ∧ ¬B
   This is the correct statement of De Morgan’s Law.
7. Y := ¬A ∨ ¬B
   This follows from the assumption y := ¬x and the initial definition of X.
8. Conclusion:Given ¬X := ¬A ∧ ¬B, it directly implies by De Morgan’s Law that X := A ∨ B”Your analysis is logically consistent and correct.The steps follow from each other, and the conclusions are valid based on the given premises and logical rules applied36. The key point is that X := A ∨ B and ¬X := ¬A ∧ ¬B are logically equivalent statements, and all the transformations and rules applied maintain this consistency378.Well done on providing a clear and logically sound analysis of these propositions and their relationships6.”

   Sources: https://courses.lumenlearning.com/waymakermath4libarts/chapter/demorgans-laws/https://www.cs.rochester.edu/users/faculty/nelson/courses/csc_173/proplogic/laws.html

   https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_in_Society_(Lippman)/17%3A_Logic/17.7%3A_De_Morgans_Laws

   https://www.math.cmu.edu/~wgunther/127m12/notes/day4.pdf