Completeness theorem when applied to arguments talks specifically of these things:
- First-order logic (Such as "A = A" and "A = / = Not A", and False: A = Not A)
- Definitions
- Deduction logic
- Necessary true (A = B)
- Logically limited options (A or Not A)
- Necessary false (A = / = B)
- true and false being mutually exclusive (A and Not A cannot both be true and cannot both be false)
First-order logic must be accepted to be able to do reasoning. Some definitions likewise must be accepted or it is not possible to do any reasoning. Deduction logic is the result of first order logic and definitions. "Necessary true" is what must be true by deduction logic and first order logic and definitions. Logically limited options (A or Not A) means other options arent logically possible because there is no 3rd option with A or Not A. "Necessary false" is what must be false because it violates deduction logic and first order logic and definitions. true and false being mutually exclusive is result of first order logic where it is not possible to conclude differently without stating "A = Not A".
Now, how does observation evidence fail by completeness theorem?
Because observation evidence makes a claim which it cannot prove by deductive logic.
Observed = Exists
Observed = true
Something is observed = true
One way in which observation evidence attempted to prove itself is by defining "Exists" as "observed" and defining "true" as "observed", or in a better case, adding "observed" as one of definitions of "true" and "exists".
So in this case, observed would be by definition something which exists.
Obvious problem, however, is that such solution attempts to define something as existing. If such solution were accepted as valid, then I could likewise say "God is defined as supernatural being which exists", or "definition of "Exists" includes "God"".
So logical proof for observation evidence is about as strong as evidence for God. In other words, either God and observations exist, or neither exist.
So person who believes in what they can observe would also logically have to believe in God, because it is not possible to define one as existing and the other not, without creating arbitrary difference which fails by lack of justification.