Übercomplicated

there is literally no way to use logic to infer existence of god.

I just gave an example of someone doing precisely that. Read Descartes first three meditations. I won’t say I agree with Descartes; I think his logic is flawed. But he does technically use logic to infer the existence of God. That’s why I used Descartes as an example.

Literally René Descartes, arguably one of the most important mathematicians and philosophers in history, spends most of his meditations trying to prove the existence of God. Religious people can be very logical and believe in their religion for logical reasons, even if you and I disagree with those reasons.