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Logic is only a construct and it has its limitations, some of which have been formally proven by Gödel.
To believe things that cannot be proven is by definition, illogical.
In that case ZFC, the most common foundation of mathematics, is illogical, because the axiom of choice can't be proven or disproven from the Zermelo-Fraenkel set theory.
In the same way that mathematicians include AC in ZF, the existence of God can be proven trivially by axiomatizing it.