Very good diagnosis. While I don’t recall where the philosophers of religion came out in this area on the survey, some of the popular arguments for the existence of God seem to rely on a very strong version of mathematical Platonism, a believe that there is a one true logic/mathematics and that strong conclusions about the world can be drawn by the proper employment of that logic (the use of PSR that you mention is a common, but not the only, example of this). Since I reject that kind of One True Logic (I’m a Carnapian, “in logic there are no morals!” guy), I tend to think that any logical proof of the existence of God (or anything else) serves only to establish that you are using a logic which has a built in “God exists” (or whatever) assumption. For example, the simple modal ontological argument which says God possibly exists, God is a necessary being, so by a few simple steps in S5, God exists; if you’re using S5, then once you’ve made one of the assumptions about God, making the other one just amounts to assuming God exists, and so in effect committing yourself to reasoning only about God worlds. Such a logic may have its uses, but it is incapable of reasoning about the possibility that God might or might not exist; for such purposes a neutral logic would be required.
Very good diagnosis. While I don’t recall where the philosophers of religion came out in this area on the survey, some of the popular arguments for the existence of God seem to rely on a very strong version of mathematical Platonism, a believe that there is a one true logic/mathematics and that strong conclusions about the world can be drawn by the proper employment of that logic (the use of PSR that you mention is a common, but not the only, example of this). Since I reject that kind of One True Logic (I’m a Carnapian, “in logic there are no morals!” guy), I tend to think that any logical proof of the existence of God (or anything else) serves only to establish that you are using a logic which has a built in “God exists” (or whatever) assumption. For example, the simple modal ontological argument which says God possibly exists, God is a necessary being, so by a few simple steps in S5, God exists; if you’re using S5, then once you’ve made one of the assumptions about God, making the other one just amounts to assuming God exists, and so in effect committing yourself to reasoning only about God worlds. Such a logic may have its uses, but it is incapable of reasoning about the possibility that God might or might not exist; for such purposes a neutral logic would be required.