No reason justifies disbelief.

[Belacqua]

why do you think logical arguments are a way to know if there is a god for whom there is little if any empirical evidence

Reasonable question! 

It's just the difference between a scientific demonstration and a metaphysical one. 

Science, as you know, very wisely limits itself to certain kinds of knowledge. Empirical, intersubjectively repeatable, quantifiable, etc. That's why it works so well. 

But that leaves a number of big questions. For example, the assertion that empirical evidence is the best or only way to know things is not itself demonstrable through empirical evidence. What experiment could be devised, using empirical evidence, that there is no other kind of evidence? 

Since Plato, serious arguments for God do not consider God to be a sensible object, knowable in the way that other sensible objects are. Sometimes people mistakenly treat God as if it were Bigfoot -- we'd get proof if we knew where to look. But God has always been considered as a non-material or even noetic thing. Actually, non-thing. Some theologians even happily agree that God doesn't exist, because God is itself existence. 

A standard although very limited analogy is to numbers. We say that the number 5 exists, though we've never seen it. Only examples of fiveness or symbols. 

I guess I should make clear that no theologian reads the Bible the way most atheists do -- literalist sola scriptura. 

A long and involved metaphysical logical argument has to do with the fact that things existing are held in existence by things that are essentially (not temporally) prior, and it appears that the chain has to go back to one non-contingent thing that holds everything else in existence. I know I'm going to get into trouble for citing this one -- people really get worked up by Aristotle. But it's just an example of what I mean by a non-empirical demonstration.

---

Then demonstrate an argument for any deity which cannot be and has not been logically demolished.

I'm not going to demonstrate them, as they tend to require more than a few sentences. 

Offhand it's easy to name two: the Thomist argument for actus purus, and Spinoza's argument. 

If these have been logically "demolished" I hope you'll point me to the relevant books. I would like to know more about them, and the only people I've found who can explain them properly feel they've lasted pretty well.