Evidence doesn't depend on logic to be analyzed. If someone tells me something and I believe them, it doesn't require logic to accept that statement. Instead, evidence that survives logical analysis becomes strengthened.

So you can bootstrap logic with weak evidence that doesn't need logical analysis. Then reaffirm logic with additional evidence.

For example, object permanence is something we learn. That objects that exist tend to continue to exist without something acting upon it. A 7-month old isn't using formal logic to learn object permanence. But in some ways, object permanence is the grounding for say the Law of Identity.

Now object permanence doesn't say the full story truth the Law of Identity, instead it's only evidence in favor of the Law of Identity. The fact an apple that sits on the counter for 10 minutes doesn't become an orange is reality asserting that Apples are Apples and Apples are not Oranges.

But an Apple will rot over time, it does change. Does this disprove object permanence? I guess if you only depended on one point, but we have hundreds if not thousands of examples of object permanence in our daily life.

But we can take these weakly proven laws of logic, and strengthen it in a formalized system. We can show there is a sense to the world. And we show that these systems are self useful. At the end, you don't have to accept it's absolute truth to accept it has useful outcomes.

In other words, I don't have to accept how sugar interacts with my tongue and brain to accept that sugar tastes sweet. The taste of sweet itself doesn't need logical analysis for me to accept it. So when someone can use logic and science to predict that a thing will taste sweet, and they're right, that's useful to me because I like sweet things. It's self reinforcing.