I am not a philosophy expert like the recently deleted EsotericLucidity, but here is what I understand: every system of logic has some premises. These premises have to be assumed. Science has assumptions. The first assumption of science is that natural phenomena have natural explanations. This is a "belief" of scientists. The logical system derived from this belief is useful. This does not prove the premise's truth, but it proves the usefulness of the premise.
So, we agree, but I think that this upsets fundamentalist Christians. Their first premise is that God exists. Their second premise is that God, directly or indirectly, caused everything.
Some of the posters here say that science can't say anything about God because God is outside the logical system that is science. In this way, God could include science as a subset, but only if you agree with the premise that God does exist. This is the position of theistic evolutionists.
Since the first scientists were Christians who believed in G-d (the inventor of the scientific method was a Christian) I would have to call this assumption a late arriver. Do you know when it became a part of science and why?
Shalom.
It's that word "natural" that's the problem. Since (as asserted above) God is not "natural," scientists doing science must assume that He plays a role in any natural process. Conveniently for the atheists, this approach guarantees that the scientific method must place itself in opposition to the possibility that God exists. It also guarantees that there is no possible "scientific" test that can demonstrate the action of God.
The logical system derived from this belief is useful. This does not prove the premise's truth, but it proves the usefulness of the premise.
The necessary and sufficient condition for the logical system is that phenomena must be explainable in terms of consistent mechanisms. There is no logical need for the additional proposition that God does not exist. The roots of that assertion rest in something other than scientific necessity.