<< I wouldn't say logic is completely free from beliefs, in order to do some common types of logic you have to assume statements have truth values and in the case of reasoning, implication and causality. So it reflects your beliefs about the natural world in that sense.
Now, in the case of this ever-popular 'existence of God' debate, it seems to me that both atheists and deists seem to base their conclusions on sometimes shaky premises and mostly faulty reasoning. Even if the argument is valid (that is, deductive), that doesn't mean the conclusion is true, because you can still draw a conclusion from false premises. A valid argument only requires that all of the information present in the conclusion exists in the premises. People crap their pants with excitement when they make a valid argument, but don't see the bigger picture in the sense that perhaps their premises are false. Not to mention that whether it is possible to prove the existence of God through formal logic. >>
Correct. That's why nothing can be fully proven.
However, even though assumptions have to be made when using logic, it's also part of a logical reasoning to select the best supported (by evidence) assumptions. See thesises and theories in science.