I've read and seen in a few places the following statement. Naturalist believe in only things that the can touched, seen or smelt. Logic can not be touch seen or smelt, therefor naturalist can not use logic.
What a load of crap. First of all logic is based on real world observations. Whether it is used to study arguments and states or the foundation of math. Logic and symbolic logic such as math are just ways in which humans use to interpret the world around us.
When we state that a=a, we don't do that because logic dictates it. We say it because the accumulation of observations have shown a = a. We use logic as a short hand to understand things we have observed thousands of times. It is universal, because something equaling itself, is independent of human observation. Just like in all cultures, 1 + 1 = 2. We don't need the hand of God to insure that a = a.
Also informal logic is just the study of arguments. Once again this is based on observation. We know that an Ad populum argument is a fallacy, because we can find cases were the majority have been wrong. Once again, these conclusions are universal. Again there is no need for the supernatural.
Athiest, Naturalist and or whatever you want to call them do limit themselves to saying that material events have material causes. It does not exclude interpeting said events using human constructs, such as logic. It just implies that the human constructs don't actually affect the physical world. Has logic ever moved a mountain?