Anything non-empirical affecting the empirical world would almost certainly be incredibly unpredictable. What experimentation could be considered valid if it's impossible to know if the conditions are the same?
I have no idea what you think "empirical" means. "Empirical" simply means it can be observed. If something is affecting the world in a manner that can be empirically measured, it's not "non-empirical", since something that is "non-empirical" by definition couldn't be observed and may as well not exist.
My definition of empirical comes from good ol' philosophy 101. Anything detectable by sight, taste, touch, smell, or hearing (or things you can build a machine to detect and give you signals in one or more of those forms) would be "empirical." So yes, observation, but specifically with those senses. Not knowledge you gain through pure logic (so knowing your own existence through "I think therefore I am" would not be "empirical" knowledge). The oft cited religious "experience" or "feeling" wouldn't count either, for those who claim revelation of some sort.
Of course, then there's the definition of "supernatural" to begin with. If something exists and can be observed, what makes it supernatural rather than simply natural?
My guess would be either it cannot affect the world we live in, or does so based in criteria outside of the world we live in. So God could exist in some higher plane of existence and come down and mess with things once in a while, and be considered entirely "supernatural." At least by my definitions.
That's your default assumption, which is fine. It might not be the default assumption of the invisible, intangible elephant in your room.
... What? It's the default assumption of basically everyone regarding basically everything, whether they believe it is or not. Otherwise, we all have to believe anything that comes to mind at any time.
Which would make a difference, how so?
Let's go back to the elephant in your room that totally exists. He's intangible, so he doesn't affect you or the world in any way. Let's look at the consequences of both assumptions (existence and non-existence):
1) You assume he exists. He does not. Net effect: You move on with your life with an imaginary pet.
2) You assume he exists. He does. Net effect: You move on with your life with a pet that doesn't do anything.
3) You assume he doesn't exist. He does not. Net effect: You move on with your life.
4) You assume he doesn't exist. He does. Net effect: You move on with your life and you have a slightly miffed elephant, assuming he can observe you at all.
In every case, you move on with your life with pretty much 0 change.
But what about tangible things that actually matter, you ask? Well guess what: people make positive assumptions all the time,
and then test them. It's called a hypothesis. Now, do they actually believe their hypothesis is correct? Maybe, maybe not. Either way, it
doesn't matter since their testing will prove it correct or incorrect anyway.
1) Assumption of truth of hypothesis. Hypothesis true. Net effect: Test finished and knowledge gained.
2) Assumption of falsehood of hypothesis. Hypothesis false. Net effect: Test finished and knowledge gained.
Blah blah I don't want to write the other two lines
Point is, for truly non-provable things via the scientific method, your assumption means squat. For things that are provable, your assumption
again means squat since your tests will either confirm or deny it.
It is only when you do
not test your hypothesis and assume truth or falsehood that there is a problem, and only for things that can be proven in the first place. In reality, most anything that can be tested has some sort of inductive argument you could make to shift your hypothesis either way. You want an example of assumption of truth being the most logical one? How about science in general? You have zero absolute proof of anything "proven" through the scientific method and experimentation. It's all "within reasonable doubt." There's always a risk of it being an anomaly, but the risk gets steadily smaller and smaller, approaching
but never reaching zero. What you define as "reasonable" is what you define as "truth," not the other way around.
You make assumptions every day. Some assumptions of truth, some of falsehood. Most based on prior experience and using an inductive argument. But for those situations without prior experience, you assumption is equally pointless no matter what it might be. Truth, falsehood, whatever -- it doesn't matter to the invisible intangible elephant. EDIT: I shouldn't have to qualify this but I will anyway: Shoving your assumption about the invisible intangible elephant down someone else's throat is bad too. If he exists, he won't like you being a jerk. If he doesn't, you're still a jerk.