A mutually exclusive, yes or no, binary type of decision can be simple to make. For some decisions we can directly observe the physical presence or absence of something and reach a definitive conclusion. But we often have no direct access to the answer and therefore we must infer a conclusion on a best overall fit with the available evidences basis. The conclusion is now going to be a probabilistic estimate, and therefore will represent a location on a continuous line joining the two end points. The binary, one or the other, proposition is thus converted into a continuum by the decision making process.
However, it is impractical to assign a particular probability number to our conclusion since we typically lack enough information to be that precise. Fortunately, Baysian probability analysis is still viable with only three discrete outcomes: Positive, neutral, and negative. We can assign the evidences one of three values, 1, 0, and -1 so that we can accumulate evidences into a single sum. Every evidence for the proposition adds to the total, every evidence against subtracts from the total. Does this conversion of our proposition from its original binary form to a discrete line form indicate that our proposition is incoherent and meaningless? No, this is merely the unavoidable outcome of the fact that we are not omnipresent and omniscient so we often need to retreat to approximate Bayesian probabilities to infer a conclusion.
Many propositions are not well defined. For example, we may ask if there is intelligent life elsewhere in the universe. We cannot even begin to reach an approximate probability estimate until we have a definition of intelligent life that includes a criteria which can be utilized for evaluating the evidences. The criteria selected to clarify the proposition will often unavoidably be somewhat arbitrary. Does this arbitrariness and ambiguity in the definition of a proposition indicate that the proposition itself is incoherent and meaningless? Lets try to answer this by identifying a criteria for intelligent life.
Our relationship with our pets, for better or worse, is limited by our inability to converse with them. So we can specify as a meaningful criteria for intelligent life the ability to symbolically communicate with an extensive vocabulary. There is still ambiguity here about what qualifies as extensive, but since inferring on the evidences is an approximation anyway we are not likely to benefit much from a more precisely defined proposition. Inferring on the evidence can be a coherent and meaningful activity even when we lack a precisely defined proposition.
We may sometimes need more than one criteria for evidencing a particular proposition. Furthermore, we may decide that each criteria by itself evidences for the proposition. This results in a complex proposition. When we apply the evidences we may find that according to one of the criteria the proposition is affirmed, but according to the other criteria the proposition is disconfirmed. Does the possibility of a such a paradoxical result mean that all complex propositions are incoherent and meaningless? Again, the answer is no. The possibility of such a paradoxical result is merely the unavoidable outcome of combining inference with a complex proposition. Nevertheless, we should try to avoid making the proposition more complex than is necessary to reduce the risk of a paradoxical outcome.
In their excellent essay "Does Science Presuppose Naturalism?" (the correct answer is no), philosophers Yonatan Fishman and Maartin Boudry propose a complex, three criteria definition of supernatural. I think their definition is a good start, but is unnecessarily complex and too weak. I am going to propose a simpler and more stringent definition using their proposal as a starting point.
I will drop the criteria that they label as (2), which is that the phenomena "exist outside the spatiotemporal realm of our universe". Although this criteria is associated with the supernatural, I want to keep the proposition as simple as possible and this criteria is not essential. I will then modify the criteria they label as (1) and reverse the order of the two remaining criteria. This results in the following two criteria for identifying supernatural phenomena: (1) They suggest that reality is at bottom purposeful and mind-like, particularly in a sense that implies a central role for humanity and human affairs in the cosmic scheme and (2) they operate in ways that fundamentally violate our current scientific understanding of what is permissible within the constraints set by natural laws.
The first criteria is important because it enables us to distinguish natural laws from supernatural laws. We need to be able to make this distinction to apply the second criteria. The second criteria is important because it defines the scope of what is possible within the natural framework. Many people greatly underestimate the capabilities of the laws of nature and as a result of this ignorance are overly dismissive of the viability of naturalism.
The existing natural laws are incomplete. Therefore the quality of residing outside the scope of natural law is an insufficient condition for identifying supernatural phenomena. Natural phenomena occurring outside the incomplete natural law framework are to be expected. This is why the criteria says that a fundamental violation of a natural law constraint must occur.
These two criteria can be combined into a single criteria. This results in the following definition of supernatural phenomena: They suggest that reality is at bottom purposeful and mind-like, particularly in a sense that implies a central role for humanity and human affairs in the cosmic scheme, and they operate in ways that fundamentally violate our current scientific understanding of what is permissible within the constraints set by natural laws. This requires an initial categorization of the existing scientific laws of the universe as being either natural or supernatural, which is done using the first half of the combined criteria. A violation of at least one natural law is now a requisite indicator of supernatural phenomena, but is by itself insufficient. Because we approach claims of supernaturalism skeptically, we will also require that the violation evince a mind-like, judgmental or supervised, purposefulness.
One common objection to any attempt to define supernaturalism is that the supernatural concept is always ruled out because it violates Occam's Razor. But Occam's Razor is a rule of thumb focused on dropping superfluous adornments, its not a fundamental law for disregarding the direction of the evidences. Another common objection is that all evidences favorable to supernaturalism should instead be interpreted as evidence of an advanced technology civilization trying to fool us. This is a mirror image of the perspective of some theists that evidences favorable naturalism are really favorable for god. Some people assert that supernaturalism entails unpredictability, undetectability, and similar attributes that place it outside the reach of evidences, rendering any attempt to evidence supernaturalism futile. This is actually false, supernaturalism entails no such particular set of attributes (the previously referenced essay by Fishman and Boudry addresses this).
Supernaturalists are inclined to try to argue that the current scientific laws of the universe are themselves supernatural. One argument that our laws are supernatural is the Fine Tuning argument. Victor Stenger, among others, argues that the premises of the Fine Tuning argument are false. But even if the Fine Tuning argument was valid, our current scientific laws predict a multiverse, and combining the Anthropic Principle with a multiverse undermines the Fine Tuning argument. Another argument that our laws are supernatural is the First Cause argument. However, the consensus of cosmologists is that modern cosmology has no need for a non-natural first cause. It is difficult to avoid concluding that our universe overall is indifferent to humanity and to life more generally. Even people who should know better nevertheless avoid this conclusion for psychological and psychology related political reasons (people prefer to believe that our universe is about us).
Merely accepting the possibility of supernaturalism being evidenced doesn't produce such evidence or otherwise render our universe any less naturalistic. On the contrary, the fact that supernaturalism could be evidenced, but is not, is all the more reason to conclude that our universe is naturalistic. As philosophical naturalists, we can and should be willing to acknowledge that if our universe did evidence supernaturalism then we should be supernaturalists. Best fit with the evidences, despite its limitations, is the only viable method we have to accurately model our universe. At the end of the day the goal of our beliefs about how the world works is to accurately model reality, not to reach any particular, fixed conclusion.