There may be considerable similarity between the modality expressed in the vague boundaries of of rough sets and the modal expressions sometimes used in theories of possible worlds in quantum mechanics. This may not be too surprising, since probability can be modelled as a calculation over sets of possible worlds. Because of this similarity, I would like to digress briefly in this next section to discuss some of the models of quantum superpositions considered as set theory and how they can deal with identity.
Under the well accepted "standard" Copenhagen interpretation of quantum mechanics, particles which may have more than one quantum state as an outcome of measurement exist only as a discontinuous probability function, which is sometimes visualized by imagining the state of the unobserved particle to be a ghostly overlay of all its possible observable states simultaneously. For example, a particle that might be observed in state A, B, or C may be considered to be in a pseudo-state where it is simultaneously in states A, B, and C. Such a particle is said to be in a superposition of states. Measurement or other interactions with the particle's environment are said to collapse the superimposed states to produce a single real state which is the result of the observation.
If we take such a concept from quantum physics, we may consider what I'll call quantum superposition sets, or QS sets. A QS set is a set of possible values, such as x = { 2, 3 ,4 } and y = { 1, 2, 3 }. A variable that is a QS set can collapse, whereupon it will take a single value, exactly one element from the set. Two variables which are QS sets are vaguely equal if they have any element in common, so that on collapse, they might take on the same value, and two such variables are vaguely unequal if on collapse they may take on different values.
Consider the sets QS_a = {A, B, C}, QS_b = {A, B, C}, QS_c = { A, D, E },and QS_d = { F, G, H }.
QS_a is both vaguely equal and vaguely unequal to QS_b and QS_c and is non-vaguely unequal to QS_d. Note that even though by the laws of identity QS_a is exactly equal to itself (since it collapses itself as one object), but, because two otherwise identical QM objects may "collapse" in different ways, QS_a is not exactly equal to QS_b when the identity function depends on measurement after collapse of the waveform, even though they are the same superposition as a set prior to "collapse."
On the other hand, QS_a and QS_d do not share any possible values upon collapse, so it is quite definite that they are not equal.
Similarly, two rough sets that share their inner and outer boundaries and have elements in the zone beyond the inner boundary should be considered both vaguely equal and vaguely unequal, and two sets that do not share their inner boundaries and are not empty sets must be definitely unequal.
How do we apply superpositions to the biology of species? Let's consider the hypothetical assignment of bears in a study sample to two different species based on three different criteria for a species decision: fur color, diet, and biome type. Let's make a chart of possible outcomes of our sampling. Our starting criteria for species G bears is that they are brown omnivores that live in the woods, and for species P bears we have the criteria that they are white carnivores that live on the tundra. Our data will mostly but not completely fit that partition. Here is hypothetical data:
outcome
label | color | diet | biome |count
| | | |
A | brown | omni | woods | 21
B | brown | omni | tundra | 1
C | brown | carn | woods | 1
D | brown | carn | tundra | 5
E | white | omni | tundra | 1
F | white | carn | woods | 1
G | white | carn | tundra | 27
H | white | omni | woods | 4
I | black | omni | woods | 3
--------------------------------------
N | 64
Here, a given bear we choose to observe is, until fully observed, an observation that may be considered to be in state { A,B,C,D,E,F,G,H,I }, until our observations "collapse" the bear's datum into one of states A through H. If we were observing a population that included pandas, we might need to include vegetarian diet and add categories allowing for this, and if we were to judge a species based only on one of two "light or dark" fur colors, we might need only an A and a B category. Thus, with rough sets, the decision as to what criteria determines our inner and outer bound determines what part of the sample is vaguely classified and what part is well defined.
Consider a researcher who hypothesizes that the sole determining factor between bear species is their biome. He has no trouble separating bears into two sets: { A, C, F, H. I } and { B, D, E, G }. According to that researcher, the P species is composed of bears fitting { B, D, E, G }. However, a researcher who sticks to the criteria that the two species are determined by differences in three traits: color, biome, and diet, would find that the P species of bear would only include 27 bears and the G category only 21 bears, leaving 16 bears to be classified, perhaps, by biome alone, about which we are uncertain. These 16 remaining bears would then fall in the "vague" zone, outside of definitely-P bears and definitely-G bears, and by then using only biome criteria to place the remaining bears, 7 would be in the vague zone for P bears, and 9 would fall into a vague zone for G bears.
If we determine that diet is determined mostly by available food in the biome and that black bears are just really dark brown bears, and so decide to collapse the species definitions into white tundra P species bears versus black or brown woods dwelling G species bears, we than have 28 P species bears and 25 G species bears, with 11 remaining in the vague zones (using biome for the remaining species criteria to assign 6 brown tundra vague P bears and and 5 white woods vague G bears, respectively).
The rough sets as used for that last classification are P = { { E G } B D } and G = { { A C I } F H }, with the inner sets contained in two sets of parentheses.
What conclusions might a biologist draw from the above data? Unless this data is collapsed to look at a single criterion such as biome, there are many data points that do not fit cleanly into our hypothetical categories. Are P and G bears really separate species? We are not classifying bears cleanly using the data we have collected, and so we may need to find better criteria for categorizing bear species.
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