*3.2. The Sirius Star*

Are there or are there not living beings on the planet in orbit around star Sirius? Let us only consider the case where the interviewed person is not an expert at all, so the case of Shafer's total ignorance, and let us consider the two different situations given in Tables 3 and 4. In the first case, total ignorance is admitted on only two sets, while in the second case, total ignorance is professed on three sets, and the two ways of forming the sets are independent.

**Table 3.** The Sirius star and the two considered sets.


**Table 4.** The Sirius star and the three considered sets.


Table 5 shows the values assigned to the belief function for the sets defined in Table 3 (first column) and for the sets defined in Table 4 (second column). It is, however, possible to compare the two cases since, by considering the sets defined in Tables 3 and 4, it can be stated that *A* = *C* and *B* = *D* ∪ *E*. The last column is the comparison of the two previous columns and shows that the assigned values in the two cases are coherent with each other.

**Table 5.** The Sirius star and total ignorance represented with the belief functions.


On the other hand, Table 6 shows the results for the probability functions and, when the two cases of Tables 3 and 4 are compared, it follows that there is no consistency at all. In fact, set *A* defined in the case of Table 3 is exactly set *C* defined in the case of Table 4, but as shown in Table 6, *Pro* (*A*) = *Pro* (*C*). Furthermore, set *B* defined in the case of Table 3 is exactly set *D* ∪ *E* defined in the case of Table 4, but *Pro* (*D* ∪ *E*) = *Pro* (*B*) since the following holds:

$$\operatorname{Prov}\left(D \cup E\right) = \operatorname{Pro}(D) + \operatorname{Pro}(E) - \operatorname{Pro}(D \cap E) = \frac{1}{3} + \frac{1}{3} - 0 = \frac{2}{3} \neq \operatorname{Pro}\left(B\right) = \frac{1}{2}$$


**Table 6.** The Sirius star and total ignorance represented with the probability functions.

Then, it can be concluded that belief functions are more suitable than probability functions to handle total ignorance, that is, all situations where an individual has no evidence/no knowledge about the considered topic and about the considered given sets.

This great interest in total ignorance is due to the fact that total ignorance is mostly present in the field of measurements, as shown in the simple practical example in the next section.
