J. Abellan and A. Masegosa Ph1(A)=Pl2(A), VA CX; implying a larger level of uncertainty for 2. This also implies that Km, CKmz, where Km, and Km, are the credal sets associated to mI and m2, respectively We consider that the situation expressed by Example I should be taken into account for a total uncertainty measure in dSt. This situation allows us to consider the following (P6) Monotonicity: a total uncertainty measure in DST must not decrease the total quantity of uncertainty in a situation where a clear decrease in information(increment Formally, let two b p as be on a finite set X, mi and m2, verifying that Km C Km, then Ez∞ TU(m1)≤TU(m2) Here, we must remark that the monotone dispensability definition of Harmanec(1995) could be used, but we prefer a more general one, which can be extended in a direct way to general credal sets Monotone dispensability always implies monotonicity axiom but not the contrary, as can be easily checked If we use the results of the works of Klir and wierman(1998), Maeda and Ichihashi (1993)and Jousselme et al.(2006), it can checked that the Ml, S and AM functions verify the following sets of requirements in DST: MI: P1. P4. P5 and P6 S: Pl, P2, P3, P4, P5 and P6. AM: P1 P2 P3 and P5 Considering the list above, we should mention the following: 1. Function MI does not satisfy the P2 and P3 requirements. Its range is [0, 2 loglXl] because it uses a clear split between the quantification of the two types of uncertainty*, each with range [0, loglXI 2. Jousselme et al. proved that function AM satisfies the P4 requirement, but recently, Klir and Lewis(2007)found an error in this proof and gave a counter example that proves that AM does not satisfy the P4 requirement 3. Function AM does not satisfy the P6 requirement. If we consider Example 1, it can BetPm ((d1))=12, BetIm((d2 )=12, BetPm ((ds) BetPm(diD Hence AM(m1)=1.078>AM(m2)=1.047 We can therefore see that only S" satisfies all the proposed requirements 4. Requirements of behaviour for total uncertainty measures in DST The paper by Jousselme et al. analyses certain shortcomings of the S function(upper entropy) in DST in order to compare this function with the AM function. ThesePl1ðAÞ # Pl2ðAÞ; ;A # X; implying a larger level of uncertainty for m2. This also implies that Km1 # Km2 , where Km1 and Km2 are the credal sets associated to m1 and m2, respectively. We consider that the situation expressed by Example 1 should be taken into account for a total uncertainty measure in DST. This situation allows us to consider the following property: (P6) Monotonicity: a total uncertainty measure in DST must not decrease the total quantity of uncertainty in a situation where a clear decrease in information (increment of uncertainty) is produced. Formally, let two b.p.a.s be on a finite set X, m1 and m2, verifying that Km1 # Km2 , then TUðm1Þ # TUðm2Þ: Here, we must remark that the monotone dispensability definition of Harmanec (1995) could be used, but we prefer a more general one, which can be extended in a direct way to general credal sets. Monotone dispensability always implies monotonicity axiom but not the contrary, as can be easily checked3 . If we use the results of the works of Klir and Wierman (1998), Maeda and Ichihashi (1993) and Jousselme et al. (2006), it can checked that the MI, S* and AM functions verify the following sets of requirements in DST: MI: P1, P4, P5 and P6. S* : P1, P2, P3, P4, P5 and P6. AM: P1, P2, P3 and P5. Considering the list above, we should mention the following: 1. Function MI does not satisfy the P2 and P3 requirements. Its range is [0, 2 logjXj] because it uses a clear split between the quantification of the two types of uncertainty4 , each with range [0, logjXj]. 2. Jousselme et al. proved that function AM satisfies the P4 requirement, but recently, Klir and Lewis (2007) found an error in this proof and gave a counter example that proves that AM does not satisfy the P4 requirement. 3. Function AM does not satisfy the P6 requirement. If we consider Example 1, it can be proved that Km1 # Km2 , and we have BetPm1 ð{d1}Þ ¼ 5 12 ; BetPm1 ð{d2}Þ ¼ 3 12 ; BetPm1 ð{d3}Þ ¼ 4 12 ; BetPm2 ð{d1}Þ ¼ 13 36 ; BetPm2 ð{d2}Þ ¼ 7 36 ; BetPm2 ð{d3}Þ ¼ 16 36 : Hence, AMðm1Þ ¼ 1:078 . AMðm2Þ ¼ 1:047: We can therefore see that only S* satisfies all the proposed requirements. 4. Requirements of behaviour for total uncertainty measures in DST The paper by Jousselme et al. analyses certain shortcomings of the S * function (upper entropy) in DST in order to compare this function with the AM function. These 738 J. Abella´n and A. Masegosa Downloaded by [New York University] at 12:09 08 November 2011