Modal intervals as a new logical interpretation of the usual lattice order between interval truth values

In the traditional fuzzy logic, we use numbers from the interval [0,1] to describe possible expert´s degrees of belief in different statements. Comparing the resulting numbers is straightforward: if our degree of belief in a statement A is larger than our degree of belief in a statement B, this means that we have more confidence in the statement A than in the statement B. It is known that to get a more adequate description of the expert´s degree of belief, it is better to use not only numbers a from the interval [0,1], but also subintervals [a?, a̅] ⊆ [0, 1] of this interval. There are several different ways to compare intervals. For example, we can say that [a?, a̅] ≤ [b?, b̅] if every number from the interval [a, a̅] is smaller than or equal to every number from the interval [b?, b̅]. However, in interval-valued fuzzy logic, a more frequently used ordering relation between interval truth values is the relation [a, a̅] ≤ [b?, b] ↔ a a̅ b & a̅ ≤ b. This relation makes mathematical sense - it make the set of all such interval truth values a lattice - but, in contrast to the above relation, it does not have a clear logical interpretation. Since our objective is to describe logic, it is desirable to have a reasonable logical interpretation of this lattice relation. In this paper, we use the notion of modal intervals to provide such a logical interpretation.