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 [a1,a2] of this interval. There are several different ways to compare intervals. For example, we can say that [a1,a2] <= [b1,b2] if every number from the interval [a1,a2] is smaller than or equal to every number from the interval [b1,b2]. However, in interval-valued fuzzy logic, a more frequently used ordering relation between interval truth values is the relation [a1,a2] <= [b1,b2] if and only a1 <= b1 & a2 <= b2. 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.