Aiming at combining the good characteristics of a differential scale in representing human cognition and the favorable properties of interval judgments in expressing decision-makers’ uncertainty, this paper proposes the interval cognitive network process (I-CNP) to extend the primitive cognition network process (P-CNP) to handle interval judgments. The key points of I-CNP include a consistency definition for an interval pairwise opposite matrix (IPOM) and a method to derive interval utilities from an IPOM. This paper defines a feasible region-based consistency definition and a transitivity based consistency definition for an IPOM. Both of the two definitions are equivalent to the consistency definition for a crisp pairwise opposite matrix (POM) when an IPOM is reduced to a POM. Two methods that are able to derive interval utilities from both consistent and inconsistent IPOMs are developed based on the two definitions. Four numerical examples are used to illustrate the proposed methods and to compare I-CNP to interval analytic hierarchy process (IAHP). The results show that I-CNP reflects the decision-makers’ cognition better, and that the suggestions provided by I-CNP are more convincing. I-CNP provides new useful alternative tools for multi-attribute decision-making problems.