A fuzzy inference system (FIS) typically implements a function f : ℝ^N → I, where the domain set R denotes the totally ordered set of real numbers, whereas the range set I may be either I = RM (i.e., FIS regressor) or T may be a set of labels (i.e., FIS classifier), etc. This study considers the complete lattice (F, ≤) of Type-1 Intervals' Numbers (INs), where an IN F can be interpreted as either a possibility distribution or a probability distribution. In particular, this study concerns the matching degree (or satisfaction degree, or firing degree) part of an FIS. Based on an inclusion measure function σ : F × F → [0, 1] we extend the traditional FIS design toward implementing a function f : F^N → I with the following advantages: 1) accommodation of granular inputs; 2) employment of sparse rules; and 3) introduction of tunable (global, rather than solely local) nonlinearities as explained in the manuscript. New theorems establish that an inclusion measure σ is widely (though implicitly) used by traditional FISs typically with trivial (i.e., point) input vectors. A preliminary industrial application demonstrates the advantages of our proposed schemes. Far-reaching extensions of FISs are also discussed.