Selecting a fuzzy logic operation from the DNF-CNF interval: how practical are the resulting operations?

Abstract

In classical (two-valued) logic, CNF and DNF forms of each propositional formula are equivalent to each other. In fuzzy logic, CNF and DNF forms are not equivalent, they form an interval that contains the fuzzy values of all classically equivalent propositional formulas. If we want to select a single value from this interval, then it is natural to select a linear combination of the interval's endpoints. In particular, we can do that for CNF and DNF forms of "and" and "or", thus designing natural fuzzy analogues of classical "and" and "or" operations. The problem with thus selected "and" and "or" operations is that, contrary to common sense expectations, they are not associative. We show the largest possible value of the corresponding non-associativity is reasonably small and thus, this non-associativity does not make these operations impractical.