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In this paper we show that a consistent logical system generated by nilpotent operators is not necessarily isomorphic to Łukasiewicz-logic, which means that nilpotent logical systems are wider than we have earlier thought. Using more than one generator functions we examine three naturally derived negations in these systems. It is shown that the coincidence of the three negations leads back to a system which is isomorphic to Łukasiewicz-logic. Consistent nilpotent logical structures with three different negations are also provided. t-norms and t-conormsFirst, we recall some basic notations and results regarding t-norms, t-conorms and negation operators that will be useful in the sequel.A triangular norm (t-norm for short) T is a binary operation on the closed unit interval [0, 1] such that ([0, 1], T ) is an abelian semigroup with neutral element 1 which is totally ordered, i.e., for all x 1 , x 2 , y 1 , y 2 ∈ [0, 1] with x 1 ≤ x 2 and y 1 ≤ y 2 we have T (x 1 , y 1 ) ≤ T (x 2 , y 2 ), where ≤ is the natural order on [0, 1].A triangular conorm (t-conorm for short) S is a binary operation on the closed unit interval [0, 1] such that ([0, 1], S) is an abelian semigroup with neutral element 0 which is totally ordered.

A new three‐parameter probability distribution called the omega probability distribution is introduced, and its connection with the Weibull distribution is discussed. We show that the asymptotic omega distribution is just the Weibull distribution and point out that the mathematical properties of the novel distribution allow us to model bathtub‐shaped hazard functions in two ways. On the one hand, we demonstrate that the curve of the omega hazard function with special parameter settings is bathtub shaped and so it can be utilized to describe a complete bathtub‐shaped hazard curve. On the other hand, the omega probability distribution can be applied in the same way as the Weibull probability distribution to model each phase of a bathtub‐shaped hazard function. Here, we also propose two approaches for practical statistical estimation of distribution parameters. From a practical perspective, there are two notable properties of the novel distribution, namely, its simplicity and flexibility. Also, both the cumulative distribution function and the hazard function are composed of power functions, which on the basis of the results from analyses of real failure data, can be applied quite effectively in modeling bathtub‐shaped hazard curves.