A gradient method for inconsistency reduction of pairwise comparisons matrices

We investigate an application of a mathematically robust minimization method -- the gradient method -- to the consistencization problem of a pairwise comparisons (PC) matrix. Our approach sheds new light on the notion of a priority vector and leads naturally to the definition of instant priority vectors. We describe a sample family of inconsistency indicators based on various ways of taking an average value, which extends the inconsistency indicator based on the "$\sup$"- norm. We apply this family of inconsistency indicators both for additive and multiplicative PC matrices to show that the choice of various inconsistency indicators lead to non-equivalent consistencization procedures.Comment: 1 figure, several corrections and precision

Probability measures and logical connectives on quantum logics

The present paper is devoted to modelling of a probabi‐ lity measure of logical connectives on a quantum logic via a G‐map, which is a special map on it. We follow the work in which the probability of logical conjunction (AND), dis‐ junction (OR), symmetric difference (XOR) and their nega‐ tions for non‐compatible propositions are studied. Now we study all remaining cases of G‐maps on quantum lo‐ gic, namely a probability measure of projections, of impli‐ cations, and of their negations. We show that unlike clas‐ sical (Boolean) logic, probability measures of projections on a quantum logic are not necessarilly pure projections. We indicate how it is possible to define a probability me‐ asure of implication using a G‐map in the quantum logic, and then we study some properties of this measure which are different from a measure of implication in a Boolean algebra. Finally, we compare the properties of a G‐map with the properties of a probability measure related to logical connectives on a Boolean algebra