Abstract
We refined the axiomatics of asymmetric logics. For logics \(X(km,k)\) of family subsets of the \(km\) -element set \(X\) , which cardinal numbers are multiples of \(k\) we completely described the cases in which \(X(km,k)\) a) is symmetric or b) is asymmetric. For an infinite set \(\Omega\) and a natural number \(n\geq 2\) we constructed the concrete logics \(\mathcal{E}^{n}_{\Omega}\) and completely described the cases in which these logics are asymmetric. For asymmetric logics \(\mathcal{E}\) we determine when both the set \(A\in\mathcal{E}\) and its complement \(A^{c}\) are atoms of the logic \(\mathcal{E}\) . Let a symmetric logic \(\mathcal{E}\) of a finite set \(\Omega\) be not a Boolean algebra, and let \(\mathcal{A}\) be an algebra of subsets from \(\Omega\) , and assume that \(\mathcal{E}\subset\mathcal{A}\) . Then there exists a measure on \(\mathcal{E}\) , that does not admit an extension to a measure on \(\mathcal{A}\) .
