Poincare-Lefshcetz duality:
Let $M$ be a compact manifold with boundary and $\dim M = n$, then we have $H_i(M, \partial M) = H^{n-i}(M)$, $H^i(M, \partial M) = H_{n-i}(M)$.
Poincare-Lefschetz-Alexander duality:
Let $M$ be a compact manifold, $\dim M = n$ and $B \subset A \subset M$ closed subsets. Then we have $H^i(M-B,M-A) = H_{n-i}(A,B)$.
It not obvious for me how to Poincare-Lefshcetz duality implies from Poincare-Lefschetz-Alexander duality. Hope for your help.