A random vector ${X} = (X_1,cdots,X_n)$ is weakly associated if and only if for every pair of partitions ${X}_1 = (X_{pi(1)},cdots,X_{pi(k)}), {X}_2 = (X_{pi(k+1),cdots,X_{pi(n)})$ of ${X}, P({X}_1 in A, {X}_2 in B) geq P({X}_1 in A){P}({X}_2 in B)$ whenever A and B are open upper sets and $pi$ is a permutation of ${1,cdots,n}$. In this paper, we develop notions of weak positive dependence, which are weaker than a positive version of negative association (weak association) but stronger than positive orthant dependence by arguments similar to those of Shaked. We also illustrate some concepts of a particular interest. Various properties and interrelationships are derived.