"Pairwise disjoint" is stronger than "disjoint"; it sometimes happens that $\displaystyle\bigcap\limits_{i\in I} A_i=\varnothing$ but for every $i,j$, or at least for some, one has $A_i \cap A_j\ne\varnothing$.
Likewise, "pairwise coprime" is stronger than "coprime".
"Pairwise independent" (in the probabilistic sense) is weaker than "independent". For example, suppose $X_1,\ldots,X_n$ are independent and uniformly distributed on the sphere $S^n$; then the great-circle distances $d(X_i,X_j)$ and $d(X_k,X_\ell)$ are independent if $\{i,j\}\ne\{k,\ell\}$ even if $i=k$, but $d(X_i,X_j), d(X_j,X_k), d(X_k,X_i)$ are not independent even though each of the three pairs of these three random variables is a pair of independent random variables.
Question: Is there a somewhat general rule that says for which sort of X the qualified "pairwise X" is stronger than "X" and for which sort it is weaker than "X"?
Here's a guess: Some sort of category-theoretic viewpoint can make some kind of sense of this. (?)