  In this paper, we introduce the notion of toric special weak Fano manifolds,
which have only special primitive crepant contractions. We study the structure
of them, and in particular completely classify smooth toric special weak Fano
4-folds. As a result, we can confirm that almost every smooth toric special
weak Fano 4-fold is a weakened Fano manifold, that is, a weak Fano manifold
which can be deformed to a Fano manifold.
