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Motivated by its links to τ-tilting theory, we introduce left weak cotorsion pairs, as a generalization of cotorsion pairs in module categories. Such pairs are also linked to co-t-structures in corresponding triangulated categories, and to cotorsion pairs in certain extension closed (and hence extriangulated) subcategories, which we call two-term categories. We give a characterization of the (left) weak cotorsion pairs which come from two-term silting.