  Text generation has received a lot of attention in computational
argumentation research as of recent. A particularly challenging task is the
generation of counter-arguments. So far, approaches primarily focus on
rebutting a given conclusion, yet other ways to counter an argument exist. In
this work, we go beyond previous research by exploring argument undermining,
that is, countering an argument by attacking one of its premises. We
hypothesize that identifying the argument's weak premises is key to effective
countering. Accordingly, we propose a pipeline approach that first assesses the
premises' strength and then generates a counter-argument targeting the weak
ones. On the one hand, both manual and automatic evaluation proves the
importance of identifying weak premises in counter-argument generation. On the
other hand, when considering correctness and content richness, human annotators
favored our approach over state-of-the-art counter-argument generation.
