  Combining symbolic and neural approaches has gained considerable attention in
the AI community, as it is often argued that the strengths and weaknesses of
these approaches are complementary. One such trend in the literature are weakly
supervised learning techniques that employ operators from fuzzy logics. In
particular, they use prior background knowledge described in such logics to
help the training of a neural network from unlabeled and noisy data. By
interpreting logical symbols using neural networks (or grounding them), this
background knowledge can be added to regular loss functions, hence making
reasoning a part of learning.
  In this paper, we investigate how implications from the fuzzy logic
literature behave in a differentiable setting. In such a setting, we analyze
the differences between the formal properties of these fuzzy implications. It
turns out that various fuzzy implications, including some of the most
well-known, are highly unsuitable for use in a differentiable learning setting.
A further finding shows a strong imbalance between gradients driven by the
antecedent and the consequent of the implication. Furthermore, we introduce a
new family of fuzzy implications (called sigmoidal implications) to tackle this
phenomenon. Finally, we empirically show that it is possible to use
Differentiable Fuzzy Logics for semi-supervised learning, and show that
sigmoidal implications outperform other choices of fuzzy implications.
