  The AI community is increasingly putting its attention towards combining
symbolic and neural approaches, as it is often argued that the strengths and
weaknesses of these approaches are complementary. One recent trend in the
literature are weakly supervised learning techniques that employ operators from
fuzzy logics. In particular, these 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, this background
knowledge can be added to regular loss functions, hence making reasoning a part
of learning. We study, both formally and empirically, how a large collection of
logical operators from the fuzzy logic literature behave in a differentiable
learning setting. We find that many of these operators, including some of the
most well-known, are highly unsuitable in this setting. A further finding
concerns the treatment of implication in these fuzzy logics, and 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 compare how different operators behave in
practice. We find that, to achieve the largest performance improvement over a
supervised baseline, we have to resort to non-standard combinations of logical
operators which perform well in learning, but no longer satisfy the usual
logical laws.
