  The AI community is increasingly focused on merging logic with deep learning
to create Neuro-Symbolic (NeSy) paradigms and assist neural approaches with
symbolic knowledge. A significant trend in the literature involves integrating
axioms and facts in loss functions by grounding logical symbols with neural
networks and operators with fuzzy semantics. Logic Tensor Networks (LTN) is one
of the main representatives in this category, known for its simplicity,
efficiency, and versatility. However, it has been previously shown that not all
fuzzy operators perform equally when applied in a differentiable setting.
Researchers have proposed several configurations of operators, trading off
between effectiveness, numerical stability, and generalization to different
formulas. This paper presents a configuration of fuzzy operators for grounding
formulas end-to-end in the logarithm space. Our goal is to develop a
configuration that is more effective than previous proposals, able to handle
any formula, and numerically stable. To achieve this, we propose semantics that
are best suited for the logarithm space and introduce novel simplifications and
improvements that are crucial for optimization via gradient-descent. We use LTN
as the framework for our experiments, but the conclusions of our work apply to
any similar NeSy framework. Our findings, both formal and empirical, show that
the proposed configuration outperforms the state-of-the-art and that each of
our modifications is essential in achieving these results.
