# Analyzing Differentiable Fuzzy Implications

arXiv: https://arxiv.org/abs/2006.03472 · HTML: https://mlsanity.com/paper/2006.03472/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2020.

## Built on by (most cited first, 15 of 15)

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- [Right for the Right Reasons: Avoiding Reasoning Shortcuts via Prototypical Neurosymbolic AI](https://mlsanity.com/paper/2510.25497/) (2025)
- [TAR: Neural Logical Reasoning across TBox and ABox](https://mlsanity.com/paper/2205.14591/) (2022)
- [Parallel Neurosymbolic Integration with Concordia](https://mlsanity.com/paper/2306.00480/) (2023)
- [Differentiable Logics for Neural Network Training and Verification](https://mlsanity.com/paper/2207.06741/) (2022)
- [Statistical relational learning and neuro-symbolic AI: what does first-order logic offer?](https://mlsanity.com/paper/2306.13660/) (2023)
- [Concise and Logically Consistent Conformal Sets for Neuro-Symbolic Concept-Based Models](https://mlsanity.com/paper/2605.18202/) (2026)
- [Scalable Regularization of Scene Graph Generation Models using Symbolic Theories](https://mlsanity.com/paper/2209.02749/) (2022)
- [Informed Machine Learning with Knowledge Landmarks](https://mlsanity.com/paper/2604.00256/) (2026)
- [SoftTri: Smooth Triangular Membership Functions for Adaptive Fuzzy Inference Systems](https://mlsanity.com/paper/2609.20194/) (2026)

## Abstract (arXiv)

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.