# A unified logical framework for explanations in classifier systems

arXiv: https://arxiv.org/abs/2105.14452 · HTML: https://mlsanity.com/paper/2105.14452/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 60 references that are in the map.
- First posted on arXiv 30 May 2021.

## Builds on (most cited first, 15 of 15)

- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Explanations based on the Missing: Towards Contrastive Explanations with Pertinent Negatives](https://mlsanity.com/paper/1802.07623/) (2018)
- [Abduction-Based Explanations for Machine Learning Models](https://mlsanity.com/paper/1811.10656/) (2018)
- [Explaining Explanations in AI](https://mlsanity.com/paper/1811.01439/) (2018)
- [On The Reasons Behind Decisions](https://mlsanity.com/paper/2002.09284/) (2020)
- [Contrastive Explanation: A Structural-Model Approach](https://mlsanity.com/paper/1811.03163/) (2018)
- [k-Nearest Neighbour Classifiers: 2nd Edition (with Python examples)](https://mlsanity.com/paper/2004.04523/) (2020)
- [On the Computational Intelligibility of Boolean Classifiers](https://mlsanity.com/paper/2104.06172/) (2021)
- [On Tractable Representations of Binary Neural Networks](https://mlsanity.com/paper/2004.02082/) (2020)
- [Ceteris Paribus Structure in Logics of Game Forms](https://mlsanity.com/paper/1310.6416/) (2013)
- [A Simple Logic of Functional Dependence](https://mlsanity.com/paper/2103.14946/) (2021)
- [Propositional Logics of Dependence](https://mlsanity.com/paper/1412.7998/) (2014)
- [Ceteris paribus logic in counterfactual reasoning](https://mlsanity.com/paper/1606.07522/) (2016)

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

- [Distance-Restricted Explanations: Theoretical Underpinnings & Efficient Implementation](https://mlsanity.com/paper/2405.08297/) (2024)
- [Delivering Inflated Explanations](https://mlsanity.com/paper/2306.15272/) (2023)
- [From Robustness to Explainability and Back Again](https://mlsanity.com/paper/2306.03048/) (2023)
- [Disproving XAI Myths with Formal Methods -- Initial Results](https://mlsanity.com/paper/2306.01744/) (2023)
- [A New Class of Explanations for Classifiers with Non-Binary Features](https://mlsanity.com/paper/2304.14760/) (2023)
- [Abductive and Contrastive Explanations for Scoring Rules in Voting](https://mlsanity.com/paper/2408.12927/) (2024)
- [Query languages for neural networks](https://mlsanity.com/paper/2408.10362/) (2024)
- [Learning Actionable Counterfactual Explanations in Large State Spaces](https://mlsanity.com/paper/2404.17034/) (2024)
- [Uncovering Bugs in Formal Explainers: A Case Study with PyXAI](https://mlsanity.com/paper/2511.03169/) (2025)
- [Axiomatic Foundations of Counterfactual Explanations](https://mlsanity.com/paper/2602.04028/) (2026)
- [Binary Spiking Neural Networks as Causal Models](https://mlsanity.com/paper/2604.27007/) (2026)
- [Rule-based Classifier Models](https://mlsanity.com/paper/2505.00474/) (2025)
- [An Information-Flow Perspective on Explainability Requirements: Specification and Verification](https://mlsanity.com/paper/2509.01479/) (2025)
- [Efficient & Correct Predictive Equivalence for Decision Trees](https://mlsanity.com/paper/2509.17774/) (2025)
- [A Modal Logic for Temporal and Jurisdictional Classifier Models](https://mlsanity.com/paper/2510.13691/) (2025)

## Abstract (arXiv)

Recent years have witnessed a renewed interest in Boolean function in explaining binary classifiers in the field of explainable AI (XAI). The standard approach of Boolean function is propositional logic. We present a modal language of a ceteris paribus nature which supports reasoning about binary input classifiers and their properties. We study a family of classifier models, axiomatize it as two proof systems regarding the cardinality of the language and show completeness of our axiomatics. Moreover, we prove that satisfiability checking problem for our modal language is NEXPTIME-complete in the infinite-variable case, while it becomes polynomial in the finite-variable case. We furthermore identify an interesting NP fragment of our language in the infinite-variable case. We leverage the language to formalize counterfactual conditional as well as a variety of notions of explanation including abductive, contrastive and counterfactual explanations, and biases. Finally, we present two extensions of our language: a dynamic extension by the notion of assignment enabling classifier change and an epistemic extension in which the classifier's uncertainty about the actual input can be represented.