# Three Types of Negation of Triple and its Elements and an Extension of Triple

arXiv: https://arxiv.org/abs/2609.08271 · HTML: https://mlsanity.com/paper/2609.08271/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 56 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [Knowledge Graphs](https://mlsanity.com/paper/2003.02320/) (2020)
- [A Review of Relational Machine Learning for Knowledge Graphs](https://mlsanity.com/paper/1503.00759/) (2015)
- [DRUM: End-To-End Differentiable Rule Mining On Knowledge Graphs](https://mlsanity.com/paper/1911.00055/) (2019)
- [Counterfactuals and Causability in Explainable Artificial Intelligence: Theory, Algorithms, and Applications](https://mlsanity.com/paper/2103.04244/) (2021)
- [Negative Statements Considered Useful](https://mlsanity.com/paper/2001.04425/) (2020)
- [Completeness, Recall, and Negation in Open-World Knowledge Bases: A Survey](https://mlsanity.com/paper/2305.05403/) (2023)
- [A Survey of RDF Data Management Systems](https://mlsanity.com/paper/1601.00707/) (2016)
- [Can large language models generate salient negative statements?](https://mlsanity.com/paper/2305.16755/) (2023)
- [Three Kinds of Negation in Knowledge and Their Mathematical Foundations](https://mlsanity.com/paper/2505.24422/) (2025)
- [Negation in SPARQL](https://mlsanity.com/paper/1603.06053/) (2016)
- [A Minimal Deductive System for RDFS with Negative Statements](https://mlsanity.com/paper/2202.13750/) (2022)

## Abstract (arXiv)

In various data models, the classical triple is a typical semantic data model. However, due to the design of the triple as a simple structure for representing positive assertions, it cannot sufficiently express different forms of negation present in the triple and its elements. This paper conceptually proposes that there are three distinct forms of negation within triples and their elements: contradictory negation, opposite negation and intermediary negation. Based on the the set SCOI and the logic LCOI+PLCOI with three kinds of negation, we propose an extension of triple that can distinguish and express these three different negations in the triple and its elements, called the TCOI triple with contradictory negation, opposite negation and intermediary negation. The TCOI triple is a semantic and structural extension of the classical triple. While retaining the ability to express positive assertions, it systematically introduces the three semantic dimensions of three negations, allowing these negations to independently act on the elements of the triple and on the whole triple. This significantly enhances the triple model capability to represent and reasoning about complex negative information. This paper also explores the expressive power and reasoning of the TCOI triple, as well as the application of TCOI triple implication reasoning in counterfactuals and counterfactual reasoning. We propose a truth-value (continuous value) algorithm for TCOI triple implication reasoning and perform its calculation through an example of the counterfactuals and counterfactual reasoning.