Figure 1: Fourteen queries used in the experiments. They do not contain personally identifiable information or offensive content. All the models are implemented in Pytorch [5] and based on the official implementation of BETAE [6]2 for a fair comparison. Forall the modules using multi-layer perceptron (MLP), we use a three-layer MLP with 1600 hidden neurons and ReLU activation. We apply dropout to the min function inCardMin and search the dropout rate in{0.05,0.10,0.15,0.20}.

One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty.

Query embedding (QE)---which aims to embed entities and first-order logical (FOL) queries in low-dimensional spaces---has shown great power in multi-hop reasoning over knowledge graphs. Recently, embedding entities and queries with geometric shapes becomes a promising direction, as geometric shapes can naturally represent answer sets of queries and logical relationships among them. However, existing geometry-based models have difficulty in modeling queries with negation, which significantly limits their applicability. To address this challenge, we propose a novel query embedding model, namely \textbf{Con}e \textbf{E}mbeddings (ConE), which is the first geometry-based QE model that can handle all the FOL operations, including conjunction, disjunction, and negation. Specifically, ConE represents entities and queries as Cartesian products of two-dimensional cones, where the intersection and union of cones naturally model the conjunction and disjunction operations. By further noticing that the closure of complement of cones remains cones, we design geometric complement operators in the embedding space for the negation operations. Experiments demonstrate that ConE significantly outperforms existing state-of-the-art methods on benchmark datasets.

We present a mechanistic interpretability study of GPT-2 that causally examines how sentiment information is processed across its transformer layers. Using systematic activation patching across all 12 layers, we test the hypothesized two-stage sentiment architecture comprising early lexical detection and mid-layer contextual integration. Our experiments confirm that early layers (0-3) act as lexical sentiment detectors, encoding stable, position specific polarity signals that are largely independent of context. However, all three contextual integration hypotheses: Middle Layer Concentration, Phenomenon Specificity, and Distributed Processing are falsified. Instead of mid-layer specialization, we find that contextual phenomena such as negation, sarcasm, domain shifts etc. are integrated primarily in late layers (8-11) through a unified, non-modular mechanism. These experimental findings provide causal evidence that GPT-2's sentiment computation differs from the predicted hierarchical pattern, highlighting the need for further empirical characterization of contextual integration in large language models.

Large Language Models (LLMs) have transformed natural language processing and hold growing promise for advancing science, healthcare, and decision-making. Yet their training paradigms remain dominated by affirmation-based inference, akin to \textit{modus ponens}, where accepted premises yield predicted consequents. While effective for generative fluency, this one-directional approach leaves models vulnerable to logical fallacies, adversarial manipulation, and failures in causal reasoning. This paper makes two contributions. First, it demonstrates how existing LLMs from major platforms exhibit systematic weaknesses when reasoning in scientific domains with negation, counterexamples, or faulty premises \footnote{Code to recreate these experiments are at https://github.com/hannahdavidsoncollege-maker/ScientificReasoningForEnvironment-MedicineWithLLMs. Second, it introduces a dual-reasoning training framework that integrates affirmative generation with structured counterfactual denial. Grounded in formal logic, cognitive science, and adversarial training, this training paradigm formalizes a computational analogue of ``denying the antecedent'' as a mechanism for disconfirmation and robustness. By coupling generative synthesis with explicit negation-aware objectives, the framework enables models that not only affirm valid inferences but also reject invalid ones, yielding systems that are more resilient, interpretable, and aligned with human reasoning.