Examining logical inconsistencies among multiple statements (such as collections of sentences or question-answer pairs) is a crucial challenge in machine learning, particularly for ensuring the safety and reliability of models. Traditional methods that rely on pairwise comparisons often fail to capture inconsistencies that only emerge when more than two statements are evaluated collectively. To address this gap, we introduce the task of set-consistency verification, an extension of natural language inference (NLI) that assesses the logical coherence of entire sets rather than isolated pairs. Building on this task, we present the Set-Consistency Energy Network (SC-Energy), a novel model that employs a contrastive loss framework to learn the compatibility among a collection of statements. Our approach not only efficiently verifies inconsistencies and pinpoints the specific statements responsible for logical contradictions, but also significantly outperforms existing methods including prompting-based LLM models. Furthermore, we release two new datasets: Set-LConVQA and Set-SNLI for set-consistency verification task.

Neural networks (NN) perform well in diverse tasks, but sometimes produce nonsensical results to humans. Most NN models "solely" learn from (input, output) pairs, occasionally conflicting with human knowledge. Many studies indicate injecting human knowledge by reducing output constraints during training can improve model performance and reduce constraint violations. While there have been several attempts to compare different existing algorithms under the same programming framework, nonetheless, there has been no previous work that categorizes learning algorithms with output constraints in a unified manner. Our contributions are as follows: (1) We categorize the previous studies based on three axes: type of constraint loss used (e.g. probabilistic soft logic, REINFORCE), exploration strategy of constraint-violating examples, and integration mechanism of learning signals from main task and constraint. (2) We propose new algorithms to integrate the information of main task and constraint injection, inspired by continual-learning algorithms. (3) Furthermore, we propose the $H\beta$-score as a metric for considering the main task metric and constraint violation simultaneously. To provide a thorough analysis, we examine all the algorithms on three NLP tasks: natural language inference (NLI), synthetic transduction examples (STE), and semantic role labeling (SRL). We explore and reveal the key factors of various algorithms associated with achieving high $H\beta$-scores.