# PAL-VMEA: A Novel Method for Enhancing Decision-Making Consistency in Maintenance Management
## 3 Paraconsistent Annotated Logic (PAL)  
Indeterminations, ambiguities, contradictions, or inconsistencies are often present in real-world situations, where Classical Logic, being binary (two truth values: true and false), can prove ineffective in solving problems. In contrast, PL is classified as a family of nonclassical logic, presenting a larger set of truth values, being, therefore, a better solution to deal with problems related to nonclassical logical systems that can present contradictions [25; 40; 41]. PL's are nontrivial and nonclassical logics and their origins are traced back to the first systematic studies that deal with the possibility of rejecting the principle of noncontradiction of classical logic [41]. The first studies on PL were developed in 1958, when hierarchical systems were proposed. In 1987, the annotated logic was developed to provide a basis for PL programming, which evolved into PAL, a more robust PL for practical application in decision-making processes [25; 42; 43; 44; 45]. An important characteristic of PL is that this nonclassical logic has found numerous applications in philosophy, quantum mechanics, artificial intelligence, traffic control, medicine, economics, finance, and computing, opening new directions for research in philosophy, science, and technology [46]. In fact, when conducting a search in the Web of Science database (May, 2020) about application studies of PL, the authors found 601 articles from several application areas, as presented in Figure 1. In this brief bibliometric analysis, the authors searched for journal articles, conference proceedings, and review articles with the following search algorithm: \"Annotated Paraconsistent Logic\" OR \"Paraconsistent Logic\" OR \"Paracomplete\" OR \"Annotated Logic\" OR \"Paracompleteness Logic\" OR \"Paraconsistency\" Logic\". The articles related to the fields of industrial engineering, management, and operations Research management science are primarily focused on the application of PAL in product quality decision-making, logistics, environmental sciences, and behavioral situations, among others. Studies carried out in the areas of electrical and electronics engineering, automation, and computer science applied, for the most part, the PAL in the development of algorithms for automation and control of industrial systems and/or algorithms for data mining (for analysis of distributed and often inconsistent databases). The PL's were also applied to examine inconsistencies from various sources in the creation and treatment of databases and knowledge bases, i.e., application of PL in data mining to improve the veracity of information [47]. In the field of physics, [PERSON] and [PERSON] [41] present a formal paraconsistent system and a corresponding intended interpretation, according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where the evidence for proposition \"A\" is understood as reason for believing that \"A\" is true. The article defines a paraconsistent and paracomplete natural deduction system, called Basic Evidence Logic (BEL), and extends it to Logic of Evidence and Truth (LETI). [PERSON] and [PERSON] [42] presented a study on the understanding of physical \"superpositions\" that exist in both classical and quantum physics. The authors presented a paraconsistent approach to quantum superpositions, which tries to explain the contradictory properties present in the interpretation of the meaning of quantum superpositions, which considers the contradiction as a key element of the formal structure of the theory. In other fields of application, studies such as [PERSON], [PERSON], and [PERSON] [48], developed a basic structure of argument (means of interaction) for Extended Generalized Annotated Logical Programs (EGAP), providing the theory of semantics and dialectical proof and proving the solidity, integrity, and equivalence of semantics well-reasoned. [PERSON] et al.