In today’s complex and data-rich environment, decision-making systems often need to effectively handle information that is uncertain, imprecise, and partially reliable. Traditional logic systems, built on binary values such as true or false, are inadequate for modeling such real-world complexities. Alt-hough fuzzy logic and probabilistic reasoning have made progress in representing imprecision and uncertainty, these approaches typically address these issues separately and cannot fully capture the dual nature of imperfect information -both its imprecision and degree of reliability.This gap has driven research into Z-numbers and Z-fuzzy relations; this approach unifies imprecision and reliability within a single mathematical framework. While theoretical foundations and practical applications exist in fields like control systems, medicine, decision making and data analysis, fully developed and system-atic methodologies for approximate reasoning based on Z-rules remain insufficient. Current approach-es often face limitations in handling bimodal information-based rule bases. Considering these chal-lenges, there is a need for new formal models that enable reasoning from Z-number-valued infor-mation, facilitating more human-like understanding and better management of uncertainty. The above highlights the analysis of existing fuzzy and probabilistic implication tools, the elimination of their shortcomings, and justifies the need to develop approximate reasoning methods based on Z- number conception.
Approximate reasoning plays a crucial role in decision-making and expert systems, especially in situations where information is imprecise and uncertain. In this context, fuzzy implication models serve as a key mechanism for deriving conclusions based on conditional rules. While classical logic expresses “if... then...” statements rigidly, fuzzy logic enables more flexible and human-like interpretation through fuzzy implications.
However, existing fuzzy implications - such as those proposed by Mamdani, Gödel, Lukasiewicz, Zadeh, Aliev, Reichenbach, Kleene-Dienes, Goguen, Yager, Weber, Fodor and others face limita-tions in practical applications. These implications consider only imprecision, but fail to adequately address important aspects such as the reliability or degree of confidence in the information.
Therefore, there is a growing need for new approaches that extend the functional capabilities of fuzzy implication by extending it to Z-number-based implications. Z-numbers allow for the simultaneous modeling of both the vagueness of information and its trustworthiness (reliability or confidence). This leads to a more expressive and realistic reasoning results.
The actuality of this work lies in the fact that it investigates the shortcomings of existing fuzzy implica-tion models, analyzes their current application potential, and justifies the development of new implica-tions based on Z-numbers. This represents a timely and significant research direction, both theoretical-ly and practically, for advancing modern fuzzy reasoning systems. The above-mentioned points de-termine the relevance of the work.
It should be noted that conditional reasoning method is the foundation of control systems and deci-sion-making systems. This article proposes a new Z-conditional reasoning method using fuzzy and probabilistic implication.