Possibility theory, first coined by Zadeh [1], is an uncertainty theory that deals with incomplete knowledge. It is closely related to probability theory as it is based on set functions. However, it differs from probability theory by using two dual set functions (possibility and necessity measurements) instead of just one. In his paper, Zadeh relates the notion of possibility to fuzzy sets by describing a possibility distribution as an elastic constraint on variable values. His aim was to provide a conceptual framework based on fuzzy sets that would properly deal with the meaning of information, rather than its measure. He concluded that this analysis required a possibilistic point of view, rather than a probabilistic one.

Fuzzy sets reflect ambiguous notions by allowing elements to have different degrees of belonging to a set. To achieve this, each element is assigned a membership degree to the set, ranging from 0 to 1, resulting in a membership function. While the concept of fuzzy logic has proved useful in resolving ambiguities and imprecision, its fundamental shortcoming appears when dealing with scenarios that contain not only uncertainty but also indeterminacy, as well as the coexistence of truth, falsehood, and indeterminacy inside a single assertion.

Smarandache [2] suggested Neutrosophy as a new field of philosophy on many-valued logics that integrated non-standard analysis with a tri-component logic/set/probability theory. Neutrosophy advocates that every idea/concept/thesis etc. possesses a degree of truth, as well as falsehood and indeterminacy, which must be considered individually. In other words, an indeterminacy assignment is explicitly defined, conjointly and independently with truth and falsity assignments. As a consequence, he introduced the theory of neutrosophic logic (NL) as a generalization of many-valued logics since fuzzy logic is thought to be incapable of demonstrating indeterminacy by itself. In a more formal definition, NL is a logic in which each proposition is estimated to have the percentage of truth in a subset T, the percentage of indeterminacy in a subset I, and the percentage of falsity in a subset F, where T, I, F are called neutrosophic components which represent the truth, indeterminacy, and falsehood value respectively [3].

Modal logic was initially defined as the logic of necessary and possible facts. The possible world semantics provides an intuitive means for reasoning about situations. Within this framework, Kripke model [4] resembles a directed labelled graph whose graph's nodes represent potential worlds from a set labelled with truth assignments π.

The aim of this article is twofold. First, we believe that it is high time to reframe and reinforce the statement made by Zadeh [1] regarding the importance of possibility theory and how it could be best examined and explained. This theory is crucial because a significant portion of the information used for human decision-making is possibilistic in nature, due to the inherent fuzziness of natural languages. It is widely acknowledged that natural languages frequently exhibit ambiguity, imprecision, and multiple interpretations, which makes them ideal subjects for a neutrosophic treatment. By incorporating neutrosophic elements, these models can better capture the multifaceted nature of language and enhance their ability to handle ambiguity and uncertainty in human communication. Secondly, our target is to extend the results presented in [1] by employing Kripke structures for neutrosophic propositions as building blocks for an extended possibility theory, thus forming a powerful and flexible mathematical framework designed to handle indeterminacy, inconsistency, and uncertainty simultaneously. This is achieved due to the concept of neutrosophy which posses no restriction on the sum of the of the neutrosophic components (T,I,F) other than they are subsets of ]-0, 1+[, thus:

-0 ≤ inf T + inf I + inf F ≤ sup T + sup I + sup F ≤3+.

This non-restriction allows room for paraconsistent, dialetheism, and incomplete information to be characterized in NL.

It is our belief that in this integrated framework, the ability of neutrosophic logic to express and manage indeterminate information supplements the ability of possibility theory to handle uncertainty. This fusion enables a more complex representation of knowledge, accounting for not only variable degrees of possibility but also the underlying ambiguity or imprecision in the information. The resultant system offers a richer and more expressive framework for reasoning in circumstances that involve uncertainty and indeterminacy. This integration could find applications in diverse fields such as artificial intelligence, decision support systems, and information retrieval, where a comprehensive treatment of uncertainty and indeterminacy is crucial for accurate and reliable analysis.

The structure of the article is as follows: Section 2 presents the basic definitions and concepts of possibility theory, neutrosophic logic, and Kripke modal logic that are necessary to build upon and construct our proposed conceptual framework. Subsequently, Section 3 will present and explain our methodology, while Section 4 will provide a brief discussion on the significance of our findings. Finally, in Section 5, concluding remarks will be stated.

In this section, we provide the basic terminology used throughout the article. While our goal is to make the current manuscript as self-contained as possible, it is important to note that there is a wealth of literature on possibility theory [5–13]. neutrosophic logic [14–19], and Kripke modal logics[20–26] that interested readers can refer to.

In this section, we will present an illustrative hypothetical scenario from the field of medical diagnosis to showcase our proposed methodology.

The findings of this study demonstrate great promise for a variety of applications across multiple fields. The integration of neutrosophic logic with Kripke structures, as outlined in this study, enhances the ability of possibility theory to address uncertainty and indeterminacy. This approach shows particular potential in areas such as artificial intelligence, decision support systems, and information retrieval, where effectively managing ambiguity is essential. The results extend previous studies in both possibility theory, pioneered by Zadeh [1], [27], and neutrosophic logic, introduced by Smarandache [2] ,[3]. While Zadeh's possibility theory effectively handles uncertain information, it does not fully address indeterminacy. Neutrosophic logic, on the other hand, excels in managing this indeterminacy but lacks robust frameworks for reasoning about possibilities. The combination of these two approaches, enhanced by Kripke models, creates a more versatile and comprehensive framework for handling complex information. Finally, the suggested paradigm enhances the theoretical understanding of possibility and indeterminacy, while also opening up new opportunities for practical applications across various domains.

In the broader context, the suggested approach has implications across various fields. For example, in artificial intelligence, this hybrid framework can improve decision-making algorithms that need to operate under conditions of uncertainty and incomplete data. It can enhance the capacity of decision support systems to provide more accurate recommendations by incorporating not just degrees of uncertainty, but also the indeterminacy present in real-world data. Within the realm of medical diagnosis, this framework can be utilized to represent complex patient data, enabling healthcare providers to navigate unclear or incomplete information more effectively when diagnosing disorders. For instance, it can assist in refining diagnostic hypotheses by assigning degrees of truth, indeterminacy, and falsehood to different medical conditions, as demonstrated in the medical scenario outlined in this study.

In this paper, we analyzed the multifaceted environment of neutrosophic Kripke structures as formalism for modeling possibility theory, providing a novel and adaptable way to deal with uncertainty and ambiguity, primarily in the context of natural language. The combination of neutrosophic logic with Kripke structures has resulted in a robust framework capable of reflecting the complex interplay between possibility, necessity, and indeterminacy.

Our investigation began with a concise yet comprehensive study of key features that characterize possibility theory and neutrosophic logic, emphasizing its potential to describe triadic links between truth, indeterminacy, and falsity. Next, we integrated Kripke structures, a conceptual tool normally associated with modal logic, into the neutrosophic paradigm. The synergy between these two formalisms not only extended the expressive power of neutrosophic logic but also facilitated the representation of dynamic and evolving systems, enriching the scope of possibility theory.

By providing examples, we aimed to demonstrate the potential influence and practical usefulness of this innovative formalism. Our findings demonstrated the ability of neutrosophic Kripke structures to handle instances where classical logic fails, such as those characterized by inadequate knowledge, contradicting evidence, or dynamic changes over time. The capacity to express and reason about possibilities within a neutrosophic Kripke framework allows academics and practitioners to approach real-world problems with greater flexibility and accuracy.

In this article, we have studied the integration of neutrosophic logic and Kripke models with possibility theory. Our work can be seen as an initial and innovative step that could stimulate new insights in this direction, as there is a lack of relevant literature on this topic. For this reason, we aimed to examine the fundamental properties of such an integrated theory. As a result, possible future work could focus on improving and enhancing neutrosophic Kripke structures as formalism for possibility theory. This includes resolving possible constraints, investigating new aspects of neutrosophic logic, and designing algorithms for tasks such as reasoning and decision-making. Domain-specific research may assess the formalism's effectiveness in fields such as healthcare and finance, while integrating it with machine learning approaches may improve model resilience.

Comparative research with other formalisms, such as fuzzy logic and dempster-shafer theory, would provide information about relative strengths. Extending the application to dynamic systems and temporal reasoning, as well as conducting thorough case studies and empirical validations, will help demonstrate the real-world usefulness of the proposed formalism. Moreover, developing algorithms to automate this framework could enhance its applicability in real-time decision-making systems. Potential applications could also be explored in finance, risk management, and cognitive computing, where nuanced treatments of uncertainty are crucial. Finally, supporting multidisciplinary cooperation can improve our knowledge of neutrosophic Kripke structures and encourage their use across fields. Pursuing these pathways will contribute to the continued progress of the formalism, guaranteeing its relevance and application in tackling a wide range of uncertainty and possibility-related concerns.

Antonios Paraskevas conceptualized the study, developed the theoretical framework for Neutrosophic Possibility Theory, and explored its modal perspectives and practical applications. He performed the literature review, conducted the analysis, and prepared the manuscript for publication.

This research received no external funding and was supported solely by the author’s resources.

The paper is theoretical in nature, and no empirical data were used. All relevant details and derivations are included in the manuscript.

The author declares no conflicts of interest concerning the publication of this work.