Neutrosopy is all about looking at the world with fresh eyes, and then tailoring the perspective to account for uncertainty. Neutrosophy offers a third logic alternative to the binary model of true or false, which goes by the name of neutrals. In summary, Neutrosophy replaces the binary method in logics by offering indeterminancy, which may also be interpreted as ambiguous, uncertain, or inconsistent. Neutrosophy was conceptualized by Smarandache et al. [1] in 1988, and development since then has rapidly grown with the use of logical extensions, such as measure, sets, graphs, and even practical applications in various areas. The field of neutrosophy has shown its power and effectiveness in a variety of contexts. This created a big backlog of contributions which were theoretical in nature and confirmed only using mathematical examples or restricted data sets. Neutrosophic logic could be used in both natural science and social science, and recently publications have been emphasized the use of the neutrosophic logic in social sciences. Neutrosophic sets are better than fuzzy sets for surveys because they provide a wider range of answers. Through its membership indeterminacy function, the former allows respondents to more clearly articulate their actual thoughts and feelings. Neutrosophy is beneficial to those who want to express themselves since it better captures their thoughts and emotions due to its embrace of indeterminacy and independent membership function of falsehood. Therefore the study’s primary goal is to use the principles of neutrosophy in social sciences, particularly in education and assessment and evaluation techniques of scale development [2–4].

The main purpose of the survey or scale development is to gather accurate and relevant data. In social sciences, the reliability and validity of scale and questionnaire formats are, therefore, used to enable to gather accurate and relevant data [5]. The data space and data range in this respect are essential parameters for developing scales since they often alter the data type, logical analytical space, methods, validity and reliability of the findings (Fig. 1).

For measurement instruments such as scales, data space refers to an independent collection of choices for a particular measurement item. For example, there is only one choice in every Likert-type scale that the individual may express his/her ideas or feelings and its data space is 1d, but there are three different dimensions about each aspect in the neutrosophic scale as undecided, agreeable, and disagreeable. The data space is 1d for any form of Likert type scale a whereas 3d for neutrosophic space. Such a extension can be done for more dimensions. For instance, the more qualitative-oriented measurement tools like providing items that require more free opinions in a paragraph like preferences are supposed to have more dimensions as well. Though n-dimensional space is more suitable for clearer and more accurate outcomes, the representation of the data in less dimensional spaces can easily be statistically analyzed. Besides, the measurement tool’s objectivity in terms of estimation of common features decreases as the dimension of space rises. The benefit of the 3-dimensional neutrosophic scale is that the participants are both involved in the degree of agreement, disagreement, and uncertainty. The difference among classical logic, fuzzy logic and neurosophic logic can be describes as in Fig. 2. In the classical logic the space is in 0 dimension where there is only discrete points as 0 and 1. In the fuzzy logic the spaces can be represented as 1 dimensional continuum or segment where there is a continuous possible admixture of the states of 0 and 1. Finally, in the Fuzzy logic there are three independent states constituting 3 dimensional logic space.

It should be noted that there is no study focusing on 2d data space in the literature because the possible combinations of the agreement, disagreement and intederminacy in the forms of two independent states such as (a, d), (a, i), (d, i). Such a 2d data space is very limited because it disregards indeterminany, agreement, disagreement dimesions. For instance if 2d scale having agreement, disagreement dimesions firstly ignore the indeterminancy dimension. Secondly, sometimes agreement, disagreement dimesions are complement to each other as in the case od classical logic or fuzzy logic but the indeterminancy is important for the analysis. Such an example can be extended into the all possible combinations of (a, d), (a, i), (d, i). The degree of freedom of 2d space may dismiss the other two parameters that cannot be ignored in the actual case. These hidden variables can lead to huge differences especially in the case of the analysis of the options of a huge number of participants and even this cannot be realized (Fig. 3).

However, in neutrosophic logic, it is impossible to dismiss three parameters since the researchers must give their opinions on them (Fig. 4).

In everyday life, humans are not confined within one dimension space in terms of the expressions of the agreement, disagreement and interdeterminancy dimensions. Neutrosophic logic is more compatible with this fact since the participants express in the three-dimensional neutrosophic space both their agreement and their contradictions and the ambiguity of the items or scale parameters. We often believe that a sentence is understood, but one term in the statement leaves us unsure if it is the ‘right message’ the source intends. We often approve of such proposals, but we sometimes disagree with the item only because of the source of the message itself. The neutrosophic scale is therefore distinct in terms of data space from the classical Likert scale (Fig. 5).

The second main point that differentiates between measuring tools is the range of the data that every scale is dependent on. The range of the data set is the difference from the highest value to the lowest value in any setting. Data may well be organized from 3 points in Likert form to 10 points on the Likert-type scale. The neutrosophic scale is, however, broader than the scales of such a Likert kind measurement tool. It contains all numbers ranging from 0 to 100. There are therefore continuous variable forms of neutrosophic scales, while Likert scales have discrete values in terms of rational numbers such that the data processing can differ. In this sense, this will help increase the sensitivity of the measuring instrument. This is actually what is called as neutrosophic Data in some recent researches is the piece of information that contains some indeterminacy. Similar to the classical statistics, it can be classified as [4]: –

Discrete neutrosophic data, if the values are isolated points.

The logic space of a measuring instrument is the third essential point. Logic space is important because “in any field of knowledge, each structure is composed from two parts: a space, and a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy of the form (t, i, f) ≠ (1, 0, 0), that structure is a (t, i, f)-Neutrosophic Structure” [6]. The logic we are focused on, Neutrosophic Logic [7], is an emergent field where the percentage (percentage) of truth in a T subset, the percentage of indeterminacy in an I subset, and the proportion of falsity in an F subset are listed. We here consider a subset of truth (or falsity or indeterminacy), rather than just a number, since in many situations we can not precisely determine the proportions of truth and falsity but we can only approach them. For example, suppose that a statement (or proposition) is between 22% and 43% true and 51% to 82% false; indeterminancy (undecided): 32% to 39% or 40% to 52% true (according to various observers) and 57% or 62% to 71% false. The subsets are not simple intervals but are arranged in line with the proposition (open or closed or semi-open/semi-closed intervals, isolated, constant, or intersected or united by previous sets, etc.). On the other hand, there are many ways to evaluate and interpret data. Some recent studies reveal important developments based on the interpretation and effective use of data [8,9].

Although in Likert-type scales, there are mostly three options as agreement, disagreement, and vagueness, classical logic is located one valued option located on the opposite sides of true and false values. The neutrosophic set has three independent components, giving more freedom for analysis so that it brings different logical operations as well. Therefore, the methodology of the analysis of the data should be changed based on the logical structure of the scale. For instance, while factor analysis is used for classical Likert-type scales, as shown in this paper, we can not directly assume that all the sub-dimensions of any neutrosophic scale directly correspond to the factor structure of the classical one. Nevertheless, it should be noted that classical analysis and methods can indeed be used for neutrosophic scales based on different analysis procedures. Hence, we can conclude that the validity and reliability of the measurement tools can change based on the logical structure of the scale. Therefore, in this study, the Five Facet Mindfulness Questionnaire which was adapted by [10] from the short form of the Five Facet Mindfulness Questionnaire (FFMQ) was evaluated and this scale into neutrosophic form was converted and the results of the scale were compared for proposing new type confirmatory analysis procedure as well as developing neutrosophic scales.

The comparison of the items in the agreement, disagreement, and vagueness dimensions in the neutrosophic and classic Five Facet Mindfulness Questionnaire gives us many clues about how the structure of any questionnaire, survey, or scale can change as the dimensions or more generally, their space change. Therefore, if we convert any classical scale into a neutrosophic one, we shouldn’t directly assume that all of the sub-dimensions of a neutrosophic scale as agree, disagree, and undecided have a similar factor structure to the classical one. This is an important point, because, for further analysis of the data, such a wrong assumption may lead to wrong conclusions since neutrosophic logic requires three independent truth values while classical one takes two dependent truth values (Fig. 9).

Interestingly, in the factor analysis of the neutrosophic scale, both classical and neutrosophic scales have the same factors, implying that the one-dimensional classical scale measures the agreement degree of the participants. When the factor analysis was conducted to disagreement and vagueness dimensions, it seemed that some factors were eliminated and even some new factors emerged, indicating that in human cognition those three dimensions can be taken as independent of each other, just as assumed by neutrosophic logic (Tab. 17).

The second important implication of the factor analysis is that the neutrosophic forms of any questionnaire can be used for the validity of the classical ones. Although it is not required that the dimensions of the neutrosophic forms of any questionnaire have the same or similar factors, since these different structures should be evaluated within their realms in terms of their structure, the classical forms of questionnaires can be checked based on neutrosophic forms. When Tab. 17 is examined, it is observed that some factors are eliminated on the neutrosophic scale while some of them stay in the same state. Since we used a valid and reliable scale having smaller items to check whether it is neutrosophic form can be used to evaluate it rather than a draft of a questionnaire having more items like 100 or 120 items, it seems that this scale is invalid, but for draft scales the similar procedure can be applied and more coherent scales having same factor structure in three dimensions with same items can be achieved so that items and dimensions can more sensitively measure the intended meaning of the items and factors. It is very similar to the Sieve of Eratosthenes, which is an ancient algorithm for finding the prime numbers up to any given limit where each prime is taken as an independent base or dimension and multiples of the selected prime in a given interval are eliminated until there are only prime numbers left.

Additionally, although it was said that this structure is deemed to be invalid for the general procedure, actually it is still used as a valid one because both factors, at least in two dimensions, were not eliminated. For instance, Act with Awareness (Factor 4) was eliminated in the disagreement dimension but it is still the same in two other dimensions as well, indicating that it has an approximately valid structure. Similar arguments can be made for items individually. For example, although Item 6 corresponds to the same structure in the classical one, indicating that it belongs to this factor, it changes its position in the other dimensions, possibly because of its dependence on other items in the realms of these two dimensions in the context of classical interdependent logic. Finally, the reliability of three independent dimensions of the neutrosophic forms of any questionnaire can also be used to check whether the measurement tool is reliable. Low-reliability results in any dimensions may imply that the scale has some problems in terms of meaning, language, or other factors.