In this study, the Five Facet Mindfulness Questionnaire which was adapted from the short form of the Five Facet Mindfulness Questionnaire 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 exploratory factor analysis was used in the analysis of the data. Besides, test results were analyzed for Kaiser–Meyer–Olkin and Bartlett values, common factor variance values, scree plot graphs, and the principal component analysis results. The sample of the study consists of 194 students in mathematics departments at Bitlis Eren University and Iğdır University in Turkey by convenience sampling method. A convenience sampling is a kind of non-probability sampling procedure in which the sample is obtained from a group of individuals easily accessible or reachable. The convenience sampling method was chosen in this study because the study aims to examine the structure of the measurement tool rather than the psychological characteristics of a particular population. First of all, it is observed that if any classical scale can be converted into a neutrosophic one. It is observed that the sub-dimensions of a neutrosophic scale as agree, disagree, and undecided might not have a similar factor structure to the classical one. Interestingly, in the factor analysis of the neutrosophic scale, both classical and the agreement part of the 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. The another important implication of the factor analysis is that the neutrosophic forms of any questionnaire can be used for the validity of the classical ones. Loads of items or their accumulation into factors are compared to the classical scale and the three-dimensional neutrosophic scale in the factor, so that the corresponding ones in the same factors and the items or factors that do not correspond to each other are eliminated. It is very similar to the Sieve of Eratosthenes, which is an ancient algorithm for finding prime numbers up to any given limit where each prime is taken as an independent base or dimension and multiples of the selected primes in a given interval are eliminated until there are only prime numbers left. 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.

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. [

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 [

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

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 (

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

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 (

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 [

Discrete neutrosophic data, if the values are isolated points.

Continuous neutrosophic data, if the values form one or more intervals.

Quantitative (numerical) neutrosophic data; for example: a number in the interval [

Qualitative (categorical) neutrosophic data; for example: blue or red (we do not know exactly), white, black or green or yellow (not knowing exactly).

The univariate neutrosophic data is a neutrosophic data that consists of observations on a neutrosophic single attribute.

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” [

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 [

Firstly, it is thought that a valid and reliable scale should be chosen that has appropriate psychometric properties such as its adequacy, relevance, and usefulness since we see the characteristics of the neutrosophic scale in the reliable and valid foundations. Otherwise, we must do the reliability and validity analysis for the neutrosophic scale, but we want to check our method based on a more solid context since there is not so much research on this subject. The exploratory factor analysis includes the determination and clustering of objects by researchers to measure the same characteristic and offers insights into the reliability of objects and the test [

This method determines the proportion of the total variation in given variables that is most likely to be caused by latent factors. If values are very close to 1.0, then one may benefit from doing a factor analysis on the data. The findings of the factor analysis are unlikely to be particularly relevant if the value is less than 0.50. To test the hypothesis that the correlation matrix is an identity matrix, Bartlett’s test can be used. Component component analysis is usually quite effective when one has small values at the significance level (less than 0.05) [

In the Factor Analysis [

The sample of the study consists of 194 students in various departments at Bitlis Eren University and Iğdır University in Turkey by convenience sampling method. A convenience sampling is a kind of non-probability sampling procedure in which the sample is obtained from a group of individuals easily accessible or reachable. The convenience sampling method was chosen in this study because the study aims to examine the structure of the measurement tool rather than the psychological characteristics of a particular population (

Age | Total | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

17,00 | 18,00 | 19,00 | 20,00 | 21,00 | 22,00 | 23,00 | 24,00 | 25,00 | 26,00 | 27,00 | 41,00 | ||

Gender | |||||||||||||

Female | 1 | 24 | 42 | 45 | 15 | 7 | 6 | 0 | 3 | 1 | 1 | 0 | 145 |

Male | 0 | 3 | 10 | 21 | 7 | 2 | 1 | 3 | 0 | 1 | 0 | 1 | 49 |

Total | 1 | 27 | 52 | 66 | 22 | 9 | 7 | 3 | 3 | 2 | 1 | 1 | 194 |

Before doing to assess the suitability of the data for the factor analysis, two methodological measures are used. KMO and Bartlett’s test are used for this [

KMO and Bartlett’s test | ||
---|---|---|

Kaiser–Meyer–Olkin measure of sampling adequacy | 0.816 | |

Bartlett’s test of sphericity | Approx. Chi-square | 1223.922 |

Df | 190 | |

Sig. | 0.000 |

After assessing if the data was appropriate for factor analysis, the data are evaluated for an exploratory factor to evaluate the factor structure in the scale. The first analysis showed that five factors had an eigenvalue of 1 and higher, which explains the total variance of 46,283 points as given in

Total variance explained | |||||||
---|---|---|---|---|---|---|---|

Factor | Initial eigenvalues | Extraction sums of squared loadings | Rotation sums of squared loadings^{a} |
||||

Total | % of variance | Cumulative % | Total | % of variance | Cumulative % | Total | |

1 | 5.011 | 25.054 | 25.054 | 4.476 | 22.379 | 22.379 | 3.167 |

2 | 2.592 | 12.961 | 38.016 | 1.851 | 9.257 | 31.636 | 2.319 |

3 | 1.573 | 7.866 | 45.881 | 1.181 | 5.903 | 37.538 | 2.879 |

4 | 1.431 | 7.157 | 53.038 | .899 | 4.497 | 42.035 | 2.949 |

5 | 1.217 | 6.087 | 59.126 | .849 | 4.247 | 46.283 | 1.375 |

6 | .966 | 4.829 | 63.954 | ||||

7 | .877 | 4.384 | 68.339 | ||||

8 | .813 | 4.066 | 72.405 | ||||

9 | .664 | 3.321 | 75.726 | ||||

10 | .652 | 3.259 | 78.985 | ||||

11 | .618 | 3.089 | 82.074 | ||||

12 | .552 | 2.762 | 84.836 | ||||

13 | .496 | 2.481 | 87.318 | ||||

14 | .453 | 2.263 | 89.581 | ||||

15 | .394 | 1.971 | 91.552 | ||||

16 | .390 | 1.952 | 93.504 | ||||

17 | .365 | 1.826 | 95.329 | ||||

18 | .346 | 1.729 | 97.059 | ||||

19 | .313 | 1.563 | 98.622 | ||||

20 | .276 | 1.378 | 100.000 |

Notes: Extraction method: maximum likelihood. ^{a}When factors are correlated, sums of squared loadings cannot be added to obtain a total variance.

The Scree plot was also examined to determine how many factors the scale consists of. The Scree plot is given in

The research was then carried out using the direct oblique rotation method. The data then proceeded. Researchers use the oblique rotation technique since the relations between factors exist [

Pattern matrix^{a} |
|||||
---|---|---|---|---|---|

Factor | |||||

1 | 2 | 3 | 4 | 5 | |

v15aIagree | 0.778 | ||||

v14aIagree | 0.759 | ||||

v13aIagree | 0.756 | ||||

v16aIagree | 0.496 | ||||

v19aIagree | 0.722 | ||||

v18aIagree | 0.706 | ||||

v17aIagree | 0.629 | ||||

v20aIagree | 0.399 | ||||

v5aIagree | −0.902 | ||||

v7aIagree | −0.606 | ||||

v6aIagree | −0.406 | ||||

v8aIagree | −0.402 | ||||

v9aIAgree | 0.773 | ||||

v12aIagree | 0.665 | ||||

v10aagree | 0.530 | ||||

v11aIagree | 0.493 | ||||

v1aIagree | 0.594 | ||||

v2aIagree | 0.480 | ||||

v3aIagree | 0.450 | ||||

v4aIagree | 0.448 |

Notes: Extraction method: maximum likelihood. Rotation method: Oblimin with Kaiser normalization.

^{a}Rotation converged in 9 iterations.

When the factor structure for agreement dimension is compared to the original classic Five Facet Mindfulness Questionnaire (FFMQ), it is observed that all items are directly correlated with the same dimensions of the original classic FFMQ (

Act with awareness | Nonjudge items | Nonreact items | Observe | Describe |
---|---|---|---|---|

(Factor 4) | (Factor 1) | (Factor 2) | (Factor 5) | (Factor 3) |

9* | 13* | 17 | 1 | 5* |

10* | 14* | 18 | 2 | 6 |

11* | 15* | 19 | 3 | 7* |

12* | 16* | 20 | 4 | 8 |

Reliability statistics show that the structure and assessment are highly reliable since reliability refers not only to the instrument itself but also to assessments obtained with a measurement tool [

Reliability statistics | |
---|---|

Cronbach’s alpha | N of items |

0.829 | 18 |

KMO and Bartlett’s test for disagreement dimension shows that data is suitable for data factor analysis (KMO = 740,

KMO and Bartlett’s test | ||
---|---|---|

Kaiser–Meyer–Olkin measure of sampling adequacy | 0.731 | |

Bartlett’s test of sphericity | Approx. Chi-square | 947.066 |

Df | 153 | |

Sig. | 0.000 |

After assessing if the data was appropriate for factor analysis, the data are evaluated for an exploratory factor to evaluate the factor structure in the scale. The first analysis showed that four factors had an eigenvalue of 1 and higher, which explains the total variance of 41.035 points as given in

Total variance explained | |||||||
---|---|---|---|---|---|---|---|

Factor | Initial eigenvalues | Extraction sums of squared loadings | Rotation sums of squared loadings^{a} |
||||

Total | % of variance | Cumulative % | Total | % of variance | Cumulative % | Total | |

1 | 3.699 | 20.548 | 20.548 | 3.115 | 17.305 | 17.305 | 2.461 |

2 | 2.928 | 16.265 | 36.814 | 2.424 | 13.468 | 30.773 | 2.138 |

3 | 1.543 | 8.571 | 45.384 | 1.033 | 5.740 | 36.514 | 2.025 |

4 | 1.381 | 7.670 | 53.054 | 0.814 | 4.522 | 41.035 | 2.413 |

5 | 0.999 | 5.552 | 58.606 | ||||

6 | 0.979 | 5.440 | 64.046 | ||||

7 | 0.872 | 4.846 | 68.893 | ||||

8 | 0.737 | 4.094 | 72.987 | ||||

9 | 0.716 | 3.978 | 76.965 | ||||

10 | 0.684 | 3.801 | 80.767 | ||||

11 | 0.646 | 3.587 | 84.353 | ||||

12 | 0.563 | 3.130 | 87.484 | ||||

13 | 0.512 | 2.842 | 90.326 | ||||

14 | 0.417 | 2.318 | 92.644 | ||||

15 | 0.405 | 2.250 | 94.893 | ||||

16 | 0.368 | 2.046 | 96.940 | ||||

17 | 0.305 | 1.693 | 98.633 | ||||

18 | 0.246 | 1.367 | 100.000 |

Notes: Extraction method: maximum likelihood. ^{a}When factors are correlated, sums of squared loadings cannot be added to obtain a total variance.

Scree plot was also examined in order to determine how many factors the scale consists of. Scree plot is given in

While one out of 20 items was removed on the draft scale and the analysis for the other 19 items was repeated. As a consequence of the study, four factors were taken into account for the remaining 19 items on the scale (

Pattern matrix^{a} |
||||
---|---|---|---|---|

Factor | ||||

1 | 2 | 3 | 4 | |

v7cIDİİSagree | 0.702 | |||

v5cIDİİSagree | 0.682 | |||

v8cIdisagree | 0.498 | |||

v9cIDİİSagree | 0.490 | |||

v12cIDİİSaagree | 0.323 | |||

v11cIDİİSaagree | 0.309 | |||

v18cIdisagree | 0.833 | |||

v19cIdisagree | 0.638 | |||

v17cIdisagree | 0.581 | |||

v20cdisagree | 0.442 | |||

v3bIdisagree | 0.808 | |||

v2cIdisagree | 0.674 | |||

v6cIdisagree | 0.457 | |||

v4cIdisagree | 0.436 | |||

v15cIDİİSaagree | 0.788 | |||

v14cIDİİSaagree | 0.759 | |||

v13cIDİİSaagree | 0.667 | |||

v16cIDİİSaagree | 0.439 |

Notes: Extraction method: maximum likelihood. Rotation method: Oblimin with Kaiser normalization.

^{a}Rotation converged in 10 iterations.

When the factor structure for disagreement dimension is compared to the original classic FFMQ, it is observed that all Item 6 is removed from the Describe dimension and it moved to the Observe dimension. It is observed that Items 9, 11, 12 were removed from the Act with Awareness they moved into the Describe. Additionally, the factor is eliminated because no items are accumulated there. It seems that as the dimension of the classical scale has changed, the general structure of the scale has also changed (

Act with awareness | Nonjudge items | Nonreact items | Observe | Describe |
---|---|---|---|---|

(Factor 4) eliminated factor | (Factor 4) | (Factor 2) | (Factor 3) | (Factor 1) |

9* | 13* | 17 | 6 (moved there) | 5* |

10* | 14* | 18 | 2 | 6 |

11* | 15* | 19 | 3 | 7* |

12* | 16* | 20 | 4 | 8 |

9 (moved there) | ||||

11 (moved there) | ||||

12 (moved there) |

Reliability statistics show that the structure and assessment are regarded as reliable (

Reliability statistics | |
---|---|

Cronbach’s alpha | N of items |

0.722 | 18 |

KMO and Bartlett’s test for uncertainty dimension shows that data is suitable for data factor analysis (KMO = 0.891,

As for the last analysis, KMO and Bartlett’s test for uncertainty dimension shows that data is suitable for the data for the factor analysis (KMO = 0.879,

KMO and Bartlett’s test | ||
---|---|---|

Kaiser–Meyer–Olkin measure of sampling adequacy | 0.879 | |

Bartlett’s test of sphericity | Approx. Chi-square | 926,646 |

Df | 91 | |

Sig. | 0.000 |

After assessing if the data was appropriate for factor analysis, the data are evaluated for an exploratory factor to evaluate the factor structure in the scale. The first analysis showed that three factors had an eigenvalue of 1 and higher, which explains the total variance of 44,498 points (

Total variance explained | |||||||
---|---|---|---|---|---|---|---|

Factor | Initial eigenvalues | Extraction sums of squared loadings | Rotation sums of squared loadings^{a} |
||||

Total | % of variance | Cumulative % | Total | % of variance | Cumulative % | Total | |

1 | 5.446 | 38.901 | 38.901 | 4.881 | 34.864 | 34.864 | 3.811 |

2 | 1.198 | 8.557 | 47.458 | 0.727 | 5.190 | 40.054 | 3.747 |

3 | 1.102 | 7.873 | 55.331 | 0.622 | 4.444 | 44.498 | 3.322 |

4 | 0.924 | 6.602 | 61.933 | ||||

5 | 0.888 | 6.343 | 68.276 | ||||

6 | 0.725 | 5.181 | 73.457 | ||||

7 | 0.639 | 4.562 | 78.019 | ||||

8 | 0.619 | 4.425 | 82.443 | ||||

9 | 0.532 | 3.799 | 86.243 | ||||

10 | 0.499 | 3.563 | 89.805 | ||||

11 | 0.400 | 2.860 | 92.666 | ||||

12 | 0.372 | 2.654 | 95.320 | ||||

13 | 0.351 | 2.506 | 97.826 | ||||

14 | 0.304 | 2.174 | 100.000 |

Notes: Extraction method: maximum likelihood. ^{a}When factors are correlated, sums of squared loadings cannot be added to obtain a total variance.

The scree plot was also examined to determine how many factors the scale consists of. The Scree plot is given in

After rotation, no items were deleted from the scale with a factor load value of under 0.30 and overlaps of more than one factor. As a consequence of the study, three factors were taken into account for the remaining 13 items on the scale (

Pattern matrix^{a} |
|||
---|---|---|---|

Factor | |||

1 | 2 | 3 | |

v1bluncertain | 0.741 | ||

v4bIuncertain | 0.598 | ||

v15bIuncertain | 0.530 | ||

v13buncertain | 0.447 | ||

v3bIuncertain | 0.423 | ||

v2bIuncertain | 0.420 | ||

v18bIuncertain | |||

v12bIuncertain | −0.805 | ||

v11bIuncertain | −0.746 | ||

v9bIuncertain | −0.555 | ||

v10bIuncertain | −0.494 | ||

v5bIuncertain | −0.782 | ||

v8bIuncertain | −0.641 | ||

v7bIuncertain | −0.510 |

Notes: Extraction method: maximum likelihood. Rotation method: Oblimin with Kaiser Normalization.

^{a}Rotation converged in 9 iterations.

When the factor structure for uncertainty dimension is compared to the original classic Five Facet Mindfulness Questionnaire, it is observed that it seems that Observe is partially merged with Nonjudge items. Therefore, Nonjudge items are eliminated. Act with Awareness is originally correlated with Factor 2 so that it does not change its position. Nonreact factor items are also eliminated because they have no items corresponding to the original classical scale. Factor 3 corresponds to Describe except Item 6 because it was removed from there (

Act with awareness | Nonjudge items | Nonreact items | Observe | Describe |
---|---|---|---|---|

(Factor 2) | eliminated factor | eliminated factor | (Factor 1) | (Factor 3) |

9* | 13* | 17 | 1 | 5* |

10* | 14* | 18 | 2 | 6 |

11* | 15* | 19 | 3 | 7* |

12* | 16* | 20 | 4 | 8 |

13* (moved there) | ||||

15* (moved there) |

Reliability statistics show that the structure and assessment are regarded as reliable (

Reliability statistics | |
---|---|

Cronbach’s alpha | N of Items |

0.875 | 13 |

Agreement |
Act with |
Nonjudge items | Nonreact items | Observe | Describe |
---|---|---|---|---|---|

(Factor 4) | (Factor 1) | (Factor 2) | (Factor 5) | (Factor 3) | |

9* | 13* | 17 | 1 | 5* | |

10* | 14* | 18 | 2 | 6 | |

11* | 15* | 19 | 3 | 7* | |

12* | 16* | 20 | 4 | 8 | |

Disaggrement |
Act with awareness |
Nonjudge items |
Nonreact items |
Observe |
Describe |

9* | 13* | 17 | 6 (moved there) | 5* | |

10* | 14* | 18 | 2 | 6 | |

11* | 15* | 19 | 3 | 7* | |

12* | 16* | 20 | 4 | 8 | |

9 (moved there) | |||||

11 (moved there) | |||||

12 (moved there) | |||||

Uncertainty | Act with |
Nonjudge items |
Nonreact items |
Observe | Describe |

9* | 13* | 17 | 1 | 5* | |

10* | 14* | 18 | 2 | 6 | |

11* | 15* | 19 | 3 | 7* | |

12* | 16* | 20 | 4 | 8 | |

13* (moved there) | |||||

15* (moved there) |

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 (

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 (

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

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.

The authors would like to thank the editor and anonymous referees who kindly reviewed the earlier version of this manuscript and provided valuable suggestions and comments.