According to Martins and Martins (2016) and Cavalcanti et al. (2013) , it was Aristotle (384 - 322 BC), a Greek philosopher and the founder of the Logic Science, who instituted strict rules to accept premises as logically valid. From this starting point, values were attributed to the statements, considering them as true or false. It was the beginning of Boolean logic, later named binary.

Campos Filho (2004) states that Boole, in 1847, in his book “The Mathematical Analysis of Logic”, applied numerical values for the statements, with 1 (one) being the true premises, and 0 (zero) being the false premises. For Sousa and Boente (2016), since then logical thinking has been conditioned to binary logic, in which a sentence is true or false, not admitting to be partially true or partially false.

Fuzzy logic had its origin in 1903, when Bertrand Russell discovered an ancient Greek paradox, known as “Russell’s paradox”, which, when solved by Aristotelian logic, always led to contradictions. It was the fuzzy logic, which accepts partial situations, which could solve the problem. In the mid-1930s, the Polish Jan Lukasiewicz made the first multilevel essays, in which he stated that contradictions were perfectly plausible from a mathematical point of view, as long as the degrees of truth were not bivalent. The Polish scholar incorporated the fraction of logical states between 0 (zero) and 1 (one) into the logic (Campos Filho, 2004; Brayan & Brayan, 1997) .

But it was in the 60s that the fuzzy logic was mathematically equated and tested by Zadeh, a professor at the University of Berkeley (California-USA), who was unsatisfied with the technological resources available for the automation demands of the time. Zadeh realized that some industrial sectors, such as Biological and Chemical, were (and still are) susceptible to ambiguous occurrences that could not be fulfilled by the 1 & 0 values of the binary (Boolean) logic (Cavalcanti et al., 2013).

Fuzzy logic followed the traditional course of technological innovations: born in the United States, perfected in Europe and widespread in Japan. In the 1980s, fuzzy logic was used especially in Europe to support decision making and was applied to information analysis on research about human decisions process (Campos Filho, 2004) . According to Toledo & Cosenza (2004) , some industrial processes involve the human reasoning and it is safer to use of a tool that efficiently substitutes human in decision making. Still in the 1980s, according to Campos Filho (2004) , companies in Europe found that the Japanese made it feasible to use the fuzzy logic in control technologies, and expanded the efforts to promote this logic in their applications.

To determine which soft skills are significant for the Engineering training, a Systematic Bibliographic Review (SBR) was conducted. To develop the SBR, the methodology developed by Pagani, Kovaleski and Resende (2015) , called Methodi Ordinatio was chosen due to its consistency and results obtained, as can be seen in Campos et al. (2018) , among others authors. The methodology steps are detailed below.

Step 1 and 2—Establishing the intention of research and preliminary search: the intention of this research is to understand the relation between two huge research areas: Humanities and Engineering. Therefore, for each of these two areas, two fronts of research were defined: Human Sciences and Human Skills for the first one, and Engineering Education and Graduate to the second one. The structure of these combinations is illustrated in Figure 2.

Steps 2 and 3—Definition of keywords and preliminary explanatory search: Keywords were defined for each research front. These keywords were grouped and crossed, forming 20 combinations as follows:

· Humanities and Engineering Education

· Humanities and Engineering Teaching

· Humanities and Degree in Engineering

· Humanities and Alumni* Engineering

· Humanities and Former Engineering Student*

· Human* Science* and Engineering Education

· Human* Science* and Engineering Teaching

· Human* Science* and Degree in Engineering

· Human* Science* and Alumni* Engineering

· Human* Science* and Former Engineering Student*

· Human* Skill* and “Engineering Education”

· Human* Skill* and Engineering Teaching

· Human* Skill* and Degree in Engineering

· Human* Skill* and Alumni* Engineering

· Human* Skill* and Former Engineering Student*

· Soft Skill* and Engineering Education

· Soft Skill* and Engineering Teaching

· Soft Skill* and Degree in Engineering

· Soft Skill* and Alumni* Engineering

· Soft Skill* and Former Engineering Student*

For the articles search, from these 20 combinations, the databases which demonstrated greater proximity to the research area were chosen, namely: Science Direct, Web of Science and Scopus. Although the European Journal of Engineering Education (EJEE) is indexed in the Scopus scientific base, it was also used as a reference for searching individual articles, due to its self-indexed scope. Furthermore, an analysis was made of the articles published from 2006 to 2015, from the Journal of Engineering Education (JEE), because it is the magazine with the greatest impact factor (2.638) within the Engineering Education area, and is not indexed in these data base.

Steps 4 and 5—Final search on databases and selected journals: After the first search with the 20 combinations, 3951 articles were obtained. The journals which were exclusively related to the health area (or very specific areas) were discarded. With this, 2788 articles remained.

As some articles were indexed in more than one databases, it was necessary to remove duplicates, which lead to 2638 articles.

The next step was to read the 2638 titles and abstracts to see if they were in line with the proposed theme. Of these, 335 were in line with the expectations of the study scope, in Appendix 7. The number of articles collected, the filters used e the final number of articles that compose de database of the present work are shown in Table 1.

From this point, categories were proposed to better analyze the articles. This categories were created based on the combination of two major areas of the Humanities field and two major areas of the Engineering field, as described in Table 2.

From this combination, 4 categories were obtained: Human Skills & Graduate; Human Skills & Engineering Education; Human Sciences & Engineering Education; Human Sciences & Graduates. Based on the scope of each of these 335 articles, they were classified into one of these 4 categories, as explained in Table 2.

Step 6—Identifying Impact factor, year and number of citations: The metrics of the article (Impact Factor) were obtained from Thomson’s Reuters/Clarivate Analytics website. The number of citations was obtained from Google Scholar. These information—metrics and number of citations, along with the year of publication—is necessary to calculate the InOrdinatio, which is explained in details in the Step 7.

Step 7—Ranking and selection of articles: In order to have a scientific support of which articles are the most relevant for the reading and elaboration of the theoretical framework, an algorithm for the relevance ranking after SBR was used, the Methodi Ordinatio from Pagani, Kovaleski and Resende (2015) , expressed in Equation (1):

InOrdinato = ( F i / 1000 ) α [ 10 − ( Year search − Year pub ) ] + ∑ ​ C i . (1)

where:

F i : Impact Factor

Year search : Year of search

Year pub : Publication year

C i : Number of citations in Google Scholar

A research in the Scielo database was also carried out, but no relevant articles were found.

As some articles were indexed in more than one scientific database, it was necessary to remove duplicates. The numbers of the articles captured, according to the keywords, are shown in Table 2.

The ranking and selection process was only necessary for the categories “Human Skills & Engineering Education” and “Human Sciences & Engineering

Source: Campos, 2019 .

Source: Campos, 2019 .

Education”. For the category of “Human Sciences & Graduate”, due to the low number of articles, all articles were read; for the “Human Skills & Graduate” no articles were selected.

To define the threshold for which articles would compose the final portfolio for the “Human Skills & Engineering Education” and “Human Sciences & Engineering Education”, the accumulated Index Ordinatio was used as a reference.

For this purpose, the articles of each of these categories were ordered in descending order according to their IO value (refer to Equation (1)). After ordering the articles, it was possible to calculate the accumulated IO (simple sum of the IO’s of each article), as well as the relative percentage from each IO—100% corresponded to the accumulated IO to each of the categories.

Table 3 and Table 4 show the ten first articles for the “Human Sciences & Engineering Education” and “Human Skills & Engineering Education”, respectively. Columns 3 to 5 show the values used to calculate the IO. Column 6 shows the IO calculated for each article, Column 7 the accumulated IO and Column 8 the percentage of the accumulated IO.

It is important to notice that both tables are just a cutout of all the articles from each of the categories (242 for Table 3 and 85 for Table 4).

From the IO, accumulated IO and % accumulated IO values, the graphics shown in Figure 3 and Figure 4 were constructed. The articles are shown in the horizontal axis, with the blue bar representing their IO. The orange curve illustrates the accumulated IO value. From this, it was established that the threshold for the articles that would compose the final portfolio would be 50% of the accumulated IO. This value is represented in both graphics by the pink horizontal line.

This percentage was a choice from the authors of the article, as per Pegani, Kovaleski e Resende (2015) statement that the threshold line is to be the author’s choice and responsibility. With this threshold, a final portfolio from 59 articles was obtained.

Bibliometric is a field of the area of the Library Science and the Information Science which applies statistical and mathematical models to analyze and construct indicators about the dynamics and evolution of the scientific information (Zupic & Čater, 2015) .

Own authorship (2019).

Own authorship (2019).

At this stage, bibliometric was performed on the 59 articles which compose the final portfolio of the research and bibliometric analysis.

The detailed study of the articles showed the soft skills, which were later classified into categories. Each time an article cited a given soft skill, this was computed into the research. Therefore, at the end of each article, the list of soft skills perceived as relevant in that work was obtained.

Moreover, for each article, it was also tracked how many times a given term, was cited either as a hard skill or a soft skill, as well as the number of times the term was said to be “necessary”, “important” or as “neutral” or “unimportant”.

Finally, it was also tracked the number of times that the definition of soft skills was given and the number of times the term appeared in the articles analyzed.

Table 5 shows the soft skills cited in the articles, as well as the values for each of the parameters mentioned in the previous paragraph.

Own authorship (2019).

Based on bibliometrics, a top-down tree-shaped model (shown in Figure 11) was elaborated with fuzzy logic to evaluate the soft skills that most appeared in the Systematic Bibliographic Review (Campos et al., 2020a) , combined with those ordered by the OECD employability reports (OECD, 2015; OECD, 2016a, 2016b) and the report P21 (Casner-Lotto & Barrington, 2006) , as already explained in the Section 2—Theoretical Reference.

The fuzzy logic operates through the intersection of fuzzy sets, considering degrees of pertinence for each set. The classical proposition differs from the fuzzy fundamentally by its truth zone values. In traditional set theory of classical logic, an element may or may not belong to a set. However, in the theory of fuzzy sets, an element of a given U Universe can be defined mathematically by a value representing its degree or level of belonging to the set. This membership value is in a range that goes from: “this element does not belong to the set” (0) to “this element belongs 100% to the set” (1) (Bonventi & Costa, 2000; Boutros & Chen, 2011; Takáč, 2014) .

An example of how fuzzy logic was used for this specific work, is in the fuzzification of the soft skill “critical thinking”, which assumes truth-values by means of fuzzy sets with the other thematic soft skills “communication”, “team work”, “ethical perspective”, “emotional control” and “creative thinking”, according to the degrees of pertinence. According to Campos Filho (2004) and Chakraverty & Behera (2013) , a fuzzy set is completely characterized by its membership vector, with individual multivalent degrees of membership within the 0.1 numerical range. These degrees of pertinence can be considered as a measurement to express the possibility of a given element being a member of a fuzzy set. Thus, a fuzzy pertinence vector is also called a possibility vector or a possibility distribution vector.

As the name already suggests, this logic is fuzzy, and it is interesting to notice that, differently from the Boolean logic, when the sets are represented using the Veen Diagram, they do not need to cross to have an intersection region. Figure 5 shows a demonstration of this phenomenon, in which the clouds of the fuzzy sets intersect, forming a fuzzy intersection.

In Figure 5, the elements of the cloud of set A intersect with the elements of the cloud of set B, forming an intersection of sets of the fuzzy type, which is not exact, but rather diffuse, similar to human reasoning.

In this sense, fuzzy logic is a way to manage uncertainties (Caiado et al., 2021; Takáč, 2014) and its use is very appropriate to treat data collected from the Human Sciences and measure the levels of uncertainty inherent to human communication.

The advantage of Boolean logic, with its crisp values, is accuracy. The disadvantage is In Figure 6, the letters “D” represent the input data, that is, the questions asked to the participants to feed the hierarchy constructed with the FuzzyTECH software The fuzzy logic calculates the existing pertinence between D3 and D4, for example, to form the soft skill “open mind” encountered in the SBR. Afterwards, the fuzzy operation calculated how much “open mind” and “problem solving” impact the fuzzy sets, to form the thematic group which is also a soft skill: “critical thinking”.

This process is repeated for all the thematic groups, forming the other soft skills of this modelling. Later, “critical thinking” will be fuzzyfied with the other 5 macro soft skills (those which compose the thematic group) to generate a result, which is the level of that soft skill within the investigated Engineering course. This illustrated in the hierarchical scheme called top-down tree.

The construction of the Top-down Tree was based on the intended result, and has 4 hierarchical levels (Figure 7). The fourth level, which represents the Result (diagnosis in soft skills development in Engineering training), comes from the aggregation of the indexes. The indices obtained from the aggregation of soft skills from thematic groups (indicators) make up the third level. The indicators constitute the second level and are the soft skills obtained from the bibliometric of SBR, broken down into guidelines inspired by the P21 report (Casner-Lotto & Barrington, 2006) , which represent the first level, as well as the tree’s input data .

This hierarchy represents a simplified cut of the Model mounted on the fuzzyTECH tree, showing the logical sequence to achieve the result, which is the diagnosis of soft skills development in Engineering training.

In order to feed the data into the software used in this research, the blocks were weighed by the “Degree of Support” or DoS, which is a calibration function of the fuzzyTECH Software itself.

This calibration occurs through the assignment of weights, according to the data obtained by the interviewees’ opinions, being 40 organizational Psychologists, chosen at random. For Gondim, Borges-Andrade and Bastos (2017) , this area of Psychology is responsible for promoting strategies for improving the work environment. These people have a greater proximity to the environment of the organizations, being then elected to assess the inferences of the soft skills among themselves.

For this purpose, the Psychologists answered the questionnaire included in Appendix 1. For the weighting of the data, the Likert scale was used, associated with the Delphi method with scores from 0 to 10, in which values were provided related to their respective perceptions about the inferences that soft skills present.

For Vieira and Dalmoro (2008) , when the answers and results are dispersed, the Likert scale may not be the most appropriate methodological tool, which suggests the need to have a complementary measurement strategy. In order to have greater reliability in the answers, the Delphi methodology was used, which, for Fagundes (2015) , associated with the Likert scale, allows the results converging, reducing the dispersion of the obtained values.

For Fagundes (2015) , the Likert scale is carried out using the weighted average, as shown in Table 6, which exemplifies the model that will be used for data valuation and insertion in the FuzzyTECH software.

Equation (3) demonstrates the calculation based on the results of Table 6.

Weightedaverage = ( 2 × 5 ) + ( 4 × 6 ) + ( 3 × 7 ) + ( 1 × 8 ) = 63

A R = 63 / ( 2 + 4 + 1 + 3 ) = 6.3 (3)

The Delphi method aims to bring responses closer to a collective consensus, reaching a more representative value than that of the isolated opinions. Leading opinions may have an influence on the results of the entire group (Giovinazzo, 2001) .

For the proper use of the aforementioned method, members of a group of experts familiar with the study themes were consulted—for the present study, the organizational psychologists were the targeted group. After the elaboration of the questionnaires, they were sent to the respondents individually, and returned to the researchers after completion. After analysis, the questionnaires were sent back to the respondents, in order to reduce the dispersion of the results (Candido et al., 2007; Wright et al., 2000) . For the scientific legitimacy of this method, three points were respected: the anonymity of the participants, the statistical evaluation of the distribution of the data obtained and the return of the responses of the group members, so that the new assessment can be made (Listone & Turoff, 2002; Wright et al., 2000; Martino, 1993) .

In the last round, the same questionnaire was used, for the same Psychology professionals, who then had the Weighted Average of the result of the previous stage, referring to all respondents, as well as the values assigned by themselves.

Source: Adapted from: Fagundes (2015) .

Based on this, they decided whether they would maintain or modify their responses from the first round. This step increases the reliability of the results, as the change of some perceptions reduces the values dispersion (Fagundes, 2015) .

Finally, the modeling developed by one of the authors of this study (Campos, 2019) , was calibrated in all its guidelines, indexes and indicators, thus providing the final result (indicator): diagnosis of the soft skills development for Engineering training, as elucidated in Table 7. This calibration is presented as complementation of the developed model and it serves to the purpose of specialists being able to determine how much a soft skill impacts another one, attending the complete development of the fuzzy logic within the present work.

The values obtained with the weighting of the Psychologists were entered in the FuzzyTECH Software to calibrate the model. Copies of the questionnaires suggested to be applied with the targeted audience composed of students, graduates, professors and employers of Engineers were also attached to this study in the Appendices 1-6. This covers the entire development of the instrument for diagnosing the soft skills development in Engineering training.