Introduction (Введение). In recent years, the mathematical modeling of social groups has been developed (Eshghi, Williams, 2017a), (Eshghi, Williams, 2017b), (Whitaker, Turner, Colombo, 2018). The specificity of social groups of the Orthodox type also receives some attention (Divisenko, 2016), (Mankoff, Miller, 2018). Some sociologists believe that religiosity is either not quantifiable at all, or very difficult (Markin, 2018). This opinion is widespread both among theorists and among practitioners, for example, those involved in state-confessional relations (Basil, 2005).

On the one hand, some authors researching religiousness in society, displace an accent from state-religious to secular-religious interaction (Kotelnikov, Lebedev, 2004), (Lebedev, 2007), (Reutov, 2008). It may be justified in current political situation characterised by declared secularity of the state. But in case of change the secular character of the state constructed theoretical model of state-religion interaction will not lose heuristic potential (whereas secular-religious one will cease to be representative), i.e. is more universal. On the other hand, nondichotomous relation between state and religious sides makes a ground for conflict: for example, believing official, that decides to transfer a religious-purposed object from state property to property of religious organisation. If he/ she knows, that it is illegal, but his/ her bishop (mufti, rabbi, pastor, lama etc.) blesses such transfer, then what norms will be carried out: state or religious? Guided by classical logic, it is impossible to answer this question, since the person in question belongs to both social groups at once. To answer this question, fuzzy logic is needed, it allows an element to belong to the set not binary {0; 1}, but with some degree of membership from the interval [0; 1].

Methodology and methods (Методологияиметоды). Examination of the possibility of measuring religiosity. But first it is necessary to consider the possibility of measuring religiosity. Fuzzy logic allows you to do this, but there are also doubters. For example, D. A. Uzlaner writes: «The question arises: on what basis is a person who believes in God but does not attend a church is considered less religious than one who goes to church? The answer to this question is obvious for a theologian based on a certain tradition, but not for a sociologist» (Uzlaner, 2008: 64). But the right question is given the wrong answer: «Consequently, the sociological interpretation of secularization, while remaining within the limits of scientificity, cannot deal with the problem of religiosity as such» (Uzlaner, 2008: 64). This does not fit with the analysis of secularization at the micro level, about which, along with the macro and meso levels, the author writes. The explanation for this discrepancy is as follows: «At the micro level, the study of secularization implies an analysis of the influence of religious ideas and religious consciousness. It's not about how religious certain individuals are, but whether their religiosity influences behavior, such as how people vote or pay taxes» (Uzlaner, 2008: 65). If you do not investigate how religious an individual is, then the sociologist has access to data on religiosity only on a nominal scale, whereas electoral or tax activity can have at least an interval character according to the electoral and economic cycles. For comparison with data on religiosity, a nominal scale will have to be used – to limit voting and payment of taxes by the fact of performing this action. But then the sociologist deliberately impoverishes his research capabilities and, moreover, perhaps reflects social reality inadequately: it is clear that a person who voted in elections only once and a person who regularly participates in them exhibit significantly different social activity, but the nominal scale obscures this.

Nevertheless, D. A. Uzlaner insists: «…to name the transformation of religiosity as secularization, it is necessary to define what is considered a religious norm, which, as shown above, is difficult if we do not take theological positions» (Uzlaner, 2008: 64). In fact, there is no difficulty – you just need not to absolutize this or that religious tradition, not to ask the same questions to respondents, regardless of their confessional self-identification. I.e. before the question about attending a church (and, therefore, before making a decision about the depth of the respondent's religiosity), it is necessary to ask the question of what tradition the respondent considers himself to. If the religion, the identity with which the respondent declares, has a requirement to attend church, but the respondent does not fulfill it, then he is less religious than the respondent who does not attend church, in whose tradition there is no requirement to attend church.

Thus, the answer to the question «What is considered a religious norm?» must be given by the respondent, not the sociologist. The researcher should only foresee in advance the possible variants of different attitudes towards religion. This removes the contradiction between sociology and theology. Of course, the sociologist must know what requirements are imposed on believers by their religion, and he can ask if the respondent fulfills these requirements – but only the respondent who declared belonging to this religion.

We completely reject the position of D. A. Uzlaner, regarding it as a deviation into agnosticism, which arose as a result of the search for universal sociological criteria of religiosity on theological grounds. Of course, to formulate specific questions of the questionnaire, one should use theological material, but not one religion, but all those whose confessions can be expected from the respondent. Thus, it is advisable to join the majority of both Russian and foreign scientists who believe that the measurement of religiosity is not only possible, but also necessary.

Association of deontic logic and morphological approach. To describe and analyze this situation, we need not only fuzzy logic, but also modal, and specifically – deontic. It is on the basis of comparing the norms of the tradition, to which the respondent considers himself, and how these traditions are fulfilled by the respondent, that one can draw a conclusion about the degree of his/ her religiosity. Let us introduce two concepts, respectively: abstract modality and concrete modality. They relate to each other as de jure and de facto: on the one hand, the normative requirements for the believer, enshrined in religious doctrine, on the other hand, the degree to which the believer fulfills these requirements. Contents for abstract modality is taken from the religious texts (specifically for each tradition, so every believer should be asked for only his/ her tradition), for concrete one – from empirical survey.

Logicians have created a strong method for describing and analysing the sphere of duty: «The normative status of an action is expressed in terms such as “permitted”, “prohibited”, “obligatory”, “normatively indifferent”, “should”, “should not”, “may”, “necessary”, etc.» (Nonclassic…, 2006: 33). However, in this list there is no single system, the concepts intersect and enter into each other, there are not only deontic, but also aletic modalities, and the list is open. In addition, if we want to compare Orthodox and Islam, like it did J. J. Sinelina (Sinelina, 2009, 2013), it will be impossible with such method, since in this religion, along with the indifferent (mubah), obligatory (fard) and prohibited (haram), the permitted (halal) is divided into desirable (wajib), neutral (i.e., the aforementioned indifferent “mubah”) and undesirable (makruh). All this means that it is impossible to use the above list of deontic modalities unchanged in our study.

Research Results and Discussion (Научныерезультатыидисуссия). The author’s position is that, using the morphological approach, two parameters should be distinguished: the nature of the norm that regulates the action, and the assessment given to the action by the rule-maker. The norm can be dispositive or imperative, and the assessment can be positive or negative. Thus, we obtain four combinations, each of which can be assigned a degree of belonging: imperative-positive “obligatory” (1), dispositive-positive is “desirable” (0.75), dispositive-negative is “undesirable” (0.25), imperative negative is “prohibited” (0).

The beginning of the scale from zero does not mean that this scale is absolute – on the contrary, in fact, it is a rank scale (Paniotto, Maximenko, 1982: 14). Thus, it cannot be said that a desirable action gives a person 25% less religious affiliation than an obligatory one, or that from a religious point of view, a desirable action is three times better than an undesirable one. But we can say that undesirable is better than prohibited etc.

In addition, the absence of negative values in the scale does not mean that there are no negative values at all. The role of negative values is played by negative assessments of actions given by the rulemaker. I.e. negative values are “dispositive-negative” (0.25) and “imperative-negative” (0). «If the rank scale provides for both positive and negative assessments, then its positions are arranged symmetrically: the number of positions with a positive value is equal to the number of positions with a negative value, and between them there is a position with a neutral (zero) value» (Gorshkov, Sheregi, 2001: 65). Therefore, a value of 0.5 can be assigned to the neutral action. Neutral point is need to reflect the situations that, on the one hand, don’t have a religious regulation or this regulation is ambiguous, on the other hand, are important in the design of the survey. Its interpretation depends on research tasks, for example, if the task is to typologize respondents, then neutral point can be an analogue for respondents who hesitate between faith and atheism.

Construction of fuzzy membership functions. The respondent's modality of the norm can be either weaker, or the same, or stronger than that established by the norm itself. It seems quite natural to assume that the degree of belonging to a group directly depends on this attitude, i.e. , where m - a membership function, А - social group, а - respondent, х - concrete modality, у - abstract modality. At first glance, the economic method of calculating efficiency as a ratio of fact to plan can be used in religious studies. However, such function cannot be used either from a formal or substantive point of view.

On the formal side, the impossibility of applying the aforementioned membership function lies in the difficulties with the analysis of the observance of prohibitions and, more generally, in the logical inconsistency. The abstract modality “prohibited” has a value of zero, which is not allowed in the denominator of the fraction. And without attention to prohibitions, the study will be deliberately incomplete. But even if we cross out the zero value from the scale, then at  and  it turns out , i.e. the value of the function is off scale - this move is inadmissible.

On the substantive side, the problem is that respondents who most accurately fulfill their socio-group norms (i.e., when a specific modality of a norm is identical to an abstract one) will turn out to be less belonging to the group than fanatics who adhere to a stronger modality. This assumption makes it impossible to study Christianity and Judaism, since the scripture repeatedly prohibits liberty in dealing with the prescribed rules. For example: «Ye shall observe to do therefore as the LORD your God hath commanded you: ye shall not turn aside to the right hand or to the left» (Deut. 5: 32), «…to keep and to do all that is written in the book of the law of Moses, that ye turn not aside therefrom to the right hand or to the left» (Josh. 23: 6), «Turn not to the right hand nor to the left…» (Prov. 4: 27). In our terms it means the condition .

A slightly more complex situation with Christianity. On the one hand, the above quotes are not only part of the Old Testament of the Bible, but are also confirmed in the New Testament by Christ himself: «Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled» (Mt. 5: 17-18). I.e. believers are still required to fulfill the condition . On the other hand, it is not the simple implementation of the law that is important, but even more enhanced: «…That except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven» (Mt. 5: 20). I.e. the condition of salvation changes significantly: . But in this case, firstly, the aforementioned formal obstacles with zero value and off-scale remain, and secondly, in practice, deviations from the norm, including exceeding the proper value of the parameter, are not always useful.

If we try to model the superiority of personal righteousness over the Pharisee as a necessary condition for belonging to Christians without the problem of division by zero, we get the membership function . The bottom line expresses a righteousness not exceeding (i.e. less than or equal to) the righteousness of the scribes and Pharisees. But with the help of this function it becomes impossible to describe the other extreme position of the scale. Both concrete and abstract modalities “obligatory” have a numerical value of 1 (or any other number - it is only important that at the top of the scale, a concrete modality a priori cannot surpass the abstract one). Therefore, the bottom line will always be fulfilled - nobody has an objective opportunity to fulfill the divine command to surpass the righteousness of the Pharisees, i.e. all are doomed to perish. This conclusion contradicts the essence of Christianity and, therefore, turns it into a simulacrum - this move is inadmissible.

Thus, the membership function we are looking for must satisfy the following requirements: firstly, it must be mathematically correct, and secondly, it must implement the following position: the closer the function arguments are to each other (i.e., the more the concrete x and abstract y modality), the greater the value of the function, the further - the less.

These requirements are met by the function . All actions associated with this function are feasible for numbers from the range [0; 1], and the value of the function does not go off scale. The modulus of the difference x and y is the degree of remoteness of the concrete and abstract modalities from each other (in other words, the inaccuracy of the respondent fulfilling his/ her social-group norms), and subtracting it from one shows that the more accurately the respondent fulfills the norms of his/ her social group, the more he belongs to it.

In accordance with our position on the quantification of religiosity, the domain of definition and the range of values of this function are discrete and have a cardinality of five. In order to comprehensively (i.e., not only analytically, but also tabularly and graphically) set the function of belonging to a particular religious tradition, the author carried out the corresponding computational and illustrative procedures (see Table.  and Figure).

Observation of churchliness in native and international contexts. Many methods have been developed for studying the religiosity of the population: on the basis of indicators of self-identification, religious consciousness, religious behavior, consciousness and behavior together, as well as the method of expert assessments (Lokosov, Sinelina, 2008). As a rule, a believer is required to recognize his/ her religious affiliation, know sacred texts and dogmas, perform rituals, celebrate religious holidays, etc. Unfortunately, different authors approach the set of operational indicators of religiosity without following any standard. For example, observance of fasts is mentioned only by L. A. Andreeva and L. K. Andreeva, but in the analysis of the data they received, there is no information about fasting (in contrast to the attendance of a church and making prayers) (Andreeva, Andreeva, 2010: 97). Whereas other authors use the post indicator (Barannikov, Matronina, 2004: 131), but don’t pay attention to prayer.

If we pay attention not only to the list of criteria, but also to their due values, then it is necessary to mention the active discussion that was conducted between the supporters of the soft (Sinelina, 2001, 2005, 2006) and hard approaches. The latter proved to be more convincing: «No matter how multifaceted the personal religious experience of the Orthodox, but if he/ she expects in the future life of rebirth, and not the judgment of God, if he/ she cannot express in words faith in the living, suffering, personal God, on whose love his/ her salvation depends, but speaks of influencing him cosmic power, it is difficult to understand what his/ her Orthodoxy consists of» (Filatov, Lunkin, 2005: 41).