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Home , Fiqh, Mubah, Risalah (fiqh), Syntax (logic)

J.Acad.(N.Y.)4,2:52-87 2014 (36 pages) Abdelwahab et al. 1

Copyright © 2016 Journal Academica Foundation. All rights reserved. With perpetual, non-exclusive license to distribute for arxiv.org Originally received December 12 2013 - accepted May 7 2014 - published May 16 2014 J.Acad.(N.Y.)4,2:52-87 (36 pages) - Theoretical - permalink: http://journalacademica.org/?smd_process_download=1&download_id=753

The Algorithm of Islamic () with Validation of an

Elnaserledinellah Mahmood Abdelwahab a*, Karim Daghbouche a, b Nadra Ahmad Shannan a makmad.org e.V., Hanover (Germany) b Faculty of Shariah, Umm Al-Qura , Makkah (Kingdom of )

Corresponding author: [email protected]

ABSTRACT The historic background of algorithmic processing with regard to etymology and methodology is translated into terms of mathematical and Computer Science. A formal logic structure is introduced by exemplary questions posed to Fiqh-chapters to define a logic query language. As a foundation, a generic algorithm for deciding Fiqh- rulings is designed to enable and further leverage rule of (vs. rule by law) with full transparency and complete algorithmic coverage of Islamic law eventually providing legal security, legal equality, and full legal accountability. This is implemented by disentangling and reinstating classic Fiqh-methodology (usul al-Fiqh) with the expressive power of of First Order Logic (FOL) sustainably substituting ad hoc reasoning with falsifiable rational argumentation. The results are discussed in formal terms of , and complexity of formal Fiqh-systems. An Entscheidungsproblem for formal Fiqh-Systems is formulated and validated.

Keywords: algorithm, Khwarizmi, , balance, algebra, Shariah, Islamic law, jurisprudence, logic, syntax, , Entscheidungsproblem, complexity, decidability, SAT-problem, transparency, security, accountability, , rule by law, Fiqh, usul al-Fiqh

1. INTRODUCTION The here presented Algorithm of Islamic Science and its underlying mathematics, Jurisprudence connects back some and a discussion (C) of the practical 1,200 years to the origin of the modern algorithmic results in computer term Algorithm which is the most programming while this introduction fundamental methodological driver of shall serve as a brief historic review: the current information technology pace It is commonly known that the term at internet speed. Algorithm is a Latin short-version of the It will be discussed in three major parts ibn Musa Al- which consist of (A) methodological Khwarizmi (163-235AH / 780-850AD), considerations in Islamic Jurisprudence who is the author of the book (Fiqh) with regard to (B) Computer Kitab Al-Jabr wa-l-Muqabala (215AH /

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830AD) [Al-Khwarizmi], i.e., The various sorts and kinds are concerned." Compendious Book on Calculation by [Al-Khwarizmi] Completion and Balancing. This book It is evident that the introduction of a was translated into Latin in the 12th new Fiqh-method back then as well as century A.D. entitled Liber Algebrae et today requires a standard of Almucabola [Chester] with algebrae persuasive in terms of and Almucabola being transliterated reproducible and proofs. into Latin from the Arabic title where Therefore, every Fiqh-problem in Kitab the term Algebra is derived from Al-Jabr wa-l-Muqabala is reducible to a Al-Jabr in the title of Al-Khwarizmi's linear equation while the more book. fundamental method of quadratic While there is widespread belief that polynomials is never used but proven. Kitab Al-Jabr wa-l-Muqabala is a But before discussing the of textbook for mathematics which, among quadratic polynomials (cf. [Al- others, introduced general rules to solve Khwarizmi]), it was necessary to algebraic problems with one conclude some preliminary research reducible to quadratic equations, it is such as introducing the calculation with first and foremost a textbook for Islamic Hindu numerals including the number Jurisprudence: zero with Kitab Al-Jam wa-l-tafriq bi- As traditionally and practically done in hisab Al-Hind (The Book of Bringing Islamic Jurisprudence (Fiqh), the first together and Separating According to half of Kitab Al-Jabr wa-l-Muqabala the Hindu Calculation - lat.: Algoritmi introduced the applied methodology and de numero Indorum) written about term , i.e., then as now the (210AH / 825AD), i.e.: Al-Khwarizmi domain of Algebra. developed and published the Hindu The remaining half of his book solves Calculation half a decade before legal questions on trade (commercial introducing the new Fiqh-method of transactions), geometry (plane surface reducing variables to quadratic distributions) as well as testimonies. equations. Based on the newly introduced While Al-Khwarizmi’s algebraic method, the by far most consisted of reducing selected Fiqh- important part of Kitab Al-Jabr wa-l- problems to linear equations, the here Muqabala deals with Islamic heritage introduced Algorithm of Islamic law which is complex with regard to as Jurisprudence proposes to cover the well as both, number of variables and complete corpus juris of Islamic number of Fiqh-rules (). Jurisprudence. Accordingly, Al-Khwarizmi’s Algebra is However, the most important just what its author says in the proposition underlying the Algorithm of introduction: "[a] work on algebra, Islamic Jurisprudence consists of one confining it to the fine and important single : parts of its calculations, such as people constantly require in cases of There exits a complete . inheritance, legacies, partition, law- suits, and trade, and in all their dealings Its mathematical and algorithmic with one another, or where surveying, formalization will be presented in part B the digging of canals, geometrical of this paper. computation, and other objects of

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This statement implies that every Fiqh- rulings on different matters, problem has a solution, i.e., that every synthetically relying on their personal religious case has a ruling,1 which is the discretion. Their rulings were of three traditional conviction from its inception: types:

A) of the and the Fiqh developed in two stages: Firstly 3 during the time of Prophet Muhammad’s (saw)2 companions when B) analogy that draws on similarities Fiqh did not rely on explicit rules. between cases (i.e., ), and Rather, it relied on synthetic C) discretion that does not rely on any understanding by the Prophet’s (saw) particular text companions as they witnessed the revelations of the Quran with a When the major works on Fiqh- situation- and procedural awareness of methodology, such as Al-Shafi'i’s Al- Islamic law. (204AH / 820AD) [Sahfii] and As the Islamic influence and Al-Ghazali’s Al-Mustasfa (504AH / responsibility grew, the Muslim 1110AD) (cf. [Hammad]) appeared, community was embraced by diversity scholars had different views on the as well as both, culturally and of the second type of scholarly intellectually. At the same time, the discretion, i.e., on analogy. personal connection of Fiqh to the Many of the works on Islamic Prophet (saw) and his companions Jurisprudence that discuss analogy 4 became more and more remote. This let include questions concerning logic . Fiqh to be studied analytically. It also Other scholars who deny the relevance required in-depth studies of the Arabic of logic to Islamic Jurisprudence language and logic that were relevant to consistently rejected the very of the study of the major sources of analogy. They deprecated the Islamic law, namely the Quran, the introduction of methods that are alien to Sunnah, ’, i.e., the unanimity of the the genuine, Arab Muslim mind. community, and , i.e., the Nevertheless, a number of leading scholarly discretion. scholars of usul al-Fiqh, such as Al- Ijtihad, which means the exercise of Ghazali and Al-Razi (240-312AH / 854- scholarly endeavor to generate the right 925AD) opposed this position (cf. rulings in different situations, started during the lifetime of Prophet 3 Sunnah is a general term that refers to the Muhammad (saw), but it was limited in Prophet’s (saw) example and guidance, whether scope and remained within the area of verbal or practical. personal matters, mainly with regard to 4 Their understanding of logic was solely based transactions and commitments (cf. [Al- on the well-known of the Greek Dawalibi]). After the Prophet (saw) had which were studied in the – then new – context passed away, his companions produced of Fiqh resulting in selections of appropriate "ways of analogy". Muslim scholars invented the of expressing syllogistic methods and their explanations is poetic form as can be found 1 Its inversion (complete induction) would in, e.g.: Al-Akhdary – Al sullam al Munawraq simply imply that every religious ruling has a fil Fiqh (cf. Al-Akhdari. (2014, March 26). In . Wikipedia, The Free Encyclopedia. Retrieved 2 saw = sall Allahu 'alay-hi wa-salaam = may 21:36, April 7, 2014, from God pray on him and grant him (a http://en.wikipedia.org/w/index.php?title=Al- traditional saying after mentioning his name) Akhdari&oldid=601304682)

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[Hammad]), citing from the We need to make certain definitions in Quran and the Sunnah to confirm the order to distinguish between the three validity of analogy as a source of disciplines: language, jurisprudence and rulings on questions that are not directly logic: addressed in the Quran or the Sunnah.5 1. A Logical Question (Qgic) is a syntactic problem that cannot be A. FIQH-METHODOLOGY (USUL) solved except through a mental link Our purpose in this part is to present between the realities of the outside some questions of Fiqh-methodology world and the natural human , (hereafter referred to as Arabic: usul al- which is totally independent of agreed Fiqh)6 and explain their relevance and terminology and belief. (cf. relation to logic and algebraic methods [Hammad]) in general. This will serve as 2. A Language Question (QLn) is a preliminary to the contemporary trends semantic problem that is solved by in logic adopted by most experimental to the meanings of the and mathematical sciences to linguistic patterns it uses, whether mechanically (algorithmically) these are words or expressions and determine whether conclusions are true sentences. These meanings are either or false. The logical trend had its roots essentially or traditionally known in in the Muslim scholarly mind that Arabic. realized that in dealing with religion, it 3. A Source Question (SF) is a search and could not exclusively rely on pure authentication problem, i.e., an logical reasoning to draw true algorithmic problem that can only be conclusions. It required two other solved by reference to a of essential elements, namely language and law. This implies that logic is a rulings that have been approved on the of Islamic Jurisprudence. basis of Quran and Sunnah. With these terms at hand we may further address and classify 5 cf. Al-Ghazali's chapter on answering the methodological questions such as arguments denying the validity of analogy in: analogy, which is the most important of [Hammad] all:7 6 Fiqh is an Arabic term meaning "deep Analogy, or qiyas, denotes the understanding" or "full comprehension". analytical reasoning process that Technically, it refers to the body of Islamic law scholars of Islamic Jurisprudence use to extracted from detailed Islamic sources (which are studied in the principles of Islamic work out Fiqh-rulings. It relies on a Jurisprudence) and the process of gaining finite set of logical rules for deductive knowledge of Islam through jurisprudence. The reasoning. historian (732-808AH / 1332-1406 As such, qiyas represents the main AD) describes Fiqh as "knowledge of the rules junction of Islamic Jurisprudence. of God which concern the actions of persons Because although the deductive process who own themselves bound to obey the law of qiyas is -preserving if applied respecting what is required (wajib), sinful (haraam), recommended (mandub), disapproved correctly on true axioms, there may () or neutral ()" [Glasse]. The Arabic term usul means lit. root denoting methodology. This is consistent 7 There may be more detailed definitions while amongst Islamic jurists. the here presented is sufficient for clarification.

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always be debate concerning the main the is true or not because this and secondary postulations used prior knowledge goes beyond that to which a actual deduction and whether these sentence refers to, this knowledge is not postulations were deduced correctly required for the logic deduction per se. from religious texts. In a precisely structured language The issue that causes such debate is system, it can be decided whether two semantics, i.e., the intention and submitted can be derived purpose of the speaker materialized as a immediately, i.e., logically deduced one dedicated linguistic expression, i.e., from the other or not (or any of several syntax. other submitted expressions) without This debate is continuous because any reference to their meaning provided that investigation about semantics is the term "immediately derivable" has generally characterized by not been clearly determined in the system in restricting the analysis to linguistic question. expressions as such and their relations Therefore, any derivation, no matter to one another. Rather, those analysis how long or complex, can be take into consideration what is being characterized syntactically (cf. referred to in the sentences of the object [Stegmüller]), i.e., the deductive part of language. And that is independent of the Shariah (qiyas/analogy) could be method chosen (i.e., the method of mechanized completely, which takes us denotation relation (cf. [Frege]) or of back to the origins of the term extension and intension (cf. [Carnap]) Algorithm with a logical connotation. because there is always the range of (cf. [van der Waerden]) those objects taken into consideration to Usul al-Fiqh discussed this finding which the expressions of the object extensively, provided with rules and language relate. All specific semantic classifications according to their predicates such as "denotes", "term xyz- validity, relevance, correct application denotes", "satisfies", "true", etc. provide to the view in question as well as how directly or indirectly (the latter as in the balanced they interpret the relevant text. case of the predicate "true") with At the same time, the Arabic vocabulary relations between linguistic expressions was investigated and classified into a and their semantics. couple of main sections, the most Going one step further by not only important of which is the one concerned abstracting away the semantics of the with the general and the specific. A speaker who uses the expressions of the detailed study of semantics was carried object language, but by also abstracting out classifying word meanings into ones from what the linguistic expressions that are specifically and expressly refer to, yields a purely formal analysis intended, and ones that are inferred where any semantics are replaced by the from the text itself. The were syntax. Hence, in the most abstract analyzed with regard to different sense we return to the fact that logic can attributes, including simplicity, operate purely syntactically, that is: complexity, limitation, extension, The basic concept of logical deduction prevention and nullification. All these is exclusively defined syntactically (as questions are purely logical (Qgic). The opposed to the concept of "truth"). essential purpose of such studies has While a formal consideration of an been the establishment of accurate expression doesn't provide knowledge criteria to ascertain the most balanced about the speaker's purpose or whether religious verdict.

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2. THE GENERAL AND THE speaks about freeing a slave without any SPECIFIC further specification. Fiqh-scholars The emphasis placed on the relation conjugated the two cases and concluded between syntax and semantics is that the male or female slave to be freed reflected in studies on the general. It in both instances must be a Muslim discusses in great detail the reasons for because both actions are forbidden. This using the general linguistic formulae reason, i.e., the prohibition of both from the point of view of the individual actions, represents the highest degree of meaning that the general term includes. of the causes (Qgic). It seeks to answer questions like: Does the text mean the least group? Does the general term have a meaning that 3. SEMANTICS AND REASONS applies universally or specifically? Does Studies of words were not limited to the general term apply to all their general linguistic or logical individuals? Obviously, all these meanings. These studies were expanded questions are common to language, to include the specifically intended whether natural or formalized such as meanings and those that are implied and algebraic logic (QLn & Qgic). understood through a pointer given in Fiqh also provides with studies about the text (Qgic), i.e., through a syntactic the case of the individual that is pre-differentiation.8 included in a general term. It was These studies provide with details about realized that the individual may be specifically intended meanings so as to referred to without using any general include what is required to ensure that a linguistic formula. In such a case, the given statement is correct, logically and reference is made through the usage of a religiously, while the meanings that applies to the individual understood through a pointer were without further details. The reference divided into what is understood through may at times be restricted by further being consistent or being inconsistent the text itself does not (Qgic). include (QLn & Qgic). The studies of reasons unveil a unique It was also concluded that reasons for a and detailed interaction between logic verdict are, in the majority of cases, closely linked to how the universal is related to the restricted case of its kind 8 It has to be taken into account that there is no (Qgic and SF). syntactic decision procedure which would allow One of the prime Fiqh-examples is the our thinking to semantically relate punishment specified for two offences, morphologically different types of terms to each other because the understanding of the meaning namely, a vow by a husband making his of a sentence in natural language is the same as wife unlawful to him () as well as knowing its truth conditions. (cf. [Eley]) manslaughter. Since we only have an explicit grammar The punishments for both are (syntax) in terms of the ontological-declarative mentioned in the Quran and include the Shariah, we need to perform a pre- differentiation of cognitively tangible facts freeing of a slave, but the relevant (perceivable facts), which indeed renders the verses mentioning the of axiomatic, i.e., the explicit character of the manslaughter specify that the slave to facts, unattainable for Shariah. The mere be freed should be a Muslim male or hypothetical nature of trial and error, however, female slave, while the verse is conditioned by a heuristic maximal- approximate predicate (cf. Figure 2 and mentioning the punishment for zihar [Daghbouche 2012])

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and language, considering, that these linguistic questions and many scholars are usually mere questions of logic. are not in favor of using analogy For illustration, we will discuss the through language (QLn). reasons of prevention and nullification, as well as the reasons derived from attributes, action, and denotement. 4. ANALOGY Scholars of usul consider the reason to Analogy or qiyas is a deductive process. be a clear and accurate description that Analyzing different aspects of analogy, may be real (Qgic), based on tradition the most important ones include (QLn), linguistically (QLn) or certainty, i.e., the need for generating a religiously (SF), while it could also be verdict on a secondary question and the non-existent (Qgic). (cf. [Hito]) clarity of the reason on which a verdict Thus, the reasons of prevention and is based. nullification are clear and balanced All these are linguistic and religious descriptions that occur during the questions (QLn & SF). An absolute performance of an action and nullify its analogy cannot be obtained without verdict (nullification), or before it is maximum certainty of the reason for the undertaken to prohibit it (prevention). ruling on the primary question and As such, they are related to a negative maximum certainty that this very reason ruling. applies to the secondary question9. An example is prayer and invalidation For example: It is explicitly forbidden of ablution (), such as urination: It to even say the word "Ugh" to one's prevents starting prayer if it occurs parents. Scholars have ruled through before it, stops prayer and nullifies it if analogy that it is also forbidden to beat it occurs during it. Another example one’s parents. It is certain that the relates to marriage when the two reason for prohibiting to saying "Ugh" partners follow different religions. If to one’s parents is that it hurts them. one of the parties follows a religion Accordingly, it is assumed that beating other than the three monotheistic book them causes greater hurt. However, we religions, marriage to a Muslim is not cannot maximize certainty in all this permitted. If this happens during an without awareness about the meanings existing marriage, it nullifies it (after of the terms: beating, ugh, and hurt. the end of a waiting period). The point In terms of logic we note that analogy is here is that although urination and the divided from a purely logical point of difference of religion are situations that view into inverse and direct analogy. have clear religious descriptions (SF), This is based on the that the giving them the of preventive and verdict of the primary case is either nullifying reasons is a verdict of logic. positive or negative in relation to the It is based on either inductive or verdict on the secondary case. according to the If it is positive, i.e., the same verdict as different scholarly approaches to rules the verdict on the secondary case, it is and differentiations. called direct; otherwise it is inverse. Reasons derived from terms become operative when the language is ascertained through analogy. Thus, whatever denote with and 9 The idea of having a solved primary questions attribute to "wine" is forbidden to drink. which forms the basis for the analytical decision This is one of the most fundamental regarding a secondary one can be traced back to the time of Greek logic.

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Thus the rule of inverse analogy is: be a necessary condition when it is specifically pledged. In the case of If the reason of the primary case leads prayer, as it is not a necessary to its verdict, the presence in the condition when not specified, it secondary case of the opposite of this remains unnecessary when pledged. reason leads to the opposite verdict on Thus, taking prayer as the primary this case. This may formally be case, the established verdict is that it represented as: is not a necessary condition for the validity of I’tikaf with the reason Fs primary ↔ R primary being that it is not obligatory when ⇔ Inverse (Fs secondary) pledged. The established verdict on ↔ Inverse (R secondary) the secondary case is that fasting is a necessary condition for the validity The analogy is not valid unless the of I’tikaf with the reason being that it reason is a necessary and sufficient becomes obligatory when pledged. condition for giving a verdict on both, The two cases differ in ruling and the primary and secondary case. reason.

We will exemplarily apply this rule to a To represent the question formally we will make the following abbreviations: case cited in books of usul:10

If a person pledges to perform the Fs ≡ I’tikaf done Prv ≡ prayed consistently during I’tikaf practice of I’tikaf11 and fast, the Pgv ≡ pledge performed before I’tikaf fasting is a necessary condition for Fv ≡ fasting observed during I’tikaf the validity of one's I’tikaf according

to all scholars, but if one pledges to A → B ≡ B necessary condition for A perform the I’tikaf and pray, Inverse (A) ≡ the predicate (A) inversed combining both acts is not necessary A B ≡ the predicate A added to B for the validity of one's I’tikaf. ʌ Scholars differ as to the condition of A v B ≡ the predicate A or B is correct fasting during the I’tikaf if the concerned person has not made a As given in the book, the case may be pledge. requires fasting presented as follows: in all cases, whether pledged or not. Al-Shafi'i does not require fasting as (Fs ᴧ Pgv → Fv) → (Fs → Fv) in the a condition. Abu Hanifa gives the secondary case following formula of inverse And the inverse ((Fs ᴧ Pgv → Prv)) → analogy: On the basis of analogy inverse (Fs → Prv) in the primary with prayer: had fasting been question. unnecessary for the validity of I’tikaf The analogy goes: It is well known that when it is not specified, it would not (Fs ᴧ Pgv → Fv), so (Fs → Fv) is arrived at through the inverse of the reason of the primary case and by 10 cf. [Al-Ghazali: p. 140] and [Hito: p. 375] substituting Prv by Fv. 11 The I’tikaf is a recommended practice during the final days of , when a person However, the presentation of the performing it spends days in a , going secondary case is incomplete. It is not out only for the necessary reasons, and devotes possible to logically deduce (Fs → Fv) all his wakeful time to worship in various forms.

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from (Fs ᴧ Pgv → Fv) because the first The second heuristic principle is the rule is more general. The fact is that the adoption of logic in formalizing the reason in the primary case is not only: Fiqh-question with the process of deduction making the mental process Inverse (Fs ᴧ Pgy → Prv) and arguments leading to a verdict transparent, i.e., fully reproducible. Had it been so, it would not have been possible to use this analogy because this reason is not necessary and sufficient 5. DEFINITIONS OF RULES FOR for: USUL AL-FIQH AND FIQH Inverse (Fs → Prv) At the beginning of the second stage of Fiqh a debate started between scholars The correct reason is: of the Hijaz12 who adhered to the literal interpretation of the texts and the Inverse (Fs ᴧ Pgy → Prv) ᴧ inverse (Fs scholars of , who had limited ᴧ inverse (Pgv) → Prv) knowledge of but were highly skilful in analytical reasoning. The This means that prayer is not essential debate motivated leading scholars to for the validity of the I’tikaf whether establish rules of usul. This complex pledged or not. Hence the secondary task was undertaken, among others, by case is: Al-Shafi'i with his book Al-Risalah. Eventually Fiqh was performed by two [(Fs ᴧ Pgv → Fv) v (Fs ᴧ inverse (Pgv) separate methods with the - → Fv)] → (Fs → Fv) school frequently relying on analogical deduction and independent reasoning, In plain language: and and generally using the Hadith instead where Shafi'i-school If fasting is an essential condition for uses Sunnah more than Hanafi and the validity of I’tikaf, whether pledged analogy more than the two others. or not, then it is an absolute condition. The first method was characterized by However, we know that it has been leaning towards analytical reasoning as stipulated in a pledge and as such it is much as possible, discussing primary an absolute condition. questions in isolation of secondary ones. Secondary questions in turn must be This example implies two heuristic subject to treatment of the rules of usul principles: but could not act as logical unless such attribution was justified by Firstly, a commitment to logical separate evidence, e.g.: analogy provides with a well-considered Quranic imperatives such as "Attend and reflected verdict. Although the prayers regularly" imply questions on discussed Fiqh-question was discussed whether the imperative form specifies a in an incomplete or not perfectly correct duty applicable to all adults, or if it way in ancient books, Abu Hanifa’s could be understood in terms of a view is correct according to the rules of recommendation. On the basis of logic. Therefore, Shafi'i scholars need to find a valid to refute the 12 Hanafi verdict. Literally "the barrier" primarily defined by the cities Makkah, , and Taif in the current "Kingdom of Saudi Arabia"

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grammatical rules and basic principles the status of the person to whom to of the Islamic faith, i.e., common sense apply.16 (QLn and SF), usul-rules established that a statement in the imperative form A rule of Fiqh is a practical component specifies an obligation, generating two of Islamic Jurisprudence that is used to aspects of evidence: a general one, i.e., identify the rulings applicable to its an obligatory order, and a specific one, details. If these details come under i.e., an order to offer prayers regularly. different themes of Fiqh, the rule is a The combination of these two generate general one, and if they come under one a Fiqh-ruling (verdict): "Prayer is section, it is a specific rule, e.g.: obligatory".13 The second method establishes the usul- "If the verdicts of permissible and rules on the basis of the rulings given by forbidden apply in one case, the leading scholars of the Hanafi-school of forbidden takes precedence." Under this thought for secondary cases. Scholars rule, the following implications apply: adopting this method undertook an extensive study of the rulings given by 1. If one part of a tree is in the their leading predecessors, induced area and another part is in the Hil17 rules and applied them to various area, the tree is forbidden to fell. questions. If they found their 2. If a Muslim butcher and an idolater conclusions conflicting with the rulings butcher take part in slaughtering an animal, that animal is forbidden for given by their predecessors, they consummation. amended the rules so as to provide 3. If meat of an animal slaughtered between the two, or else according to the Islamic tradition is they excluded specific questions from mixed with carrion, nothing of it is their rules. Depending on the field of permissible to eat. application, there are two major categories for rules which concern usul and Fiqh. The following will provide with some preliminary definitions which will be further differentiated in the course of the then following 15 chapters: In other words: how rulings are to be derived from it when there are conflicting or equally valid considerations, such as: "The text takes A rule of usul is a general verdict that is precedence over the apparent state; a report used to identify the elements of Fiqh- transmitted by many reporters at every stage is evidence,14 how it may be used,15 and given precedence over one reported by single transmitters; the unspecified is explained in the light of the restricted, and the general in the 13 Another example is the rule that says: "A light of the specific, etc." noun with an indefinite article implies general 16 This applies to a scholar who is competent to application when it occurs in a context of exercise scholarly discretion, i.e., ijtihad, and negation." This rule is generated from the one who follows a particular school of thought, Quranic verse that says: "This is the Book, there and the criteria applicable to each. is no doubt about it." [Quran 2:2] When this rule 17 The Haram area is the area surrounding is applied, the verse means that no type of doubt Makkah extending a few km in some places and can apply to the Quran. up to 20km in others. It is subject to certain 14 Such as "unanimity is a valid source of restrictions, including the prohibition of felling rulings"; "An order signifies obligation while a its trees. The Hil area is the complement of the negative order signifies prohibition". Haram area.

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6. FIQH AND LOGIC 4. Rules as basis to look at questions Having briefly discussed usul-rules and having similar reasons which were not how they were anticipated, a closer look given rulings or discussed by earlier at the rules of Fiqh may shed light on scholars. their practical signification and their The last point is the one that is relation to logic. particularly important for our present Fiqh-rules were developed organically. discussion, as it means that Once reaching maturity, scholars contemporary cases can be examined on studied the foundations with its rules the basis of these rules. Undoubtedly drawing analytic comparisons between they cannot be put together unless there different schools of thought,18including: are plenty of secondary questions that

have similar bases of deduction. When 1. Looking into Fiqh-questions in a researching the outlined in regulated way and to facilitate their old books on subject matter we observe: memorization. The rules were Firstly, most books list Fiqh- formulated in easy and precise rules together with usul-rules and Fiqh- language minimizing room for criteria. The reason may be that it was confusion and making it easy to not possible to practically separate these memorize. Examples are: "What is two sets of rules. Moreover, the permitted for a specific cause is no definition of usul-rules is too broad and longer permitted when the cause is lacks precision. Besides, they apply to inapplicable"; "There shall be no many Fiqh-rules as well. To give an infliction of harm on oneself or example according to this definition, the others"; and "Necessities permit what rule that requires reference to both, is forbidden". habit and social tradition, would be an 2. Development of Fiqh-insights by usul- and Fiqh-rule at the same time. enabling the grouping of questions Secondly, most scholars who list with similar aspects as well as separate such rules do not cite evidence in dissimilar ones. support of them because most of the 3. Understanding the objectives of rules were set by quasi-logical 19 Islamic Jurisprudence. Knowledge of induction. The inductive process uses detailed or a general rule that applies to a large particular cases to infer a general rule number of questions giving a clear that may be complete or incomplete. view of the objectives of the legislator. Hence, we suggest drawing the line of This, e.g., motivated Al-Izz ibn Abd Al- demarcation between usul-rules and Salam to include the whole Islamic Fiqh-rules in a mere formal sense, i.e.: Jurisprudence under the major rule of Usul-rules are finite, "Bring benefit and prevent harm". Al- denumerable, and constant sets of Subki, e.g., even abstracted further axioms inductively deduced from suggesting the rule simply to be "Bring source texts (QLn). They treat (in benefit", because the prevention of almost all cases) semantics such as the harm is already implied in this meaning of a statement using the formula. imperative form, or the meaning of a

18 This is a natural development that applies to 19 i.e., it pretends logic reasoning while it is most modern natural sciences. mere ad hoc .

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general/specific statement. As discussed "Paying in advance of its due earlier, we denote linguistic questions date", "Paying zakat-al-fitr (which is with QLn attributing to be generic due at the end of Ramadan) at anytime expressions and serving as deductive during Ramadan", "Paying the bases, i.e., as axioms. compensation of accidental killing once Fiqh-rules look at the causal the victim is injured, but before his reasoning behind verdicts with a variety death", and "Paying the atonement for of Fiqh-details and exemptions the violation of an oath after saying the provided with logic variables (Qgic). In oath but before the violation." anticipation of the implications for Since these financial matters are all due and as a summary of if one of two different reasons apply,21 the foregoing chapters, it is important to a generalized rule would look like: note that logic variables of Qgic are pre- "A financial obligation which can be differentiated by QLn. This implies not triggered by two different reasons may only Qgic to be finite and denumerable, be settled even if only one of the two but also the possible set of values. reasons applies." In this formal relation usul-rules act as In the first example, the relationship axioms for detailed rulings with the between the axioms "order signifies Fiqh-rules processing the logic obligation" and "there shall be no implications to accomplish the axioms infliction of harm on oneself or others" (see Figure 1 and 2). Thus, if constitutes a hierarchy i.e., in the cases analogically the "order signifies of a dog licking a plate and a fly falling obligation" is set, every case that into it, the second axiom provides an includes an order implies an obligation, implicit reason for the first one.22 e.g.: That finding yields two heuristic Methodologically, one may find Sunnah propositions formulated with regard to texts such as: "Pray as you have seen mathematical logic terms which will be me pray", "Recite the Quran", "Order deepened in the next section: your children to start praying when they are seven years of age", "If a fly falls Proposition 1: In Islamic into your plate, immerse it all", or "If a Jurisprudence, there exists a Fiqh-rule dog licks your plate, wash it seven for the deduction of any legal verdict, times, one of which should be with dust deduction being done using a formal and water." These texts are imperative logic system, i.e., Islamic Jurisprudence in terms of pray, recite, order, immerse constitutes a Consistent Legislation. and wash. If the axiom order signifies an obligation applies all implications in The corresponding concept in terms of imperatives are mandatory mathematical logic is correctness which actions (vs. voluntarily actions). If there is the ability of a to are similarities between the reasons calling for such imperatives, as with the cases of immersing the fly and washing harm on oneself or others", which makes it a the plate, specific logic formulas can be logical Fiqh-rule. 20 formalized (variables), e.g.: 21 The two reasons making the payment of zakat due are having more than the threshold and a year is completed. 22 The other cases may be attributed to the 20 The axiom common to both, the immersing axiom "bring benefit" because prayer and the and washing is: "There shall be no infliction of recitation of the Quran provide benefits.

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deduce new assertions which are 7. MATHEMATICAL LOGIC AND consistent within given axioms; FORMAL LOGIC As a brief overview, far from being Proposition 2: For every formal, mathematical logic comprises of query/legal Fiqh-question there exists a two constitutive features for the here legal verdict, i.e., Islamic Jurisprudence presented: constitutes a Complete Legislation. Firstly, it intends a complete and concise formulation of formal logic The corresponding concept in (incl. FOL)23 in so far that complex mathematical logic is completeness mathematical can be which is the ability of a formal system expressed in simple, singular formal to produce a of any assertions propositions. (cf. [Whitehead/Russell]) related to the domain of discourse. This has the reductionist advantage that

secondly, mathematical axioms in Lemma 1: If proposition 1 arithmetic, algebra, geometry, etc., can and 2 both proof valid, Islamic be described by countable many logical Jurisprudence would be complete with calculi in terms of statements about respect to a set of given Fiqh-axioms classes, relations, and syntax. (cf. and an intrinsic formal system of [Frege]) deduction. In this sense, the second constitutive

feature of mathematical logic according

to the "western"24 understanding was B. LOGIC & COMPUTER SCIENCE first understood by Leibniz (1646-1716) Since the inception of algorithmic in his "Characteristica Universalis". processing, namely with computer (cf. [Gerhardt]) science in the early 40ies of the past The symbolism, the hierarchy of classes century, it was realized that any of statements, the syntactic linking of technology to automate logical rules, simply, the of a system could have tremendous of concepts constitutes all necessary potential to solve problems by drawing areas of human mental activity, which automated conclusions from any given in turn, using this very system, becomes fact. Up-until-today, the most powerful transparent. (cf. [Wittgenstein] and and expressive formal method for [Tarski]) describing and analyzing information is Thus, the constitutive role of represented by First Order Logic (FOL). mathematical logic may be introduced In this abstract and formal context, logic as preceding all other sciences, or in the shall just be the concept of what follows words of Leibniz: from what, e.g., if two statements x, y "…that humanity would have a new are true, then one can infer a third kind of an instrument increasing the statement z from it while it doesn't matter if any of the statements is actually true! The formal, i.e., syntactic 23 More specifically, the axiomatic . quality is simply that if x and y are true, The classes of formalized areas are: then z must also be a logically valid propositional logic, predicate logic with statement. (incl. syllogistic), predicate logic with identity and identity theory [cf. Daghbouche 2013: p. 68]

24 "western" is hereinafter referred to as connotation for "non-Islamic"

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powers of reason far more than any The structure of formal logic languages, optical instrument has ever increased particularly the simple ones that are the power of vision." intended for the study of mathematics, [Whitehead/Russel: pp. XXVIII] should include five sets of symbols25: For the furtherance of our discussion it is largely irrelevant whether 1. Symbols of logical statements (QLn); mathematics is regarded as a further 2. Symbols of conjunctions (Qc); development of logic (Logicism, 3. Symbols of variables (Qv); represented by G. Frege, B. Russell, R. 4. Symbols of rules (Qg); Carnap) or if it consists of calculi that 5. A set of grammatical and dictation are formed out of formal systems by rules that distinguish structurally sound preceding axioms using rules sentences for erroneous ones (Gmr). corresponding to theorems (, represented by D. Hilbert, W.v.O. Furthermore, we may assign the Quine, H.B. Curry) or whether (Sm) to indicate an of mathematics represents basic mental Cartesian products of the components processes where the critical path of the sets QLn, Qc, and Qv. This yields consists of what can be constructed a set () which includes all effectively due to these processes, but possible relations between sets and their not what was raised as object of elements, independent of their observation by the mathematician magnitude. (Intuitionism represented by L. Thus Sm represents the set of all Brouwer, A. Heyting, L. Wittgenstein, possible sentences, correct or incorrect, Lorenzen). that result from the use of the symbols In order to provide with sound of QLn. foundations for mathematics, logicians Obviously, Sm is infinite even though of the 19th Century AD pursued the the sets used to generate it are finite. construction of a special formal system: This observation stresses the regulatory importance of Gmr because without proper procedures of how to combine 8. THE STRUCTURE OF LOGIC sentences, the formal system would SENTENCES even be structurally infinite. This formal system, as a whole, We will illustrate this with the represents an artificial language that is formalized Fiqh-example (see section similar to natural languages in having a 4), which contains: system of spelling and grammar that identifies possible sentences, QLn ≡ (Fs, Prv, Fv, Pgv) and Qc ≡ (→, distinguishing them from wrong ᴧ, v, inverse) expressions. For example, we cannot If we abstract primary and secondary say in English, "A tree and ran I", cases we realize that the following two because the conjunction "and" cannot rules apply: occur between a noun and a verb. Likewise, the expression (FS → → Fv) 25 The description of the structure of languages is meaningless because it contains an used here is semi-formal and intends only to error of symbols, as the arrow symbol convey the main of the topics on hand (denoting material implication) cannot without dragging the reader into the burden of be used twice in succession. precise formalizations as those formalizations can be found in any textbook of mathematical logic.

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((Fs ᴧ Pgv → X) v (Fs ᴧ inverse (Pgv) Usually this set contains two types of → X)) → (Fs → X) rules: (a) Axioms of the formal system and (inverse (Fs ᴧ Pgv → X) ᴧ inverse used (Qg’) and (b) Axioms of the (Fs ᴧ inverse (Pgv) → X) → inverse (Fs logical theory processed (Qg’’). The → X) classical syllogistic system traditionally used by ancient Muslim scholars X represents a variable that admits both, denoted S is the logic corresponding to Prv and Fv. Therefore Qv ≡ X the following fragment of English:

The set of grammatical rules Gmr, Every p is a q: ∀x( p(x) → q(x))∀( p,q) which is not explicitly mentioned in our example because we assumed its Some p is a q: ∃x( p(x)^(x))∃( p,q) presence implicitly, may include for No p is a q: ∀x( p(x) →~ q(x))∀( p,~ q) example QLn x Qc x QLn26. This means that an item of QLn is Some p is not a: ∃x((x)^ ~ q(x))∃( p,q) processed together with an item of Qc and another item of QLn following it; As this system is very well understood another example of Gmr might be: since ancient times and has been (QLn x Qc x QLn) x (Qc) x (QLn x Qc investigated recently in the shed of x QLn) modern advances in formal logic, comparing its expressive power to FOL These formulae mean – for example - subsets used in computer science and that we allow only expressions such as: investigating the complexity of algorithms based on it (cf. (Fs → Fv) or (Fs → Pgv) → (Fs → Fv) [Lukasiewicz]), emphasis in this paper shall be put on Qg’’ which is the set of The deduction in the example of section axioms of the logical theory being 4 can thus be formally represented as: formalized and processed, i.e., Fiqh- axioms and rules27. Thus for the sake of Sen. 1: (inverse (Fs ᴧ Pgv → X) ᴧ simplification but without loss of inverse (Fs ᴧ inverse (Pgv) → X) ↔ generality, Qg = Qg’’ in what follows. inverse (Fs → X) Reaching a conclusion C through the ⇔ Sen. 2: ((Fs ᴧ Pgv → X) v (Fs ᴧ (syntactical) rules of Qg will be inverse (Pgv) → X)) ↔ (Fs → X) depicted as the symbol ♦Qg C. We then write: ♦Qg (Fs → Fv) to We replace X with Fv so that it indicate that the sentence (Fs → Fv) has becomes: ((Fs ᴧ Pgv → Fv) v (Fs ᴧ been deduced from the sentences Sen. 1, inverse (Pgv) → Fv)) ↔ (Fs → Fv) Sen. 2, Sen. 3.

As we know that Sen. 3: (Fs ᴧ Pgv → Fv) is realized.

Hence (Fs → Fv) (Q.E.D.)

26 The notation A x B x C ... etc. is used for 27 Note that the set of rules Qg‘‘ in the previous Cartesian products example = {Sen1, Sen2, Sen3}.

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9. For every X from QLn: ((Fs ᴧ Pgv → SENTENCES X) v (Fs ᴧ inverse (Pgv) → X)) ↔ (Fs On the basis of the foregoing we → X) conclude that sentences are initially of two types: As such, it allows formulae such as:

The first type contains variables of the Sen. 4: ((Fs ᴧ Pgv → Fs) v (Fs ᴧ inverse type Qv, called general sentences, such (Pgv) → Fs)) ↔ (Fs → Fs) as Sen. 1 and Sen. 2. The second type has no variables. It In natural language: "I’tikaf is an only contains predicates and logical essential condition for I’tikaf whether conjunctions, such as Sen. 3. We will with or without a pledge, and all this is call this type detailed sentences28. a reason because I’tikaf is an essential condition for itself". There is a marked difference between these two types and it clearly appears This statement yields . when we want to express the semantics, i.e., the values "true" or "false". Should X therefore be limited to certain cases of the set QLn, or should it be left Let CS represent the set of true to include it all? If we were to limit X, sentences in the formal system. It is what would be the procedure to be clear that it will not be difficult for us to followed for this purpose? generate the meanings of the detailed The second problem lies in the fact that sentences on the basis of CS, even if if we were to consider the sentences that they are logically conjoint, because we yield new statements (or if we were to know the meanings of symbols through allow X to derive its values from CS QLn and also the meanings of the logic instead of QLn) we would encounter conjugations between these symbols. logic formulae that involve circularity However, when we try to establish the yielding even more nonsense, e.g.: meaning of the following formula, we If the following predicate is true M ≡ are faced with some important for every X from CS inverse (X) problems: M claims that all predicates from CS are false, but it is nevertheless one of these Sen. 2: ((Fs ᴧ Pgv → X) v (Fs ᴧ inverse predicates. (Pgv) → X)) ↔ (Fs → X) When we replace X by the value M we come up with the inverse (X) ≡ there is We will deepen two of these problems: a true X from CS (X). This contradicts the assumption that M The first is that this sentence has a is true meaning that the fact that M is general purpose, which means that its true leads to being false, which is origin is: contradictory. If we were to establish a pyramidal structure (hierarchy) so as to prevent general sentences from being uncontrollably mixed with other 28 In mathematical logic those sentences are sentences and to prohibit in particular called "instantiated". We prefer here the term any definition that involves circularity, "detailed" to provide more clearer context to Fiqh-questions

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we may avoid such nonsense For every X from QLn: ((Fs ᴧ Pgv → statements.29 X) v (Fs ᴧ inverse (Pgv) → X)) ↔ (Fs Therefore, mathematical logic works → X) with two sets:30 it is necessary to restrict the variable X 1. SymG the set of the meanings of with one value (at a time). predicates 2. VV the set of the values of This is done by using the set VV. variables Let us say VV ≡ (X=Fv, X= Prv). We SymG includes all the predicates from thus get: the set QLn with the values "true" and "false". For example: ((Fs ᴧ Pgv → Fv) v (Fs ᴧ inverse (Pgv) → Fv)) ↔ (Fs → Fv); and SymG ≡ (Fs = true, Pgv = true, Prv = ((Fs ᴧ Pgv → Prv) v (Fs ᴧ inverse (Pgv) true, Fv = true) → Prv)) ↔ (Fs → Prv)

Let us express the case of a person who This shows that individual sentences has performed I’tikaf, pledged, fasted yield different meanings according to and prayed all at the same time. If we the variation of the pair (SymG, VV) define the set SymG in this way, the applicable to the sentence. meanings of the detailed expressions From this perspective, sentences are such as (Fs ᴧ Y) will easily be defined classified into true sentences for any as we understand the meaning of the possible pair (SymG, VV)32 and are symbol .31 ᴧ called Valid Truth. Evidently, there are However, if we want to know the sentences that do not meet this meaning of: condition. In other words, there will be at least one pair of (SymG, VV) that yields the value "false". We assign the abbreviation {(SymG, 29 This pyramidal structure shall have steps that VV) ♥ Sen} if we wish to say that the are numbered from 0 to infinity, with the step 0 pair (SymG, VV) assigns sentence Sen representing expressions that include the value "true" and say "The pair characteristics of predicates, while the logical (SymG, VV) validates Sen". systems of step 1 include such expressions, and step 2 includes characteristics of these expressions, and step 3 characteristics of these characteristics, etc. 10. FORMAL INCOMPLETENESS 30 Standard textbooks on mathematical logic We now look into the relationship use the slightly more complex concepts of between formal (syntactical) deduction "assignment" and "interpretation" to describe and model-based deduction 33. correctness and completeness in a formally ♦ ♥ correct way. The description used here is For any formal system which yields for informal and intends to clarify those two every Qg=(Sen1.... SenN) and for every concepts intuitionally without going into the M from CS: unnecessary details of a mathematical formalism. 32 31 The meaning here is "true" because both Fs Such as the sentence Fs → Fs and Y are true predicates according to the said 33 The relation ♥ is sometimes called "semantic SymG. entailment relation"

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♦ Qg Μ ⇒ there is a pair (SymG, VV): principles of contemporary computer (SymG, VV) ♥ M science as it directly impacts the methods of general programming. Such a formal system is said to be Nevertheless, complete (and 36 correct (=consistent). decidable) formal systems with less expressive power than FOL are known And if it yields for every M from CS to exist and can be used in the context every pair (SymG, VV): of Fiqh:

(SymG, VV) ♥ M ⇒ there is a set Qg= 11. COMPLETENESS AND (Sen1... SenN): ♦ Qg M DECIDABILITY OF FIQH- SYSTEMS Such a formal system is said to be complete. Some famous formal systems which are weaker than FOL, but both complete Therefore, the equality of the two and decidable are among others: relations ♦ and ♥ would yield a , the set of FOL complete and correct formal system. validities in the signature with only Fundamental mathematical results show equality, the set of FOL validities in a that even Arithmetic cannot be signature with equality and one unary described using a complete formal , and the set of first-order system34. Boolean algebras. It is also known that Some conditions rendering formal every complete recursively enumerable 37 systems inconsistent/incomplete and first-order theory is decidable . relevant for the current work shall be summarized as: 36 The term decidable refers to the (Entscheidungsproblem) which 1. Circular definitions of logic sentences imposes the question of the existence of an (which we have already exposed). effective method for determining membership in They lead to inconsistency because the a set of formulas, or, more precisely, an correctness of a predicate is algorithm that can and will return a Boolean true or false value (instead of looping necessarily linked to its incorrectness. indefinitely). Logical systems such as propositional logic are decidable if membership 2. The use of semantic sets with non- in their set of logically valid formulas (or denumerable cardinalities, such as theorems) can be effectively determined. A those required by general mathematical theory (set of sentences closed under logical logic functions.35 consequence) in a fixed logical system is decidable if there is an effective method for The principle of incompleteness of FOL determining whether arbitrary formulas are is considered one of the most important included in the theory. Decidability should not be confused with completeness. For example, the theory of algebraically closed fields is 34 This famous result is attributed to Kurt decidable but incomplete, whereas the set of all Goedel's incompleteness (cf. [Goedel]). true first-order statements about nonnegative 35 The notion "denumerable" sets (cf. [Cantor]) integers in the language with + and × is is related to the important notion of complete but undecidable. "constructability" of an algorithmic approach, 37 This does not contradict the famous i.e., all cases of problems where the underlying incompleteness result, since it only says that one domain sets are not denumerable yield can start with any consistent first order theory T undecidability results (cf. fn 36) and construct a "logic completion" of T by

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As seen earlier, syllogisms were used Complete Legislation means, as we by ancient Muslim scholars to express have already noted, that every religious and use rules of Fiqh in real world question has a ruling (verdict). This settings (c.f. e.g.: [Al-Akhdari])38. The means: completeness of various formulations of syllogistic logic has been demonstrated For every M of SF: there is R of QR: in [Lukasiewicz]. Although syllogistics (M → R) provide with some quantification properties, there is a lack of predicates Consistent Legislation means that every which have -arity many more than one religious ruling has a reason. This as well as the possibility to consider translates into: functions. Thus they can be seen as subsystems of monadic first-order logic For every M of SF and for every R of (MFO), which is also less expressive QR: ((M → R) → there is E of QE: (E than full FOL. → R)) It is worth noting that a thorough study of the formal systems for Fiqh needed, We may choose taking into account the unique formal properties of this domain, may result in QLn ≡ QE ∪ SF ∪ QR and Qc ≡ (→, ᴧ, the choice of any of the above inverse) and Qv ≡ (X1, X2... Xn) and restrictions of FOL. Such a study is due. Gmr and Qg as discussed in section 8. Referring back to Fiqh, we can now In other words, QLn incorporates the discuss how to represent the previously totality of the predicates of all three suggested characteristic of Consistent sets. and Complete Legalization by using formal systems. Proposition 3: If a formal system For the set of Fiqh-questions we will is chosen in such a way that (QLn, Qc, use the symbol SF, for the set of rulings Qv, Gmr, Qg) is both logically complete QR, and for the set of reasons QE and decidable, then the question of respectively. Complete and Consistent Legislation

(Fiqh) is decidable (i.e., it is decidable Thus SF ≡ (M1, M2, M3...Mn), QR ≡ whether Lemma 1 holds or not). (R1, R2, R3...R,), QE ≡ (E1, E2,

E3...E,)39 Proof: Complete and decidable logic

system is equal to calculate the meanings of all possible logic expressions. As such, it is necessarily adding logical consequences of the formulas in possible to also know the meaning of T in a constructive way. Incompleteness states the expressions Consistent and that this process cannot be done for sufficiently Complete Legislation. expressive T. 38 The most apparent reason for choosing syllogisms to express Fiqh-rules by ancient scholars is represented by the fact that those logic systems were the only ones known at that time.

39 El represents the predicate "The reason l is a valid religious reason". The same applies to Rl and Ml.

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12. COMPLEXITY OF general formulas from a set of Fiqh- DEDUCTION IN FIQH-SYSTEMS axioms - in Propositional Logic is an 42 An important aspect with respect to any instance of a SAT problem . formal logic system, especially if the intention is to implement it on modern Proposition 4 and 5 have a direct computer systems, is the complexity of practical consequence for the efforts to its deduction procedures. As is the case realize Fiqh-deductions on with many interesting sub-sets of FOL contemporary computer systems: While which are complete and decidable: The the amount of computation needed for best known deduction methods require answering concrete queries/questions is exponential number of steps in the relatively small if the Fiqh-sub-system worst case. A prominent example of used is complete, i.e., if there are Fiqh- such systems is Propositional Logic. rules to be applied to each query Although the decision concerning the (queries = detailed sentences), deducing best formal system to use for Fiqh must Fiqh-rules from others may be be subject to a thorough investigation computationally challenging. eventually yielding a tailored formal The algorithm described below accounts system specifically designed for Fiqh, for both cases, i.e., while describing a we will consistently assume procedure of automatically solving user- Propositional Logic. Accordingly, the defined Fiqh-queries, it also describes a following two trivial propositions heuristic way of finding "generic Fiqh- related to general and detailed sentences rules" starting from a set of possible of Section 9 hold: queries in the different relevant chapters. Proposition 4: The deduction of detailed sentences (by into 13. ALGORITHMIC VS LOGIC general ones40) is efficient, i.e., in the worst-case polynomial with regard to PROGRAMMING length of sentences. The computer is one of the most important applications for : Straightforward substitution of mathematical systems. Perhaps one of values into a logic formulae using its most evident advantages is its factual simple procedures of linear complexity presentation of logic, i.e., many abstract in the length of formulas. are materialized by computer systems yielding visible results. Proposition 5: Deduction of Yet not everything can be described general sentences from a set of general logically. There are meanings that can and detailed ones is NP-Complete41 only be referred to by terminology systems, such as everything that is Proof: It is well known that general described by a series of values that may deduction - like the one needed to infer be infinite. Such series may not contain any logical structure and it may have

40 Which are basically Fiqh-rules. 41 cf. NP-complete. (2014, February 20). In 42 cf. Boolean satisfiability problem. (2014, Wikipedia, The Free Encyclopedia. Retrieved April 30). In Wikipedia, The Free Encyclopedia. 20:58, February 28, 2014, from Retrieved 21:01, May 2, 2014, from http://en.wikipedia.org/w/index.php?title=NP- http://en.wikipedia.org/w/index.php?title=Boole complete&oldid=596284955 an_satisfiability_problem&oldid=606502483

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values which have no apparent pattern, By contrast, logic-programming as is the case with the arbitrary numbers languages rely on a simple logic (12, 155, 76.4).43 representation of problems and have the In order to accommodate non-logical advantage of being able to predict the structures, a method is needed which results of application when the logic relies on representing the problems to system represented by those programs is be solved in terms of successive both, complete and decidable. commands in form of series that start Those facts are mentioned here to and finish in a controlled way underline the importance of the (algorithmic programming). Here, logic algorithmic method for entering and only determines the number and order resolving practical questions of Fiqh. of the solutions. This method can Although Fiqh meanings follow represent any series, whether logical or incrementally in a logical way that illogical, e.g., the Algorithm for Islamic admits formal definition, people’s heritage law in Kitab Al-Jabr wa-l- actions, represented by possible Muqabala. religious questions, are not necessarily The most important features of subject to immediate logical algorithmic programming languages implementation. Hence an algorithmic compared to their logical counterparts method is necessary in addition to the are: logical one to represent such cases.

1. Sentences do not have a direct logical structure44. In other words, the 14. ALGORITHMIC structure does not include predicates IMPLEMENTATION linked by standard attributes. It relies The need for looking at Fiqh in a formal on the principle of a chain of logic framework is complementary to commands (solution steps). the urgent need of re-writing the Fiqh 2. Semantics are not simply logic reference works in a style that can predicates being true or false. They are easily be understood by contemporary internal states of a machine according readers. Moreover, the re-writing should to operational semantics or address the need to provide solutions for applications of mathematical functions practical problems faced in an according to denotational semantics. increasingly complex and differentiated They follow one another according to world. Among others, this attitude is the order of commands in the series. supported by Al-Zuhayli (1350AH- / 3. There is no mathematical or logical 1923AD -): possibility to check the correctness of There is nothing to preclude re- writing Islamic Fiqh in the form of the results of general algorithms. articles that make it easier for a Indeed it is not possible to predict judge, or ordinary person to whether the algorithms will stop or not refer to its rulings by article and after the start of the solution operation. paragraph. This is indeed one of the urgent needs of our time. Old books 43 e.g., most statistical and mathematic of Fiqh are not easy to refer to for a denotations as well as the digit distribution of ruling on a particular question, all members of the set of real numbers (per definitionem) except by a specialist. Indeed the whole body of Fiqh needs to be re- 44 although they can be translated to non- intuitive logical sentences when needed organized and re-arranged into

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chapters, in accordance with the 15. PROGRAMMING METHOD systems that are familiar to today’s The method relies on generating all students and scholars. It is well possible questions under any Fiqh- known that Fiqh presents its material chapter or sub-chapter by means of an in detailed questions. It does not algorithmic machine. To translate the follow the approach of explaining its result of formal completeness and general theory before studying its decidability into a practical application, contents. This presents real the following steps were necessary: difficulties in identifying the religious rulings. (cf. [Al-Zuhayli]) (1) Define the formal structure of a simple Fiqh-question in the While Al-Zuhayli stresses the need for following way: re-arranging Fiqh-chapters he lags proposing or pointing to any specific The algorithmic process of method. The re-organization of legacy induction yields a religious systems, however, has become a full- question45 composed of four fledged discipline which is directly essential elements: related to computer science incorporating new approaches that were 1. subject unknown and impossible in earlier 2. tool times. Legacy methods of chapter 3. reason arrangement relied on a library system 4. method that arranged manuscripts on bookshelves according to their main or In case for example of a (non- subsidiary headings. Students could trivial) human subject, initial kind- only do research with a single indexing specific qualities for purification feature while enhanced methods classify may consist of: Fiqh-information with combinatorial and semantic variety providing with 1. anatomy more efficient research experience. 2. conviction (religious affiliation) The here suggested classification is an 3. properties inherent result of the algorithmic 4. action approach taking the user experience an etc… important step further by re-arranging the books using all possible These qualities may interact so as to combinations of Fiqh-questions produce numerous possibilities and generated from sets of terminology trees complex interactivity. Moreover, the valid in different chapters and/or sub- main qualities may contain secondary chapters allowing thus to address a qualities which may also interact question directly without assuming its within or outside the main qualities. answer beforehand. 46 Hence, a compound Fiqh-question is

45 Such as: a man performed wudu using water that has already been used to remove simple impurity. 46 Such as: a man performed wudu, then urinated, then washed his private parts, then performed ablution.

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a conjugation that gives a ruling based which may be as seen in on a sequence of Fiqh-actions/events proposition 5 computationally generating previous ones in the light of intensive. This process is explained the conditions of actions being linked in detail in the next section. to what is essential, recommended, etc. (5) Based on this state of formal (2) Construct a tree of defined terms completeness the next and last for each question which is stage consists of implementing a devised to contain all aspects of the complete algorithm which contains addressable Fiqh-questions (Figure the formalized, logic Fiqh-theory. 3 with a tree sample for all four This algorithm comes with two elements of a classic Shafi'i-school basic flavors: one which answers taymammum sub-chapter). simple questions and one which answers compound questions as (3) This stage is followed by what may seen above. An example of the be called generation of practical purification chapter, i.e., Taharah, questions. As illustrated in Figure will illustrate this step: 3, a potentially large number of possible questions is generated by a straightforward combinatory 16. PRACTICAL INSTANCES OF algorithm. As trees get more FIQH-RULES (NEGATIVE AND detailed in describing the different POSITIVE) aspects of a Fiqh-decision in the The formal concept of negative and particular chapter, the combinatory 47 positive Fiqh-rules was introduced to becomes enormous . reduce the apparent combinatory explosion resulting from step 3 above. (4) To reduce the combinatory Figure 4 illustrates the revision explosion seen in step 3 practical procedure needed to generate those rules are derived by excluding all rules. entries are inspected possibilities providing for the possibility of (manual) algorithmically of null-results, such abstraction of rule patterns. The result is as the performance of wudu by a two sets of negative- and positive rules baby or the performance of substituting the entire truth table48. taymammum without reference to a Having exemplarily referred to valid reason. This process yields Al-Suyuti (849-911AH / 1445-1505AD) negative practical rules while the (cf. [Al-Suyuti]), 12 applicable Fiqh- complements (inverses) are rules were divided during our practical generated on-the-fly as positive work into negative and positive ones. practical rules. The process of Some of them are not applicable to one deducing negative and positive chapter only but to many. For some of rules is an efficient practical the generated we could not (heuristic) alternative to classical find known primary rules in the Shaffi'i- deduction of general formulas

47 The constructed trees for taymammum (as 48 This is at least the objective of this per Shafi'i) yield an amazing potential abstraction exercise. It may happen that parts of combinatory of 15G questions (i.e., 15 Billion the truth table (i.e., single questions) do not fall questions) while the same number was 2000G under any category (whether +ve or –ve) of questions for the whole Tahara book. abstraction.

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school indicating a gap in the ancient 5. to perform an obligatory scholars descriptions of their used prayer may not be done before that intrinsic logic49. prayer is due. This rule is a part of a Processing the rules related to rule mentioned by Al-Suyuti: purification ultimately generated a "Tayammum to perform an obligatory number of secondary rules: prayer may not be done before that prayer is due and it cannot validate Negative rules: recommended prayers". This rule mentioned by Al-Suyuti is derived from 1. Tayammum becomes invalid when a primary rule that says: "If a special water becomes available. This rule is case is no longer valid, does the general derived from a primary rule that says: case apply?" (cf. [Al-Suyuti]) "What is permissible as a result of a certain reason becomes invalid when 6. Sparkling water to remove the that reason no longer applies." impurity caused by a girl’s urine is not valid. (No primary rule). 2. Nothing other than water removes an impurity that is beyond a person’s 7. In the case of taymammum the private parts. This is derived from a intention statement "I intend to perform primary rule that says: "When the two taymammum" is not valid. This rule is rulings of permissible and forbidden an exception from the rule that says: apply to the same thing, the forbidden "Whatever is derived from an intention takes precedence." It is also possible to statement that applies to a section of say that the use of a solid object, such as Islamic law should be expressly clear." a stone, is permissible for the removal of the impurity from the private parts as 8. When a sufferer of incontinence is a matter of necessity. This relies on the cured, his prior ablution is invalid. (No rules: "Necessities permit what is primary rule). forbidden," and "Necessities are measured according to need." Positive rules

3. It is not permissible to use something 1. When the lesser impurity (i.e., of value or something eatable to remove hadath) and ceremonial impurity (i.e., the impurity from one’s private parts. janabah) apply together, the grand This is derived from a primary rule that ablution (i.e., ) is enough to says: "Concessions may not be remove both.50 exercised by what is forbidden". 4. It is not permissible to use water that 2. When the ceremonial impurity (i.e., has been used (no primary rule). janabah) and menstruation apply together, one grand ablution (i.e., ghusl) is enough to remove both. 49 It is here worth noting that such gaps – if confirmed – indicate a breach of proposition 1 and thus the Fiqh-consistency assertion for the Shafi'i-school (related to this specific chapter) and would require from Shafi'i scholars a 50 The lesser impurity occurs as a result of a revision of logic reasons and/or intentions discharge through one’s private parts and it is behind those gaps. Negative- and positive rules removed by wudu while the ceremonial are thus a very important tool to investigate impurity occurs as a result of an ejaculation or completeness of Fiqh-theories. sexual intercourse. It requires a full bath.

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These two rules are derived from a 17. REALIZED ALGORITHMS primary rule that says: "When two Algorithms were realized in conceptual matters of the same type apply and their prototype terms for two different purpose is the same, they become Tahara sub-chapters: Wudu and mostly concurrent." taymammum. For taymammum, both flavors of the method described herein 3. The discharge of semen does not were realized. The one replying to make wudu necessary. This is derived simple questions and the one replying to from: "What necessitates a greater compound questions. For wudu only the action in particular, does not require the compound question part was realized. lesser one in general". The User Interface (UI) was implemented for Arabic as its 4. It is sufficient to use a dry object to translation to any other language remove the impurities of madhi and required consensus on what wadi. This is derived from the rule: terminologies to use for different Fiqh- "Should what is rare be treated expressions (nomenclature). In what separately or attached to what is of its follows only the simple question kind?" answering procedure for taymammum and the compound question answering Transferring this theoretical information procedure for wudu shall be described. into practical Fiqh by computing all The conceptual schematic of the simple possibilities that can be derived from questions answering procedure is the set of terms yields the complete set depicted in Figure 5. The method of secondary questions, i.e., an generated a complete truth table which intermediate step towards Fiqh- included all possible Fiqh-questions in completeness as discussed above. the sub-chapter of taymammum. The In contrast, a traditional Fiqh-scholar truth table was then reduced (manually) without computer-assisted aid cannot to two sets of rules which formed the consider the totality of secondary backbone of the logical engine. A questions, hence, remains in an restricted logic compiler understanding intuitive, artistic anticipation of the 51 the format of Fiqh-rules (which is that most balanced ruling. of propositional calculus formulas) was written (in Visual basic) to enable matching of User queries with rules and providing intelligent results (i.e., results with explanations) as illustrated in Figure 5. Based on the principle of compound questions, the general wudu-algorithm shown in Figure 6 was also realized. As the main difference between the two flavors (simple and compound) is the fact that sequences of actions have to be 51 This is not due to the difficulty of the task, taken into account in the latter case, the but because the mere magnitude and complexity algorithm was realized in form of a in terms of combinatory. Hence, continuous follow up is required to place such secondary questions under a general- or specific rule and treat them with suitable rulings.

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finite state automaton52. Finite state C. RESULTS AND DISCUSSION automata have exactly the required In 1928, the German mathematician expressive power to model sequences of David Hilbert (1862-1943) formulated a Fiqh-related actions/events. mathematical question in terms of the As illustrated in Figure 6, this algorithm Entscheidungsproblem. Its essence asks takes into account the following six for an algorithm to decide whether a obligatory actions to be performed in given statement is provable within the order to reach a valid state of Tahara set of axioms using the rules of logic (Shafi'i): (cf. [Church] and [Turing]). Although the narrow context of this scientific • Correct intention assertion project was to try to solve important • Washing the face questions of fundamental mathematics, • Washing the arms up to the ankles its broader context was much more • Wiping parts of the head important, namely: To formalize a • Washing the feet including the heels generic tool for informed, precise and • Doing all this in the above given transparent decision-making. sequence Muslim philosophers and scholars didn’t go a much different path in the As preserving the order of one’s actions dawn of Islamic civilization. As is required by the Shafi'i-school, the mentioned in the introductory remarks, realized automaton is a deterministic fundamental properties of Islamic Law 53 one . The UI provides action buttons (Shariah) and its Fiqh-sub-systems for actions from 1-5. All these actions were investigated with such a precision affect the state of purification where the that inventing new, previously unknown algorithm changes the cases step-wise methods and concepts to enable answers as the user enters another action. If a to Fiqh-questions became necessary. In user misses to enter an action or if this context, fundamental mathematics he/she changes the order of some played a vital role. Not that much as a action(s), the system is able to advise subject of investigation but rather as a and correct him/her (pre- means to study the newly created Fiqh- differentiation). The process of wudu machinery. Muslim scholars knew that can also be conjugated with its they needed to be sure that their invalidation (hadath) algorithm emerging, previously unknown tool of generating rulings on whether the wudu Fiqh was precise enough to answer even remains valid or has been invalidated by the most sophisticated questions. some action/event sequences. This has Although their Entscheidungsproblem been realized in the taymammum- was not explicitly formulated (asking version of the compound algorithm and for an algorithm to decide whether a is not described here. given statement is provable from the Fiqh-axioms using the Fiqh-rules), it 52 cf. Finite-state machine. (2014, May 3). In was implicitly understood that they Wikipedia, The Free Encyclopedia. Retrieved were working in the direction of a 21:18, May 3, 2014, from positive solution54. http://en.wikipedia.org/w/index.php?title=Finite -state_machine&oldid=606933991 53 A Hanafi wudu-Automaton is not necessarily 54 Many ancient Fiqh-books contain claims of deterministic since preserving the order of completeness of sets of generic Fiqh- or usul- wudu-actions is not obligatory according to the rules. Some of the scholars used to reduce basic Hanafi-school. principles of their school to only a small amount

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As such, it is not entirely new to express 18. SUMMARY an Entscheidungsproblem for Fiqh and The results of this work shall be to determine criteria under which it can summarized as follows: be positively solved (cf. Proposition 3). Pre-requisite for doing so was the One: Usul- and Logic: formulation of the properties consistent and complete for Fiqh- 1- Consistency and completeness (Propositions 1 & 2). properties of Fiqh-legislations Since the attribute of completeness are formalized in mathematical depends on how far Fiqh-rules and their logic terms (Propositions 1 & 2 reasons can fulfill the underlying formal as well as Lemma 1) principles, it is evident that formally 2- An Entscheidungsproblem is complete legislations of Fiqh were formulated with the criteria for never discussed before, neither by its positive solution clearly leading scholars of usul nor by ancient determined (Proposition 3) Muslim scholars of logic. No matter 3- Complexity of Fiqh-algorithms how comprehensive a scholar’s ability (detailed- and generic queries) is to memorize, classify and induce, he investigated (Propositions 4 & cannot claim to have looked into all 5) possible questions within a certain area and verified them all consistent with the Two: Fiqh: rule applied to them. Today, we are in a much more comfortable position as we 1. Reclassification of the chapter on can leverage machines deploying logic. Tahara according to the Shafi'i-school With the help of these machines we can converting it into secondary cases consider looking into all these pertaining to questions that have the possibilities, at least the ones related to four essential elements a particular set of terms. Complete Legislation appears certain at 55 2. Drawing logical trees linking the four first sight, but a careful examination essential elements to Fiqh-decisions in shows that there is no extensive study of the chapter how far the usul-rules stated in scholars’ books give all rulings. Hence, attention 3. Introduction of negative and positive should be paid to regulate this aspect rules (+ve & -ve) formalized in mechanically, as manual is propositional calculus form which are practically impossible. essentially a way of compressing Fiqh- chapters

Three: Application:

1. Use of wudu/taymammum prototype algorithms for applications that answer user-defined Fiqh-questions with no or very little Fiqh-background of those rules claiming their sufficiency to answer all related questions and provide a rational argumentation for that answer. 2. Assistance to scholars by identifying negative and positive rules in a chapter 55 It is commonly assumed for the Islamic Jurisprudence to give a ruling on everything. to investigate consistency and

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completeness of the Fiqh-theory of a 13. Enable continuous ijtihad given Fiqh-school and identify logical processing on 24x7 basis gaps in classical Fiqh-arguments 14. Make Islamic Jurisprudence (all schools) available to audiences Future objectives may be summed up as around the world by translating follows: Fiqh-algorithms to all major languages 1. Fully leverage the work of earlier 15. Get most balanced religious answers scholars defining many ancient to people’s current and future concepts more precisely especially questions those related to consistency and 16. Train students of to completeness give sound and most balanced 2. Investigate syllogistic and rulings using formal Fiqh-systems propositional systems for their 17. Establish a rational basis for adequacy to be used as a basis for correspondence between Islamic modern, algorithmic Fiqh-machines and secular legislations (especially SAT problems occurring in both systems and their possible Together with broadly built-up internet solutions) infrastructure, a re-algorithmization of 3. Revisit the current definitions of Islamic Jurisprudence with an usul- and Fiqh-rules yielding a more appropriate cloud-presence not only precise logical framework includes the possibility for access by the 4. Re-arranging ancient books in a people, but with: system taking questions/queries into account instead of themes only a) Transparency and accountability 5. Ensure that questions are classified on the axiomatic level, i.e., which properly under the largest controlled axioms were considered, how and number of specific- and general why? Fiqh-rules b) Completeness and consistency as 6. Explore new yet inherent Fiqh-rules discussed in the preceding chapters 7. Check the applicability of Fiqh- yielding a total consideration of rules to present and future cases axioms and rules. 8. Ascertain how a Fiqh-rule truly c) Legal security and legal equality operates while defining it in a solid with rule of law, i.e., any verdict way.56 must account for the axioms which 9. Attain Fiqh-completeness for are completely transparent to the chapters and/or sub-chapters whole world, for Muslims and non- 10. Identify and put together all Fiqh- Muslims alike. reasons mentioned in ancient books 11. Confirm the importance of logic in the study of Fiqh-limitations 12. Operate the process of analogy (qiyas) fully mechanically

56 Is the rule merely collation of secondary cases, or does it operate like the general rules? Are they flexible to accommodate new cases?

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19. FUNDING

This research was supported by Al-Zuhayli, Wahba Mustafa, Juhud Tekcentric Corporation, 155 P. Moffett Taqnin al-Fiqh al-Islami, Muassasah al- Park Drive, Sunnyvale, 94089 Risalah, Beirut, 1987 California, USA; and Mir Amir , Belmont, 94002 California, USA. Cantor, Georg, Beiträge zur Begründung der transfiniten Mengenlehre, in: Mathematische 20. ACKNOWLEDGEMENT Annalen, 1895, 46:481-512, and ibid, 1897, 49:207-246 A special appreciation is dedicated to

Adil Salahi, London (UK), who Carnap, Rudolf, Logische Syntax der translated important parts of this paper Sprache, Springer, Wien (A), 1934/36 (Arabic into English) from a previous concluded technical whitepaper (2001) Chester, Barnabas Bernard Hughes and to Sami M. Angawi, Jeddah (KSA), Robert, Latin translation of al- who provided with critical philosophical Chorezmis al-Jabr. A new critical discussions. edition, Stuttgart 1989

Church, Alonzo, An unsolvable problem 21. REFERENCES of elementary number theory, American Al-Akhdari, (2014, March 26): In Journal of Mathematics, 1936, 58:345- Wikipedia, The Free Encyclopedia. 363 Retrieved 18:34, April 4, 2014, from http://en.wikipedia.org/w/index.php?titl Church, Alonzo, A note on the e=Al-Akhdari&oldid=601304682 Entscheidungsproblem, Journal of Symbolic Logic, 1936, 1:40–41 Al-Dawalibi, Muhammad Ma`ruf, al- Madkhal ila `Ilm Usul al-Fiqh, Beirut: Daghbouche [2012], Karim, The Dar al-`Ilm li’l-Mala’in, 1965 Ontological Principle, J. Acad. (N.Y.) 2012, Vol. 2, 4:160-163 Al-Ghazali, al-Mustafa min ‘ilm al-usul (The Essentials of the Islamic Legal Daghbouche [2013], Karim, The Theory), : al-Matba'ah al- Secularization Ability of Islamic Law: Amiriyyah, 1904-1906 An Epistemological Analysis, J. Acad. (N.Y.) 2013, Vol. 3, 2:67-116 Al-Khwarizmi, The Algebra of Mohammed ben Musa, translated and Ebbinghaus, Heinz-Dieter, Flum, Jörg; edited by Frederic Rosen, Hildesheim: Thomas, Wolfgang, Einführung in die Georg Olms Verlag (originally mathematische Logik, pp. 209-217 published 1831), 1986, p. 3 Eley, Lothar, Philosophie der Logik, in: Al-Suyuti, Jalal al-Din, Ashbah wa al- Erträge der Forschung Bd. 230, Nazair Fi Furu al-Shafiiyah, Beirut: Wissenschaftliche Buchgesellschaft, Dar al-Kitab al-Arabi, 1987 Darmstadt, 1985, pp. 238-239

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Frege, Gottlob, Funktion, Begriff, Tarski, Alfred, Der Wahrheitsbegriff in Bedeutung, Vandenhoeck & Ruprecht, den formalisierten Sprachen, Studia Göttingen, 1975, reprint, ed. Dittes, Philosophica, Bd. I, 1935 Günther Turing, Alan, On computable numbers, Gerhardt, C.J. (Eds.), Die with an application to the philosophischen Schriften von G. W. Entscheidungsproblem, Proceedings of Leibniz, Vol 7, 1890 the London Mathematical Society, Series 2, 1936-7, 42:230–265 Glasse, Cyril, The New Encyclopedia of Islam, Altamira, 2001, p. 141 van der Waerden, Bartel L., A History of Algebra From al- Khwarizmi to Goedel, Kurt, Über formal Emmy Noether, Springer, Berlin, unentscheidbare Sätze der Principia Heidelberg, New York (USA), Tokyo Mathematica und verwandter Systeme I, (Japan), 1985, p. 7 Monatshefte für Mathematik und Physik 1931, 38:173-198 Whitehead, Alfred North, Russell, Bertrand, , Hammad, Ahmad Zaki, Abu Hamid, Al Cambridge University Press, 2nd Ghazali's juristic Doctrine in 'Al- edition, 1963 Mustafsfa Min 'Ilm Al-Usul' with a translation of Volume One of 'Al- Wittgenstein, Ludwig, Tractatus Mustafsfa Min 'Ilm Al-Usul, Phd. Logico-Phliosophicus in: Ostwalds Thesis, The University of Chicago, "Annalen Naturphilosophie", 1921 1987

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22. FIGURES

Figure 1

Usul-rules act as axioms for detailed rulings with the Fiqh-rules processing the logical implications to accomplish the axioms.

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Figure 2

Together with primary sources (Quran and Sunnah), usul- and Fiqh-rules reside on the axiomatic side (left part). The fact-base and generated verdicts (database) reside on the hypothetic side (right part) yielding balance between static & constant axioms and dynamic & variable hypothesis. The process of pre-differentiation reduces complexity by heuristically increasing query-efficiency with a syntactically- and semantically closed terminology system defined by primary sources.

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Figure 3

chapters - sub Taiammum in in Terminological decision tree sample for sample tree decision Terminological question elements question

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Figure 4

rules reducing reducing rules - Fiqh combinatory possibilities of questions possibilities combinatory Positive and negative negative and Positive

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Figure 5

Answering simple questions Answering simple

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Figure 6

Automation) - Wudu ( Answering compound questions Answering compound

36