Understanding Prescriptive Texts: Rules and Logic as Elaborated by the M ī m āṃ s ā School *

The M ī m āṃ s ā school of Indian philosophy elaborated complex ways of interpreting the prescriptive portions of the Vedic sacred texts. The present article is the result of the collaboration of a group of scholars of logic, computer science, European philosophy and Indian philosophy and aims at the individuation and analysis of the deontic system which is applied but never explicitly discussed in M ī m āṃ s ā texts. The article outlines the basic distinction between three sorts of principles —hermeneutic, linguistic and deontic. It proposes a mathematical formalization of the deontic principles and uses it to discuss a well-known example of seemingly conflicting statements, namely the prescription to undertake the malefic Ś yena sacrifice and the prohibition to perform any harm.


The Mīmāṃsā School and Its Primary Focus
Like most Indian philosophical schools, the Mīmāṃsā school has a fundamental text and a fundamental commentary on it. The first extant text of the Mīmāṃsā school of Indian philosophy, the so-called Pūrva Mīmāṃsā Sūtra (henceforth PMS 1 ), can be approximately dated to the last centuries BCE. This has been commented on in the first centuries CE by Śabara in his Śābarabhāṣya (henceforth ŚBh), which together with the PMS constitutes the common ground for all successive developments within the Mīmāṃsā. 2 In fact, even after the Mīmāṃsā split into the two sub-schools of Prābhākara and Bhāṭṭa Mīmāṃsā (around the 7th c. CE), the PMS and the ŚBh remained the shared foundation of the schools. After having been at the philosophical centre of innumerable debates for centuries, the Mīmāṃsā sees its importance declining in contemporary India. The Mīmāṃsā focuses primarily on the exegesis of sacred texts (called Vedas), and more specifically on their prescriptive portions, the Brāhmaṇas. These contain sacrificial prescriptions accompanied by mythical 2

Rules For Interpreting Deontic Texts
The Mīmāṃsā thinkers were not the only experts on the Vedic rituals. By contrast, there were a number of ritualists who only focused on the praxis of the ritual. Other types of texts, the Śrautasūtras and the paddhatis, described the rituals in much more detail than did the Mīmāṃsā texts. But what distinguishes Mīmāṃsā authors is their theoretical interest in the way the Brāhmaṇa texts work. It was this interest which led them to elaborate on and/or systematically implement nyāyas. Little research has been done on the Mīmāṃsā nyāyas. Apart from Freschi (forthcoming), there is an alphabetic list of these rules in an appendix of Kane's History of Dharmaśāstra. Ancient and Mediaeval Religious and Civil Law in India. This however does not distinguish between metarules and topics of each section (adhikaraṇa) of the PMS and ŚBh, so that, e.g., "the meaning of the verbal root" is listed along with proper metarules such as "the maxim of the application of the [procedure as found in the] majority of cases." More generally, as for the Mīmāṃsā approach to prescriptive texts, several scholars, e.g., Yoshimizu 6 and Marui, 7 have noted that the Mīmāṃsā has a chiefly deontic focus but they did not attempt any formalization of Mīmāṃsā prescriptions. A first attempt to use mathematical logic in the analysis of the Mīmāṃsā deontic systems is found in Horn's work (Horn 2001) 8 which used one of the few English translations available of a Mīmāṃsā text (Edgerton 1929) 9 , and discussed the analysis of prohibitions in Mīmāṃsā. But in spite of the interest raised by the opportunity to formalize Mīmāṃsā principles, a comprehensive endeavour in this direction has not been attempted. Indeed apart from Horn's work, which addresses only one Mīmāṃsā prescription (and the analysis of a few prescriptions using Standard Deontic Logic in Freschi 2012 10 ), no Mīmāṃsā deontic principle has been formalized using the methods of mathematical logic. A possible explanation for this situation is that Mīmāṃsā has never been intensively studied by scholars familiar with both Sanskrit and mathematical logic. By contrast, two other central schools of Sanskrit philosophy, namely Nyāya and Vyākaraṇa, have been investigated also by scholars of linguistics and logic (for formal analyses of Navya Nyāya texts, see, e.g., Chakraborty et al. 2008 and especially Ganeri 2008 therein 11 ). The studies about such schools have greatly benefited from these investigations.
The present paper was inspired by the attempt to understand and, as far as possible, re-construct the system of rules which was operating beyond the Mīmāṃsā interpretative strategies. The project was developed as a collaboration between the authors of this article and thus involved the elaboration of a shared methodology between philosophers, logicians and philologists. The first necessary step was the discussion of whether such an approach was suitable for the material examined.
Although it is possible that Mīmāṃsā authors merely used an ad hoc approach and thought of specific solutions to each problem instead of general rules for solving classes of similar problems, the structure of the ŚBh suggests a different interpretation. In fact, the ŚBh displays a clear five-fold structure: 1. enunciation of the topic (viṣaya) 2. enunciation of the problem (saṃśaya) 3. prima facie view of the problem (pūrvapakṣa) 4. antithesis to the prima facie view (uttarapakṣa)

conclusive view (siddhānta)
Steps 3-5 can be repeated several times if the problem is particularly complex and needs a detailed discussion. More important from our point of view is that the upholder of the prima facie view, the upholder of the antithesis and the upholder of the conclusive view (who can be identified with Śabara himself) all resort to the application of rules. In fact, the discussion is mostly all about which rule should be applied and why or why not. Thus, it appears that the Mīmāṃsā authors were hoping to create a system that would be independent of the arbitrariness of specific ritualists. Mīmāṃsā authors had in front of them an already existing corpus of ritual prescriptions and most probably also of ritual praxis but they nonetheless tried to detect an order in that corpus that could be described by the set of rules they were developing. During this process they probably also imposed new interpretations on the ritual corpus (which were not always accepted by ritualists; see Daya Krishna 2001 12 ).
A further, albeit ex post piece of evidence for the general nature of Mīmāṃsā rules is the fact that they were massively implemented in the philosophical school of Vedānta and in the field of Dharmaśāstra (jurisprudence). This application is even more significant insofar as it occurred as if the possibility of applying the Mīmāṃsā rules outside the context of the specific Vedic injunction in connection with which they were first developed was obvious and did not need to be questioned at all.

Sources for the Identification of Metarules
The main sources of the present paper are the ŚBh and the PMS. In the case of major disagreements between the two main schools of Pūrva Mīmāṃsā, the texts of their respective founders have also been consulted, namely Kumārila's Ślokavārttika, Tantravārttika and Ṭupṭīkā, which together comment on the entire ŚBh, and Prabhākara's Bṛhatī, which also comments on the ŚBh but is incompletely transmitted. In some cases, other sources (such as the late primer Mīmāṃsā Nyāya Prakāśa 'Light on the Rules of Mīmāṃsā,' henceforth MNP, for which see Edgerton 1929) have also been examined in order to find rules which seemed to be presupposed by passages in the PMS and ŚBh. 13 The first problem for identifying the basic principles, the ones presupposed by the majority of the other rules, is the intersection of at least three sets of principles. These are not explicitly distinguished in Mīmāṃsā texts and thus their identification is completely a posteriori and only aims at making the structure of the Mīmāṃsā reasoning clearer for its contemporary interpreters: 1. The hermeneutic principles are needed in order to recognize the boundaries of a given prescription and the way it has been formulated; 2. The linguistic principles are needed in order to discuss the interpretation of linguistic peculiarities of the Sanskrit form of the various prescriptions; 3. The deontic principles are used in order to explain the conceptual structure of the Brāhmaṇas' obligations.

Figure 1: Principles in Mīmāṃsā
The three sets of principles overlap only in part (see figure). In the next sections, we will discuss these three kinds of principles and go into detail about some of them, albeit without attempting to produce a comprehensive list of rules implemented by Mīmāṃsā authors.

Hermeneutic Principles
Hermeneutic principles are those that directly examine the Brāhmaṇa texts and whose significance usually cannot be automatically extended outside them, e.g., to a different corpus of texts. If they are extended, this can be done only with specific caution and only to another textual corpus (as it was done in the case of Mīmāṃsā principles being adopted for the Dharmaśāstric jurisprudence). 14 These principles often refer to a shared background knowledge, e.g., to the way the various classes of Śūdras or Brahmans differ when it comes to the study of the Veda. Mīmāṃsā authors had to develop these principles first of all due to epistemological concerns, namely because they considered the prescriptive portion of the Veda authoritative and thus needed to distinguish such portions of the Veda from the rest. In order to make sense of complex texts like the Brāhmaṇas, in which it is not at all easy to distinguish between what belongs to a certain ritual, what to another, and what is just a mythological excursus, Mīmāṃsā authors needed to be able to distinguish the boundaries of a given prescriptive passage. Consequently, some basic hermeneutical rules deal with the identification of single prescriptions through syntax as well as through the unity and novelty of the duty being conveyed. The closer one comes to automatizing the process of distinguishing a ritual from a complex textual passage, with intermingled mythological explanations and excursuses, the more complicated the enterprise appears to be, so that the rules easily multiply.
In the following lists we tried to enumerate the cornerstones of the Mīmāṃsā hermeneutic principles. The first list regards the nature of Vedic prescriptions as outlined in these principles.
1. No Vedic prescription can be meaningless. If it appears to be meaningless, it is not a prescription (vidhiś cānarthakaḥ kvacit tasmāt stutiḥ pratīyeta, PMS 1.2.23). (meaningfulness) 2. Since Vedic prescriptions cannot be meaningless, each prescription must be construed as prescribing a new element (vidhir vā syād apūrvatvāt, vādamātraṃ hy anarthakam, PMS 1.2.19, see Kataoka 2003 15 ). Text passages which seem to only repeat a given content must have a deeper and different meaning, e.g., enhancing the value of the sacrifice to be performed. (novelty) 5. Each prescriptive text, which may entail several sentences, is assembled around a principal action to be done (conveyed by the principal prescription); all the rest is subsidiary to that action (PMS 1.2.7). A sacrificial substance may lead to a result resting on an already prescribed act, like a king's officer can achieve a certain result only insofar as he relies on the king's authority (homam āśrito guṇaḥ phalaṃ sādhayiṣyatīti, yathā rājapuruṣo rājānam āśrito rājakarma karotīti, Vṛttikāra within ŚBh ad PMS 2.2.26). (auxiliarity) As apparent in the next diagram, we interpret the rule regarding the meaningfulness of the Veda as being presupposed by all other ones, and thus as having the aspect of meta-rules: The second list, by contrast, deals with the identification of Vedic prescriptions: a. The most powerful instrument of knowledge for knowing the meaning of a prescription is what it states directly (śruti), which is more powerful than its implied sense, context, syntactical connection, etc. (niṣādasthapatinyāya PMS 6.1.51-52). (instruments of knowledge) b. Only what is intended (vivakṣita) is part of the prescription. For instance, in the rule "The [sacrificer] should wipe the goblet," the singular in "goblet" does not count, since "goblet" only calls to memory something one already knew (in linguistic terms, a topic), and thus amounts to "the goblet or goblets you already know of, however many they are" (grahaikatvanyāya, PMS 3.1. [13][14][15]. By contrast, if the linguistic peculiarities are part of the comment (e.g., they are directly related to the prescribed duty), then they do count. For instance, in the rule "The [sacrificer] should kill a piece of cattle," the singular in "piece of cattle" is significant, since "piece of cattle" is part of what is newly prescribed (in linguistic terms, a comment), and thus the term amounts to "a single piece of cattle" (paśvekatvanyāya PMS 4.1.15, the same as ārtyadhikaraṇanyāya, PMS 6.4.22). Thus, whether something is intended or not is determined by its link to the sentence's principal duty. (topic vs. comment distinction) c. Any prescribed action must have a result. If a prescribed action seems to have no result, happiness is to be postulated as the general result (viśvajinnyāya, ŚBh ad PMS 4.3.10). (postulation of result) Some of these rules appear characteristic of the Brāhmaṇa hermeneutics (e.g., the strict distinction between the prescriptive parts of the Veda and the non-prescriptive, auxiliary parts). Others may be more general linguistic principles. Yoshimizu (2006) 16 has, for instance, suggested connecting the principle (b) (immediately above) with the linguistic distinction between topic and comment, and with the consequence that only what is part of the comment is intended, whereas what is part of the topic is not intended. The hermeneutic nature of these principles is shown by the fact that such distinctions disappear if one tries to express them in logical terms.

Linguistic Principles
The linguistic principles can be considered a subset of the hermeneutic ones, but they examine more specifically how Sanskrit works even outside the Veda, e.g., the import of accusative endings. Some of these principles have had a deep impact on Sanskrit linguistic thought, as is especially the case for the following definition of a (prescriptive) sentence: In other words, a single prescriptive sentence is defined through the fact of its conveying a single purpose. If there is syntactic expectation (i.e., if there are elements in the alleged sentence requiring further complements to be understood), this indicates that the sentence must be split into two. The linguistic principles had their origin in the practical need of detecting distinct prescriptions within the continuous text of the Brāhmaṇas (as discussed also in the previous section), but within the history of Mīmāṃsā and subsequently also of Indian linguistic thought it grew into a general rule for the definition of sentences in general.
Similarly important for the history of the methodology of exegesis within Mīmāṃsā is the assertion (discussed in ŚBh ad PMS 1.1.1) that words should be interpreted as far as possible according to their ordinary (laukika) meaning and not as technical terms (paribhāṣika). The topic of establishing the correct meaning of words is dealt with using various principles (e.g., the aśvakarṇanyāya in the Ṭupṭīkā ad PMS 4.4.1, similar to the principle enunciated in ŚBh ad PMS 6.1.44-49; the bahirnyāya in ŚBh ad 3.2.1), all focusing on the idea that the meaning of words in the Veda should be the same as in ordinary language. 18 Further principles deal with the semantic interpretation of longer passages, as in textual linguistics, e.g., with the conditions for metaphorically interpreting a given prescription.
On the grammatical level, a rule notes that in order to express the meaning of the suffix both the root and the suffix must be uttered together: The root and the suffix together express the meaning of the suffix (prakṛtipratyayau pratyayārthaṃ saha brūta iti, ŚBh ad 3.4.13).
Similarly, several rules are dedicated to the semantics of case endings. All such rules pertain to the functioning of morphology.

Deontic Principles
Deontic principles were introduced by Mīmāṃsā authors in order to make sense of deontic texts, primarily of the Brāhmaṇas. As already mentioned, none of the three distinct sets of principles was the object of a systematic description by Mīmāṃsā authors. They were rather implemented whenever needed, so that one is confronted with the task of identifying and classifying them a posteriori. This is particularly true for the deontic principles, since Mīmāṃsā authors were chiefly interested in the hermeneutics of Vedic texts and deontic problems per se remained in the background. They, too, must be extracted by scholars from the main hermeneutic enterprise. Accordingly, as yet no adhikaraṇa has been found that discusses only deontic rules, although several sections are dedicated to long and intense debates on the application of one or the other hermeneutic or linguistic principle (alone in the last lines of TV ad 2.1.12 the objector or upholder of the conclusive view evokes six different metarules about the passage in question).

Why Mathematical Logic?
Given that deontic principles are not clearly outlined but just implemented by Mīmāṃsā authors, a rigorous mathematical formalization would greatly benefit the understanding of them. Indeed, such formalization is exactly meant to provide instruments for the representation, analysis and discussion of such principles. Thus it will be possible to investigate the conceptual core of arguments and principles, distinguishing it from contextual details. Moreover, such a formalization might provide a different perspective on Mīmāṃsā arguments and improve our ability to explain the seemingly conflicting obligations in the Vedas. However, one might still ask why one should attempt to examine the Mīmāṃsā deontic principles precisely according to mathematical logic. This question ultimately deals with the universality of logic. While the investigation of this very interesting issue would exceed the limits of the current paper, which is presenting a casestudy of what happens once such formalizations have been provided (see Gabbay, Abraham, and Schild 2013 for a similar approach to the Talmud 19 ), a few words on this may nonetheless be appropriate. At the core of logic lies the assumption that, in correct reasoning, conclusions follow from reliable premises via rigorous (and verifiable) inferential steps. It can be easily detected among Mīmāṃsā authors that they also (at least implicitly) held this assumption, due to both their theory of inference (anumāna) and their attention to possible violations of it. For instance, Kumārila emphasizes the fact that a text is not epistemically reliable if its beginning is not reliable, even if its whole chain of transmission is reliable. The classical example of this kind of unreliability is that of a chain of truthful blind people transmitting information about colours (andhaparamparānyāya, TV on PMS 1.3.27).
Given that the attitude of the Mīmāṃsā school towards rational and inferential reasoning justifies a formal approach to its study, the problem amounts to which system of reasoning (i.e. logic) one should adopt. The best known logic-classical logic-focuses on assertions, not prescriptions, and thus is clearly not adequate. George Henrik von Wright and many other authors have developed a number of systems of deontic logic exactly in order to address prescriptions (see e.g.  for an overview 20 ). The best known among such systems of logic is Standard Deontic Logic, or briefly . Without going into further details, the main feature of this logic is that it extends classical logic by introducing the additional operator , which stands for "It is obligatory that...," thus allowing the formalization of prescriptive statements.
While is reasonably simple and well-behaved as a form of logic, it is not suited to deal with conflicting obligations (e.g., ), something that is found in parts of the Vedas; this is one of the points that render insufficient to deal with the intricacies of the reasoning employed by Mīmāṃsā authors. An example of this is the well-known case of conflict regarding prescriptions arising in connection with the Śyena sacrifice, which is broadly discussed by Mīmāṃsā authors (see section "Logic at Work: the Śyena Sacrifice" for its formal treatment in our Mīmāṃsā-inspired logic): A. "One should not harm any living being" (na hiṃsyāt sarvā bhūtāni, a prohibition found in the MBh ( The problem is that while the first prescription prohibits any sort of violence, the second enjoins it. Again, without going into details, this intuitive reasoning can be paralleled in , so that from a reasonable formalization of the statements A. and B. we can conclude by the principles of both that it is obligatory to harm a living being and that it is obligatory not to harm a living being. But this is in contradiction with one of the fundamental principles of , which states that nothing can be obligatory at the same time as its negation. Given that the Mīmāṃsā authors embrace the principle of noncontradiction (see section "Formalizations of Mīmāṃsā Deontic Rules" below) and that they invested all their efforts in creating a consistent deontic system, we conclude that does not provide an adequate system of reasoning to handle, e.g., the problems associated with the prescription to perform a Śyena sacrifice.

Formalizations of Mīmāṃsā Deontic Rules
Establishing as the default choice for formalizing Mīmāṃsaka arguments about prescriptions leads, when applied to the prescription concerning the Śyena sacrifice, to an inconsistent system of reasoning. It thus should not be adopted. But then we are faced with the question of which reasoning system, i.e., logic, to adopt instead. With this in mind, in the present section we are introducing a new logic, which we call basic Mīmāṃsā deontic logic ( for short). This logic, which has been created by formalizing Mīmāṃsā deontic principles, is used in section "Logic at Work: the Śyena Sacrifice" to examine prescriptions related to the Śyena controversy.
In general, when introducing or describing a logic, two levels must be considered, the syntax and the semantics. The former deals with the formal language of the logic under consideration and provides the rules for constructing its formulae, i.e., certain strings of symbols. It does not interpret or give meaning to these symbols. This is done by the semantics, which give a meaning to the formulae and determine their behaviour with respect to some notion of "truth." Once the syntax has been settled, the semantics of the logic is often defined by adding properties (e.g., algebraic properties or Hilbert axioms 21 ) to the base system. This is the approach we follow for the logic , which extends classical logic with suitable modal and deontic operators whose properties are directly extracted from first principles, i.e., the Mīmāṃsā deontic rules. These principles are transformed into Hilbert axioms.
We start by defining the formal language of our logic, which is planned to serve as a tool for the formalization of the deontic principles extracted from Mīmāṃsā texts. We will first discuss and justify the choices about the non-modal part of the base logic and then introduce and explain the modal operators and present the grammar for the formulae of the language. Apart from propositional variables, p, q, r, … which stand for propositions, the first elements needed for the definition of the language are the propositional connectives. These are the logical symbols, ∧ (conjunction), ∨(disjunction), ¬ (negation), → (implication), ⊤ (verum), and ⊥ (falsum). The intuitive meaning of a compound formula built using one of these connectives depends on the meaning of the component formulae. Given two formulae and , the compound formula ∧ stands for ' and '; ∨ is for ' or '; and → is for 'if , then .' Given a formula , the formula ¬ is for 'not .' The symbols ⊤ and ⊥ can be considered connectives of a particular sort; they are not applied to formulae, but are instead treated as formulae themselves.
The symbol ⊤, also called verum, is regarded as a formula that is unconditionally true, the symbol ⊥, also called falsum, as one that is unconditionally false.
The first issue that arises is the interpretation of the above language in classical logic, which has been chosen as our base system. It is important to note that when it comes to basic assertive reasoning, the two main alternatives are classical and intuitionistic logic. An exhaustive account of the differences between these two logics would be a complex endeavour and certainly falls outside the scope of this article; one view on the subject is that truth values of formulae in intuitionistic logic refer to the epistemic justification of what is expressed by those formulae, while truth values in classical logic refer to the actual truth of what is expressed. In this sense, for example, in intuitionistic logic it is not licit to infer that something is true knowing only that its negation is not. This is called reductio ad absurdum (or proof by contradiction) and is in contrast allowed in classical logic. More formally, in classical logic if ¬¬ is true then must be true, but in intuitionistic logic this does not hold.
Although we could not find stipulative definitions of classical logic principles, when directly confronted with an objection concerning the alleged violation of the law of non-contradiction, 22 a Mīmāṃsā author found in Jayanta's Nyāyamañjarī 23 describes a principle implying the legitimacy of reductio ad absurdum arguments: As for what has been said [by the Buddhist opponents], namely that "it is illogical that in a single real entity two contradictory aspects simultaneously occur," this is also wrong.
Also, in the case [of our theory] there is no mutual contradiction, because no such [contradiction] is grasped | [In fact,] it is not the case that one knows the one once the other is excluded, as would be the case with mother-of-pearl and silver (where a shiny object can either be an instance of silver or of mother-ofpearl) ||

When there is a contradiction, upon the denial of one [alternative], the other is known [to be true]. But in the topic under consideration it is not so, hence, what is the contradiction? 24
The last sentence of this passage indeed says that when a contradiction is considered, denying one alternative makes the other true.
To see why this sentence implies the legitimacy of reductio ad absurdum, consider the contradictory expressions and ¬ . If one of the two is denied there are two possibilities: either is denied or ¬ . If is denied, then the sentence implies that ¬ is true, thus (identifying the fact that is denied with the fact that ¬ is asserted) from ¬ follows ¬ . This is clearly consistent with classical logic but does not tell us much about the opportunity of adopting a logic that is different. If in contrast ¬ is denied, the same sentence yields that , the other alternative, is true; and this means (again identifying the fact that ¬ is denied with the fact that ¬¬ is asserted) that from ¬¬ follows . And this is exactly the reasoning of reductio ad absurdum. In our opinion this provides sufficient reason to adopt classical logic rather than intuitionistic logic as the propositional logic underlying . This choice contrasts with one of the few other attempts to reconstruct the deontic logic underlying the traditional reading of a given sacred text, namely the analysis in Gabbay  Having established the purely assertional basis for our logic, we now consider the additions necessary for capturing the Mīmāṃsā reasoning. Since we want to reason about deontic principles, we need to account for various spheres of discourse: the normative one, the one of actual truth and the one of necessity. While the sphere of actual truth is captured by the propositional reasoning of the base logic, i.e., classical logic, to deal with the other spheres, we need to consider modal operators. Therefore, we will add two modal operators to the language: the binary modal operator ( / ) from dyadic deontic logics and the unary modal operator from alethic modal logics.
This particular choice of operators is not the only one possible, and hence warrants some discussion. As mentioned in section "Why Mathematical Logic?" the unary deontic operator is usually used to represent an unconditional obligation in a formalized system of norms. By applying this operator to an expression, it asserts that what is meant by the expression is obligatory, with no further requirements. However, since adopting the unary version of the operator in our case leads to similar complications in the formalization of conflicting obligations, such as those mentioned in the discussion of (section 5), we follow an alternative (but also reasonably standard 25 ) approach and employ in the binary (or dyadic) version of . This allows us to impose conditions on the obligation describing the situation in which the obligation holds. In this case the syntactic structure is ( / ), i.e., the operator is applied to two formulae: one as the injunction, the first, and one as the condition, the second: The formula ( / ) can accordingly be read as "it is obligatory that given that ." Trying to account for the Mīmāṃsā analysis of the Vedic prescriptions, in our formalization the conditions represented by the second formula under the binary operator will usually be of the form "it is desired that…," which identifies the addressee of a given prescription. The standard example for this is "The one who desires heaven should sacrifice with the Full-and New-Moon Sacrifices," which would be formalized as ( _ _ _ / _ ), where _ _ _ stands for performing the Full-and New-Moon Sacrifices and _ for the fact that one desires heaven. To keep the basic Mīmāṃsā deontic logic as simple as possible, we have chosen not to further analyze statements of the form "one desires X," but rather to model them using separate propositional variables. While this leaves room for refinement, we believe it to be an appropriate compromise, in particular since Mīmāṃsā texts are mainly concerned with prescriptions and not with analyzing the notion of desire.
The chosen operator refers to the normative level in the sense that the truth of a proposition corresponding to a formula in the form ( / ) does not depend in general upon the actual truth of and at all, but on the status of and with respect to a corpus of prescriptions.
Finally, the operator for necessity is used in to formalize auxiliary conditions of general deontic principles. When the relationship between two prescriptions is formally asserted in one of these principles, the correlation between what is mentioned by the parts of the two prescriptions needs to be stronger than mere accident: it must be stable with respect to contingencies and must concern characterizing features of what is meant by the expressions in the formulae. An instance of this difference is, for example, that it is usually contingent whether or not someone harms someone else, but it is necessary that if someone harms a man then they are harming a living being. The operator allows us to assert the necessity of the truth of such an expression: the formula means that is true in all relevant circumstances or, in other terms, that it cannot be the case that does not hold; accordingly we read the formula as "it is necessary that ." To summarize, the formulae of are inductively defined as follows: • any propositional variable, ⊤ and ⊥, are formulae of ; • if and are formulae of , then ∧ , ∨ , → , ¬ , and ( / ) are formulae of ; . The selected principles fix the logical ground on which the basic Mīmāṃsā reasoning stands, whereby it is possible that they capture the structural traits characterizing the common logical core of Mīmāṃsā arguments.
To define a first axiom, we will analyze and attempt to formalize the following excerpt. This principle affirms that if something is enjoined by a prescription and it has some requirements that are not prescribed separately, then the single prescription is also valid for all the requirements. Hence one could infer that all prescriptions provide sufficient grounds to prescribe all the necessary consequences of what is enjoined, if they fall under the same circumstances. Thus enough grounds are given for accepting the axiom where , and can be any formulae. This formula expresses that if it is necessary that is true when is true then, if is obligatory, also is obligatory, given the same conditions. Saying that is a prerequisite of means that it can be the case that only if it is the case that too, and therefore that if is true then must be true as well, which is formalized as ( → ). The operator is used here to guarantee that the correlation between the truth of and the truth of is not just accidental, but a permanent relationship between what is represented by the two formulae.
Another principle that provides grounds for the previous axiom is the one presupposed by Śabara in the context of the discussion on dharma (i.e., Mīmāṃsā itself). 27 Śabara explains that one cannot start an enquiry into the dharma prior to the study of the Veda, because during the enquiry into the dharma there will be various discussions of Vedic texts, a requirement that can only take place after the Veda has been studied. In a more abstract pattern: If the obligation to perform X presupposes the accomplishment of Y, the obligation to perform X prescribes also Y.
A third principle supporting the acceptance of Axiom (1) is the so-called kaimutikanyāya (a form of the a fortiori argument), which is found at work in instances such as the following passage discussing the fact that even Brāhmaṇas who commit murder should be executed: The mention of the word 'Brāhmaṇa' [in the passage specifying that murderers who are Brāhmaṇas should be sentenced to death] has the purpose of [implementing] an a fortiori argument. If one ought to sentence to death even a murderer who is a Brāhmaṇa, then even more so in the case of another [murderer] (that is, one who is not a Brāhmaṇa). 28 What is attested here is the use of the a fortiori argument for obligations. The reasoning is as follows: If something of a certain generality is enjoined, also all of its necessary consequences are enjoined too, since if Y is necessarily a consequence of X, either X and Y are equivalent, or Y is an unavoidable aspect of X.
In the endeavour of choosing axioms for the system, another excerpt that has been considered is the following.
Therefore, if these (Smṛti 29 texts) do not have any authority | they must be rejected in all their aspects, and there should not, by contrast, be a 'half-hen'-situation (i.e., a partial acknowledgement of their authority) || 30 The principle of the half-hen is widely used in different contexts. As is customary for Mīmāṃsā principles, it is only used and never defined. Nonetheless, an abstract representation of it could be as follows: Given that purposes Y and Z exclude each other, if one should use item X for the purpose Y, then it cannot be the case that one should use it at the same time for the purpose Z.
This principle stresses the incongruity of enjoining someone to act in contradiction with himself with regard to some object. It is formalized as the axiom where , and can be any formulae. It is possible to read this as prohibiting the separate prescription of two contradictory statements under the same conditions. Finally, the discussion found in the commentary on the chapter about the eligibility to perform sacrifices (adhikārādhikaraṇa, PMS 6.1) was considered, paying special attention to the passages where the eligibility of the people of different classes is discussed. There, Śabara explains that one should perform sacrifices if one has heard the Vedic injunctions enjoining them. And, since only the people of the higher three classes (out of four classes) can read or hear the Veda, one has to perform sacrifices if one belongs to the higher three classes.
As already stressed, Mīmāṃsā authors did not define the principles they were using and rather assumed they could freely use one element or another from a store of available principles shared by all. Nonetheless, Śabara seems to imply the following scheme of reasoning: If the condition X and Y are equivalent, given the duty to perform Z under the condition X, the same duty applies under Y.
From this one can derive the following axiom: where , and can be any formulae. This axiom states that a prescription is not only applicable under the mentioned conditions, but also under any conditions that hold in exactly those cases in which the mentioned ones do. If and are necessarily equivalent, it states that what is obligatory if holds, is obligatory also if holds. This last deontic axiom establishes the generality of prescriptions with respect to logically equivalent formulae. The absence of this axiom would make a formalized prescription dependent upon the particular form of the formula that conditions the prescription. This would clearly prejudice the possibility of reasoning effectively about the conditions of the prescription. After having described the properties of the ( / ) operator in (i.e. its deontic axioms), we need to settle the properties of the necessity operator . While the properties of ( / ) are formalizations of principles extracted from Mīmāṃsā texts, the same cannot be done for because, due to their polemics with Buddhist epistemologists, Mīmāṃsā authors do not conceptualize necessity as being separate from epistemic certainty. 31 However, as discussed above, the introduction of can be derived from the implicit need of a non-accidental link between the variables of a prescription.
The standard and well-established choices for a logic for the alethic operator of necessity are and , with being stronger than (see Garson 2014: section 2). 32 Of these, the concept of necessity represented by seems to match the concept used by Mīmāṃsā authors best. In particular, we have not found any principle motivating the additional properties of . Hence, we have chosen to implement the logic adding the following standard axioms to our logic: the axiom ( → ) → ( → ), usually known as , which states that if something is necessarily a prerequisite of something else and the latter holds necessarily, then also the prerequisite holds necessarily; the axiom → , usually known as , which states that whatever is necessary is actually true; to transgress any Vedic prescription-and this is possible because, as stressed also by Mīmāṃsā authors, it is nowhere said in the Vedas that the Śyena sacrifice must be performed under any condition. Through the use of a dyadic operator our analysis highlights that Vedic prescriptions are commands addressed to those who want to attain a certain desired outcome rather than absolute imperatives.
It now remains to be seen whether this solution maps well with the Mīmāṃsā solution of the Śyena conundrum. This necessitates a further restriction, insofar as different solutions of the conundrum were offered by different authors and sub-schools within Mīmāṃsā. Within Prābhākara Mīmāṃsā, the well-known Śālikanātha (9th c.), the main Prābhākara, remarks that the Veda prescribes that no violent act should be carried out (na hiṃsyāt sarvā bhūtāni). However, Śālikanātha continues, while the Veda can impart the knowledge about what must be done, this does not limit one's free will to do it or not. And since one wishes to perform an action because one desires its result, if one does not desire to harm anyone, one should not perform the Śyena sacrifice. Thus, the Veda merely lets us known that if we would like to harm someone, we should perform the Śyena sacrifice. It does not say that we must long for such a result. Śālikanātha thus solves the seeming conflict by pointing out that the Śyena prescription only addresses people who fall in the category of "those who desire to harm their enemy." In the Mīmāṃsā terminology the fact of belonging to a determinate category is called adhikāra, as already mentioned, which corresponds to the condition expressed by the dyadic operator introduced in section "'Formalizations of Mīmāṃsā Deontic Rules." Śālikanātha can thus make the following conclusion: Hence, only one who desires to harm an enemy is obliged to fulfil such an obligation (i.e., the Śyena one). But, since such a desire is itself forbidden, no one should ever be obliged by it: In the case of an adhikāra related to a wish, the prescription does not say that the sacrifice must be performed, because its performance is accomplished only because of the result. Although the prescription is known as something to be fulfilled and the sacrifice is [known] as the instrument for this (fulfilment of the prescription), nevertheless since in the case of a wish-related adhikāra the sacred text is [only] valid until the accomplishment of a wished result, the prescription does not cause the person to act for its (the prescription's) own fulfilment. For people accomplish the action only because of the result, since they undertake actions because of desire. 35 In conclusion, the formal solution provided by matches rather closely the reasoning underlying the solution by Śālikanātha.

6
Hierarchy of Rules

Evidence Against a Radical Hierarchy of Rules
A further problem connected with the nyāyas is the hierarchy of their application. The Mīmāṃsā sources do not explicitly state that there is a hierarchy between these rules. Although some rules are clearly directed only to localized cases, such as the understanding of specific terms in certain prescriptions (e.g., rathakāra in PMS 6.1.44), the ŚBh never denies the general value of individual rules. The discussion in the ŚBh focuses on the legitimacy of the application of given rules, not whether they are overruled by more important ones. But one cannot argue that Mīmāṃsā authors neglected the issue of hierarchies altogether since they do discuss in detail the hierarchical application of the various means of knowledge to ascertain the subordination-links between the various elements prescribed in relation to the same ritual (see MNP: 67-181). A further means of avoiding the need of an explicit hierarchy is the fact that each obligation only addresses a specific group of people (see above, sections "'Formalizations of Mīmāṃsā Deontic Rules" and "Logic at Work: the Śyena Sacrifice," for a longer discussion of such dyadic conditions), so that conflicts among obligations can frequently be avoided by taking into account these different addressees (see section "Logic at Work: the Śyena Sacrifice" for the case of the Śyena sacrifice). ones. Our formalization led to the introduction of , a new deontic logic which is used to reason about the seemingly conflicting duties of avoiding harm and performing the malefic Śyena sacrifice.
The present paper is the result of a novel collaboration between scholars of logic, computer science, European philosophy and Indian philosophy. It is our hope that this interdisciplinary dialogue will provide (i) interesting suggestions for developing new tools for the philosophical analysis of various sacred texts and their prescriptions 40 and (ii) new systems of deontic logic, drawing on ideas elaborated by Mīmāṃsā authors.

Further Directions of Research on Logic and Mīmāṃsā
The basic logic introduced in this paper is only a first step towards mapping the structural elements of the Mīmāṃsā deontic system onto a formal framework. We will hence point out two main directions that further research should or might take. In primis, since important features expressed in the deontic principles cannot yet be captured in , (i) the logic should be extended in different ways. For instance, it would be interesting to enrich in order to cover the Mīmāṃsā analysis of prohibitions as well, formally distinguishing them from obligations (see also Chapter 4 of Gabbay, Abraham, and Schild 2013 for similar work with regard to Talmudic prescriptions).
Moreover, while our preliminary investigation has focused on the easiest (and less expressive) level of logic, namely propositional logic, some deontic principles seem to hint at the need of introducing predicates and quantifiers (thus shifting from propositional to first-order logic). This is for instance the case of the adhikāranyāya (discussed in PMS 6.1.1-3 and 4-5), which states that the agent of a duty should be the one identified by a given prescription. The basic structure of this rule can be easily represented by means of predicate and quantifiers, namely, "Given the group of people who are agents of duties and the group of people identified by a prescription, it applies for all members of the first group that they are also members of the second." The language of might also be extended with temporal operators, as suggested by the principles of the padārthānusamayanyāya (ŚBh ad 5.2.1-2) and the kāṇḍānusamayanyāya (ŚBh ad 5.2.3). Supposing one needs to perform three or more actions on three or more auxiliaries (i.e., sprinkling, threshing and reciting a mantra over rice, barley and millet grains), according to the padārthānusamayanyāya one performs the first action on the first item, then on the second item and then on the third item, and then one repeats the same procedure with the second and the third action. According to the kāṇḍānusamayanyāya, in contrast, one performs all three actions on the first item, then moves to the second, performs all three and then moves to the third and performs all three.
A further useful extension could be an additional conjunction connective that, in contrast to the ∧ connective of classical logic (see the example in section 5), asserts that the repetition of a prescription might be, in specific circumstances, meaningful, as stated in the aṅgabhūyastve phalabhūyastvanyāya, discussed in PMS 10.6.62 and 11.1.15. According to this rule, in fact, if one wants more than one result, one needs to repeat the auxiliary of an optional act (kāmya karman), i.e., a sacrifice undertaken because of the desire for the result. However, one does not need to undertake the auxiliaries of a permanent act (nitya karman) again, i.e., a sacrifice undertaken merely because it is due.
Moreover, one might want to extend our system with a mechanism to distinguish subordinate concepts, as hinted by the antaraṅgabahiraṅgayor antaraṅgaṃ balīyaḥ (ŚBh ad PMS 12.2.27; see section 6 above).
Finally, we would like to find a way to handle conflicts between different authorities, as suggested by the vedānām anarthatvaprasaṅge āmnāyātikramanyāya (ŚBh ad 1.1.1), which prescribes that if there is a conflict between authoritative texts, the Vedas prevail over the Smṛtis. Thus, if the consequence of a Smṛti-passage is that a passage of the Veda becomes meaningless, then one ought to transgress that Smṛti text. This latter feature is also needed in many applications of artificial intelligence. This intersection also points to the possibility of certain issues discussed in Mīmāṃsā leading to the development of crucial concepts in the area of automated reasoning for deontic logics, which in the context of "ethical machines" has never been more important than today (see Chaudhuri and Vardi 2014 41 ).  The standard edition for PMS and for its commentaries ŚBh and TV is Śrīmajjaiminipraṇitaṃ Mīmāṃsādarśanam, ed. Subbāśāstrī (Poona, 1929(Poona, -1934. 2 In the following, text passages of the ŚBh commenting on a given aphorism of the PMS will be indicated through the number of the PMS aphorism, e.g. ŚBh ad PMS 1.1.5. 3 For the terms "epistemic" and "deontic" authority, see Józef Maria Bocheński, Was ist Autorität? (Freiburg: Harder, 1974 The Smṛtis are a class of texts whose prescriptive contents, according to Mīmāṃsakas, are derived from the Vedas. 30 tenāsāṃ yadi vā naive kvacid asti pramāṇatā | sarvatrāvyāhatā vā syān na tv ardhajaratīyatā || (TV ad PMS 1.3.3, p. 85).