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Over the course of more than two millennia the philosophical school of M¯ımam ¯ s¯a has thoroughly analyzed normative statements. In this paper we approach a formalization of the deontic system which is applied but never explicitly discussed in M¯ımam ¯ s¯a to resolve conﬂicts between deontic statements by giving preference to the more speciﬁc ones. We ﬁrst extend with prohibitions and recommendations the non-normal deontic logic extracted in Ciabattoni et al. (in: TABLEAUX 2015, volume 9323 of LNCS, Springer, 2015) from M¯ımam ¯ s¯a texts, obtaining a multimodal dyadic version of the deontic logic MD. Sequent calculus is then used to close a set of prima-facie injunctions under a restricted form of monotonicity, using speciﬁcity to avoid conﬂicts. We establish decidability and complexity results, and investigate the potential use of the resulting system for M¯ımam ¯ s¯a philosophy and, more generally, for the formal interpretation of normative statements. Keywords Deontic logic · Non-monotonic inference · Speciﬁcity · M¯ımam ¯ s¯a · Sequent systems · Legal representation We thank E. Freschi for her invaluable help with the Indological aspects of this work. Funded by the Vienna Science and Technology Fund (WWTF) under project MA16-28 and by BRISE-Vienna (UIA04-081), a European Union Urban Innovative Actions project. Björn Lellmann lellmann@logic.at Francesca Gulisano francesca.gulisano@sns.it Agata Ciabattoni agata@logic.at TU Wien, Vienna, Austria Scuola Normale Superiore Pisa, Pisa, Italy 123 352 B. Lellmann et al. 1 Introduction The M¯ımam ¯ s¯a is a philosophical school which originated in ancient India in the last centuries BCE and whose main focus was the exegesis of the prescriptive portions of the Vedas—the Sacred Texts of (what is now called) Hinduism. Together with Nyaya ¯ and Buddhist epistemology, M¯ımam ¯ s¯a is one of the fundamental schools of Indian philosophy, and the only one centered on deontic concepts. In order to read the Vedas not as a religious text, but as a set of precepts, and to explain “what has to be done” in presence of seemingly conﬂicting obligations, M¯ımam ¯ s¯a authors have thoroughly discussed and analyzed normative statements. They have proposed a rich body of deontic, hermeneutical and linguistic principles of interpretation, called nyaya ¯ s, which are so modern, rational, scientiﬁc, and systematic (Bathia 2010) that they are still applied in Indian jurisprudence, e.g. Katju (2006). Although not well known to the logic community, the resulting theories are rightly considered early deontic logic (Huisjes 1981). Among the deontic nyaya ¯ s, some can be transformed into properties (Hilbert axioms) for the operators corresponding to the deontic concepts in M¯ımam ¯ s¯a; this method led to the introduction of the non-normal dyadic deontic logic bMDL (basic M¯ ımam ¯ s¯ a deontic logic), which was used in Ciabattoni et al. (2015) to formally ana- lyze a famous controversial passage in the Vedas. However, in the construction of bMDL only nyaya ¯ s concerning the obligation operator were considered. Here we extend bMDL with new operators for prohibitions and recommendations (or weak obligations); we call the resulting logic MD+. We extracted the properties for these operators from additional nyaya ¯ s that were translated from Sanskrit and interpreted only recently. Similar to the situation in Talmudic logic as investigated in Abraham et al. (2011), the M¯ımam ¯ s¯a deontic operators are not interdeﬁnable. Intuitively, the most evident difference between them is in the achievable results: obeying Vedic obligations yields good karma which leads to eternal happiness; in contrast, follow- ing Vedic recommendations yields only speciﬁc immediate results; ﬁnally, following Vedic prohibitions only prevents the accumulation of negative karma, see, e.g., Fres- chi (2012, 2017). However, the main difference between the two deontic concepts of prescription (vidhi in Sanskrit) and prohibition (nisedha ˙ ) is not properly dependent on the results of complying with commands or disregarding them. The conveyed idea (buddhi) at the base of the concept of obligation is “activation” or “being impelled to act”, while, in case of prohibition, it is “inhibition” or “being prevented from taking an action”, hence prescription and prohibition represent two genuinely different notions of duty, one irreducible to the other Although speciﬁcally targeted at formalizing M¯ımam ¯ s¯a reasoning, these operators can be applied in different contexts. For instance, in line with the argument for using deontic notions in the formalization of legal texts given in Jones and Sergot (1992), Royakkers (1998), obligations and prohibitions could be used for comparing moral and legal duties (see also Example 4 in Sect. 6), and the distinction between obliga- tions and recommendations could be adapted for representing the difference between Since the Vedas are assumed not to be contradictory, M¯ımam ¯ s¯a authors invested all their efforts in creating a consistent deontic system. 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 353 prescriptions to fulﬁl duties within mandatory terms and within indicative terms, in some legal frameworks. Not all the nyaya ¯ s can be converted into Hilbert axioms though. Some of these offer indeed more general interpretative principles to resolve apparent contradictions in the Vedas; prominent examples of such nyaya ¯ sare gunapradhana ¯ (also known as s¯ amanya-vi ¯ ses ´ a) and vikalpa, which are investigated in this paper. The vikalpa principle states that when there is a real conﬂict between obligations, any of the conﬂicting injunctions may be adopted as option: this principle is known in deontic logic as disjunctive response (Goble 2013) and is similar to the phenomenon of ﬂoat- ing conclusions in nonmonotonic reasoning (Makinson and Schlechta 1991, see also Remark 7). Introduced by Sabara (3rd–5th c. CE), the gunapradhana ¯ principle states that more speciﬁc rules override more generic ones; it is widely used, e.g., in Artiﬁcial Intelligence, where it is known as speciﬁcity principle, and in Law as the principle “Lex specialis derogat legi generali”. These principles are also used to capture defea- sible reasoning in the context of non-monotonic logics (see e.g. Delgrande and Schaub 1997;Nute 2003; Hage 2003; Straßer and Antonelli 2016). Different methods and systems have been introduced in the literature to deal with deontic conﬂict resolutions using speciﬁcity. Although some are close to ours [e.g. Horty’s syntactic approach in Horty (2012)], none of the various proposals can be used “out of the box” for representing M¯ımam ¯ s¯a reasoning. Indeed they are either based on logics different from MD+ (e.g. Straßer and Arieli 2019), or are implemented within general frameworks that do not allow us to distinguish between Vedic commands and human deductions [e.g. the argumentation-based approach in Prakken and Sartor (1999), and Deontic Default logic in Horty (1993, 2012), see Remark 3], use explicit priorities among rules—which are not present in M¯ımam ¯ s¯a–[e.g., Defeasible Deon- tic logic in Governatori and Rotolo (2004) and Input/Output logic in Makinson and van der Torre (2000)], or a different way to apply speciﬁcity [e.g. Horty’s approach (Horty 2012), see Example 2]. Due to its relevance to legal reasoning, a number of these approaches to conﬂict resolution have been applied in that area, and often imple- mentations are available. Good recent overviews and comparisons are given, e.g., in Batsakis et al. (2018) and Calegari et al. (2019). The aim of this article is to extend the basic deontic logic MD+ for obligations, prohibitions and recommendations with reasoning from deontic assumptions using speciﬁcity and vikalpa. We further explore the usefulness of the resulting system for the evaluation of different competing formalisations in M¯ımam ¯ s¯a and beyond. To this end we introduce a sequent-based approach to deal in MD+ with speciﬁcity and vikalpa as well as a system implementation. This work is a signiﬁcantly extended version of the conference paper Ciabattoni et al. (2018), which only concerned obli- gations, and did not discuss potential applications. Resolving conﬂicts using speciﬁcity, our calculus derives enforceable and appli- cable commands from the explicit prescriptions contained in the Vedas (sr ´ auta in Sanskrit) and from a ﬁnite set of propositional facts. The calculus presented here is also shown to satisfy vikalpa (disjunctive response). As, e.g., in Goble (2013), and van der Torre (1994), here we interpret the notion of a conditional obligation being more speciﬁc than another one as the conditions of the former implying those of the latter. Our calculus is built on the sequent calculus for 123 354 B. Lellmann et al. the -free fragment of bMDL from Ciabattoni et al. (2015), which turns out to be the dyadic version of the non-normal deontic logic MD considered, e.g., in Chellas (1980) (cf. Proposition 1 and Freschi et al. 2019), extended with new rules for recom- mendations and prohibitions. Additional rules to derive all possible prescriptions are deﬁned using limited (downwards) monotonicity on the conditions of the (non-nested) prescriptions in the Vedas (prima-facie deontic statements) “up to conﬂicting deontic statements” relative to the given set of facts. These rules offer the technical advantage that the consequences of a set of prima-facie deontic statements can be constructed iter- atively instead of by a ﬁxed-point construction, as, e.g., in Horty (2012). Importantly, since the prima-facie injunctions are assumed to constitute a closed set, the additional rules contain statements expressing the underivability of a formula; these statements do not compromise the decidability of the system and do not affect its complexity, which remains PSPACE, i.e., as for deciding theoremhood in intuitionistic or many standard modal logics. The central technical result ensuring these properties is the cut elimination theorem (Theorem 2). Similar underivability statements are present, e.g., in the sequent calculi for non-monotonic (non-modal) logics of Bonatti and Olivetti (2002), but in contrast to that work here we do not need to develop a full-ﬂedged cal- culus for these statements. Other sequent-based calculi capturing non-monotonicity in the context of normative reasoning have been developed and applied in deontic logics (e.g. Governatori and Rotolo 2006) and in argument-based systems (e.g. Straßer and Arieli 2019). The design of our system is motivated by the particular interpretation given by most of M¯ımam ¯ s¯a authors and made explicit by the later author Medhatithi ¯ (9–10th c. CE), that more speciﬁc srauta ´ precepts provide exceptions to more general ones and that the latter apply to all circumstances but those indicated in the exceptions (or implied by them). Apart from the nonmonotonic inferences from prima-facie to actual deontic statements, all derivations use the monotonic system MD+. Keeping the inferences of the logic at this level deductive (i.e., monotone) is inspired by the effort of Indian philosophers—in particular the M¯ımam ¯ s¯a author Kumarila—to ¯ keep their arguments not defeasible “as much as possible”, see Taber (2004). Applications of our system to M¯ımam ¯ s¯a philosophy, and, more generally, to the formal interpretation of normative statements, e.g., in legal representation, are also provided. Prima-facie (srauta) ´ injunctions and statements about factual conditions can indeed give rise to many interpretations, each of which corresponds to a group of prima-facie commands and global assumptions about facts. For instance, in Sanskrit the same word is used both for “obligations” and “recommendations”, and the correct meaning has to be inferred by scholars of Indian philosophy. Our system can be used to derive a set of consequences for each of these groups. Using these consequences to compare the different interpretations, it is then possible to choose the most suitable one. In the case of M¯ımam ¯ s¯a philosophy, one criterium which was heavily used in this comparison is to minimize applications of the vikalpa principle between prima- facie deontic statements; this is due to the fact that M¯ımam ¯ s¯a authors considered applications of vikalpa to be “the last resort” and hence to be avoided as much as possible. In our system this criterium can be evaluated by checking how many of the prima-facie deontic statements are actually derivable. This check can be performed 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 355 with the help of our Prolog implementation of the system, available at http://subsell. logic.at/bprover/deonticProver/version1.2/. The rest of the paper is organized as follows: Sect. 2 recalls the base logic bMDL. The simpliﬁed set of sequent calculus rules, from Ciabattoni et al. (2018), to reason about obligations in presence of speciﬁcity is given in Sect. 3. The base logic is extended with prohibitions and recommendations in Sect. 4, and the complete set of rules for reasoning using speciﬁcity is introduced in Sect. 5. Consistency, decidability and complexity results for the system are presented in Sect. 5.1, while its potential use for M¯ımam ¯ s¯a philosophy and beyond is described in Sect. 6. The technical proof of cut elimination is contained in the “Appendix”. 2 The base logic: bMDL Basic M¯ımam ¯ s¯a Deontic Logic bMDL was introduced in Ciabattoni et al. (2015)asa ﬁrst step towards mapping the structural elements of the M¯ımam ¯ s¯a deontic system onto a formal framework. The idea was to deﬁne a logical system following a bottom-up approach of extracting deontic principles from the M¯ımam ¯ s¯a texts. The logic resulting from the analysis of circa 50 such principles extends the alethic system S4 with the following axiom schemata for the deontic operator O( A/ B), which intuitively reads as “ A is obligatory under the condition B”: 1. (( A → B) ∧ O( A/C )) → O( B/C ). 2. ( B →¬ A) →¬(O( A/C ) ∧ O( B/C )). 3. ((( B → C ) ∧ (C → B)) ∧ O( A/ B)) → O( A/C ). Axioms (1)-(3) arise by rewriting some of the M¯ımam ¯ s¯a deontic interpretative prin- ciples (nyaya ¯ s) as logic formulae, while the choice of modal logic S4 over S5 was suggested by some statements found in the texts as well as technical convenience, in particular the existence of cut-free sequent calculi. See Ciabattoni et al. (2017, 2015) for more details. Note that deontic statements in M¯ımam ¯ s¯a can also be analysed on a more detailed level in the context of speciﬁc sacriﬁces, taking into account the nature of the latter. Since here we are interested in the general properties of M¯ımam ¯ s¯a reasoning, we do not consider this distinction and refer the reader to Freschi et al. (2019) for details. Remark 1 bMDL is weaker than most known deontic logics, e.g., the logics considered in von Wright (1964), von Wright (1965), Hansson (1969), van Fraassen (1972), Prakken and Sergot (1997), Goble (2019); in particular it has neither any deontic aggregation principles like O( A/C ) ∧ O( B/C ) → O( A ∧ B/C ) nor any form of factual or deontic detachment, i.e., O( A/ B)∧ B → O( A) and O( A/ B)∧O( B/C ) → O( A/C ) respectively. In part this is due to our bottom-up methodology: so far indeed we have not found any mention of corresponding principles in the texts. However, the absence of (factual) detachment principles is also in line with the statement by one of the main authors of M¯ımam ¯ s¯a, Prabhakara, ¯ that “A prescription regards what has to be done. But it does not say that it has to be done”(Brhat¯ ı I, 7th c. CE). We read this as stating that a prescription states what is obligatory under certain conditions, but not that this is unconditionally obligatory if these conditions hold. 123 356 B. Lellmann et al. Fig. 1 The modal part of a Hilbert-style system for dyadic MD Fig. 2 The sequent calculus G for dyadic MD MD Fig. 3 The modal part of the sequent calculus G for bMDL from Ciabattoni et al. (2015) bMDL Here for simplicity we only consider the box-free fragment of bMDL. Formally, the set of formulae is given by the grammar A ::= p |⊥| A → A | O( A/ A). We treat the remaining propositional connectives ∧, ∨, ¬ as deﬁned by {⊥, →} in the usual way, i.e., ¬ A :≡ A →⊥ as well as A ∨ B :≡ ¬ A → B and A ∧ B :≡ ¬(¬ A ∨¬ B). We will show in Proposition 1 below that the box-free fragment of bMDL coincides with the dyadic version of the logic MD (see Chellas 1980) axiomatized as in Fig. 1 In this paper we will consider an extension of a sequent calculus for this logic. Here, a sequent is a tuple of multisets of formulae, written as Γ ⇒ Δ.The rules of the base sequent calculus G are given in Fig. 2, those of the calculus G MD bMDL for bMDL from Ciabattoni et al. (2015)inFig. 3, where Γ denotes Γ in which all formulae not of the form A are deleted. As usual, a derivation is a ﬁnite labelled tree where every node is labelled with a sequent such that the labels of a node follow from the labels of its children using the rules of the calculus. In particular, the leaves are labelled with conclusions of the zero-premise rules init or ⊥ , see also Troelstra and Schwichtenberg (2000). For G one of G , G we write Γ ⇒ Δ if there is a MD bMDL G derivation of Γ ⇒ Δ in G. The following proposition gives a proof-theoretic proof of the equivalence of the box-free fragment of bMDL and MD. For the original proof using semantical methods, see Freschi et al. (2019). Proposition 1 If Γ ⇒ Δ does not contain , then Γ ⇒ Δ iff Γ ⇒ Δ. G G MD bMDL Hence the box-free fragment of bMDL is MD. Proof One direction of the equivalence follows from changing the rules of G into bMDL the corresponding rules of G possibly followed by the weakening rules W , W . L R MD Note that since we have contraction on both sides of the sequent we could alternatively consider sequents as tuples of sets instead of multisets. To make the role of contraction explicit and to facilitate a less error- prone cut elimination proof, we chose to use multisets. 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 357 The other direction follows since a derivation in G is a derivation in G with the MD bMDL addition of the structural rules of weakening and contraction Con , Con , which are L R admissible in G (Ciabattoni et al. 2015, Lemma 1). Completeness and soundness bMDL of G for MD follow from general methods (e.g. in Lellmann and Pattinson 2013) MD for constructing sequent calculi from axioms and proving cut elimination. Remark 2 Following Ciabattoni et al. (2018), in this paper we employ a mechanism for handling propositional facts that differs from that in Ciabattoni et al. (2015): whereas there we encoded such assumptions as boxed formulae in the conclusion of a derivation, here we treat them as leaves in a derivation. E.g., for deriving that the conditional obligation to not perform violence implies the conditional obligation to not kill from the assumption that killing implies violence, using the mechanism from Ciabattoni et al. (2015) we would try to derive the sequent (kill → violence), O(¬violence/C ) ⇒ O(¬kill/C ) using only inital sequents at the leaves of the derivation. Here, instead we will try to derive the sequent O(¬violence/C ) ⇒ O(¬kill/C ), where the non-logical axiom or ground sequent kill ⇒ violence may occur at a leaf of the derivation (see Deﬁnition 1 below for the formal details). This has the welcome consequence that we can avoid the alethic modality including any question about its axiomatisation, in line with the view that M¯ımam ¯ s¯a authors did not distinguish between necessity and epistemic certainty. 3 Reasoning with more speciﬁc obligations in M¯ımam ¯ s¯a Here we continue the proof-theoretic approach initiated in Ciabattoni et al. (2015)to reproduce M¯ımam ¯ s¯a reasoning in a formal framework. Before considering the full language, we ﬁrst illustrate the main ideas behind the sequent calculus approach to deal with speciﬁcity/gunapradhana ¯ in the simpliﬁed context of Ciabattoni et al. (2018), i.e. using the (dyadic version of the) logic MD with the obligation operator only. Speciﬁcity is used in M¯ımam ¯ s¯a to resolve apparent contradictions which may occur in the set of Vedic (srauta) ´ prescriptions or can be derived via the facts. For example, consider the srauta ´ injunctions: (a) You ought not to study the Vedas if you are a S¯udra (i.e., a member of the lower class), (b) Not studying the Vedas implies not performing the Agnihotra sacriﬁce, and (c) You ought to perform the Agnihotra if you are a chariot maker. The additional fact (d) A chariot maker is a S¯udra, (apparently) leads to the conﬂicting obligations that you ought to study the Vedas if you are a chariot maker and that at the same time you ought not to do so, as extensively discussed by the M¯ımam ¯ s¯a author Jaimini (2nd c. BCE). The following example illustrates the way M¯ımam ¯ s¯a authors reason to solve such kinds of conﬂicting obligations. Moreover it shows that inferences which aim to mimic M¯ımam ¯ s¯a reasoning do not satisfy some of the principles identiﬁed as key properties of non-monotonic logics in Gabbay (1985). Remark 3 A central feature of our calculus is the distinction between prima facie and derived obligations, needed for differentiating Vedic commands from human deduc- tions. This distinction is instead missing in Deontic Default logic (Horty 2012), that also does not satisfy the Vikalpa principle. 123 358 B. Lellmann et al. From now on we will denote prima-facie obligations or deontic assumptions (sr ´ auta obligations) with O ( A/ B) to distinguish them from derived obligations (written pf O( A/ B)). Formally, the language of deontic assumptions is obtained from the lan- guage of MD by replacing the operator O with its prima-facie variant O . pf Example 1 We formalize the above statements concerning the Agnihotra sacriﬁce as follows : (a) O (¬ved/sdr),(b) agn → ved,(c) O (agn/chmk), and (d) pf pf chmk → sdr, where ved denotes the act of studying the Vedas, agn the perfor- mance of the Agnihotra sacriﬁce, sdr the fact of being a S¯udra, and chmk being a chariot maker. Using the monotonicity of the deontic operator in its ﬁrst argument, from (b) and (c) we derive the obligation (e) O(ved/chmk) (“You ought to study the Vedas if you are a chariot maker”). On the other hand, if it were possible to use indiscriminately monotonicity in the second argument of the deontic operator, from (a) and (d) we would derive (f) O(¬ved/chmk) (“You ought not to study the Vedas if you are a chariot maker”), obtaining a conﬂict between (e) and (f). By applying the speciﬁcity principle, we “limit” the monotonicity in the second argument of the operator; hence, following the above example, the derivation of (f) is blocked by the presence of the prima-facie obligation (c). The derivations from prima-facie injunctions are non-monotonic, as adding more premisses can change the derived result. However, they do not satisfy for exam- ple cautious monotonicity (if Γ ϕ and Γ ψ, then Γ, ϕ ψ), one of the crucial properties of non-monotonic logics. Indeed given the prima-facie injunctions O (ved/) and O (¬ved/sdr), both O(ved/tch) (“You ought to pf pf study the Vedas if you are a teacher”) and O(¬ved/tch ∧ sdr) are derivable, but, if one of these conclusions is considered as a premiss, the result changes, i.e. {O (ved/), O (¬ved/sdr), O (ved/tch)} O(¬ved/tch ∧ sdr). pf pf pf We extend below the sequent calculus G for the logic MD with special rules MD O (C / D) O (C / D) pf pf O , O to derive conditional obligations of the form O( A/ B) from L R prima-facie obligations (i.e. srauta ´ prescriptions) written as O (C / D), adopting lim- pf ited forms of monotonicity. (Sect. 3.1). The extension to the full language will be considered in Sect. 4. 3.1 Sequent calculus for specificity/gunapradhana ¯ In order to extend the sequent calculus for MD to capture the speciﬁcity principle, loosely following Goble (2013), p. 281, we interpret the notion of speciﬁcity as entail- ment in the presence of (global) propositional assumptions. I.e., given a set F of propositional facts about the world we say that proposition A is at least as speciﬁc as proposition B,if F entails A → B. Given this interpretation, the speciﬁcity principle can be understood as limiting monotonicity of the operator O in the second argument in the following sense. Given a list L of non-nested prima-facie obligations, and a proposition B, we should be licensed to infer the actual obligation O( A/ B) if (A) there is an injunction O ( A/ D) in L which is applicable i.e. we can infer using pf F that B → D, and there is no O ( X /Y ) in L such that B is at least as speciﬁc pf 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 359 as Y , Y is at least as speciﬁc as D, and the formulae A and X are inconsistent, i.e. we can infer ¬( A ∧ X ). However, while this implements the notion that more speciﬁc srauta ´ obligations over- rule less speciﬁc conﬂicting ones, this only resolves conﬂicts between propositions O (G/ H ) and O ( I / J ) in L for which the conditions are comparable in the sense pf pf that either H implies J or J implies H. Hence, to make the resulting theory consistent with MD, following the M¯ımam ¯ s¯a reasoning in Example 1 we add a further condition stating that (B) there is no obligation O ( X /Y ) ∈ L such that B is at least as speciﬁc as Y,the pf enjoined A and X are inconsistent, and which is not overruled by a more speciﬁc obligation O ( E / F ) from L. pf At this point, in order to make this intuition formally precise we need to take a fundamental design decision: given that our logic MD includes monotonicity in the ﬁrst argument, whenever we can derive an obligation O( A/ B), we should also be able to derive the obligation O( A ∨ C / B) as well. Given a list of prima-facie obligations, the question then essentially is whether we ﬁrst eliminate all the conﬂicts from this list, and then saturate under monotonicity in the ﬁrst argument (as, e.g., in the suggested procedure of removing conﬂicts from NDSICs in Libal and Pascucci 2019), or we ﬁrst consider all the consequences of the original list under monotonicity, and then eliminate all the obligations which would yield a conﬂict. We clarify this with the following example. Example 2 Consider the list containing exactly the two prima-facie obligations O ( A ∧ B/C ) and O (¬ A/C ). Since A ∧ B and ¬ A are inconsistent, the approach pf pf of ruling out conﬂicting obligations ﬁrst and then saturating under monotonicity in the ﬁrst argument would yield an empty set of obligations. In the second approach, instead, we ﬁrst saturate under monotonicity, giving, e.g., the obligation O(( A ∧ B) ∨¬ A/C ), and only then rule out conﬂicts. Since ( A ∧ B) ∨¬ A is contradicting neither A ∧ B nor ¬ A, we thus would keep the obligation O(( A ∧ B) ∨¬ A/C ). Also, if we added the prima facie obligation O (¬ A/C ∧ D) we would get both O( B/C ∧ D) and pf O(¬ A/C ∧ D), in contrast with the ﬁrst approach which would ﬁrst eliminate the assumption O ( A ∧ B/C ) and hence would not yield O( B/C ∧ D). pf While both approaches have their uses, in this work we choose to follow the second one, because of two main reasons. It allows us to preserve the power of deontic assumptions (Srauta obligations) as much as possible, by suspending only the part of an obligation which is in conﬂict with another one. Moreover, it naturally implements the M¯ımam ¯ s¯a principle of vikalpa. Such a principle corresponds to the disjunctive response: given two incompatible prima-facie obligations O ( A/ B) and O (C / D), pf pf in a situation where both apply, i.e., where B ∧ D holds, one may choose which one to follow, corresponding to the obligation O( A ∨ C / B ∧ D). We now make this formally precise. In the remainder of this paper we assume that F is a ﬁnite set of sequents containing only propositional variables, which is closed under cuts, i.e., whenever Γ ⇒ Δ, p and p,Σ ⇒ Π are in F, then so is Γ, Σ ⇒ Δ, Π, and closed under contractions, i.e., whenever Γ, p, p ⇒ Δ or Γ ⇒ p, p,Δ are in F, then so are Γ, p ⇒ Δ and 123 360 B. Lellmann et al. Fig. 4 A graphical representation of the conditions for O( A/ B) being derivable. Areas can be taken as formulae with containment representing entailment, i.e., more speciﬁc formulae are contained in less speciﬁc ones Γ ⇒ p,Δ respectively. We call F the set of (propositional) facts. Note that, since every propositional formula is equivalent to a formula in conjunctive normal form and sequents containing only propositional variables correspond to clauses of a formula in conjunctive normal form, using this deﬁnition we can stipulate arbitrary propositional formulae as facts. We further assume a ﬁnite set L of non-nested deontic assumptions, i.e., a ﬁnite set L of formulae of the form O ( A/ B) where A and B do not contain pf the O-operator. We call these formulae prima-facie obligations. Remark 4 The facts expressed by the sequnts in F areassumedtobetruestate- ments about the world: we do not consider the reliability of information conveyed by such statements. Indeed, in contrast with Nute (1997), we do not distinguish actual obligations—in force under conditions that are actually veriﬁed—from apparent obli- gations, which are in force given all we know about morally relevant circumstances, that is under conditions that are not necessarily veriﬁed, but only believed to be true. To capture the intuition for the speciﬁcity principle given above in a well-behaved sequent system, we ﬁrst need to make the notion of implication used there formally precise. In particular, we would like to deﬁne a notion of inference from the facts in F depending on the set L, such that we can derive a sequent ⇒ O( A/ B) if and only if both of the following hold, corresponding to the conditions (A) and (B) above: – there is O (C / D) ∈ L such that F B ⇒ D and F C ⇒ A and for all pf O ( X /Y ) ∈ L we have: ( F B ⇒ Y or F Y ⇒ D or F X , A ⇒) pf – for all O ( X /Y ) ∈ L we have: F B ⇒ Y or F X , A ⇒ or there is a pf O ( E / F ) ∈ L such that: (F B ⇒ F and F F ⇒ Y and F E ⇒ A). pf To ensure that these conditions hold, we will simply turn them into premisses of the corresponding sequent rules. At ﬁrst this might seem rather problematic, because we use underivability () to deﬁne derivability (). However, we will show below that the resulting notion of derivability is well-deﬁned, using the technical tool of the cut elimination result. Graphically, these two conditions can be visualised as in Fig. 4. The ﬁrst condition requires that the derivable obligation O( A/ B) is implied (via upward monotonicity on the ﬁrst argument and downward monotonicity on the second argument) by a less speciﬁc deontic assumption O (C / D) ∈ L and that there is pf 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 361 no O ( X /Y ) ∈ L conﬂicting with O( A/ B) which is more general than that and pf more speciﬁc than O (C / D). This condition includes the choice mentioned above. pf Indeed, it requires that any O ( X /Y ) ∈ L, which is more speciﬁc than O (C / D) pf pf and more general than O( A/ B), does not conﬂict with O( A/ B), instead of requiring that it does not conﬂict with O (C / D). This means that the speciﬁcity principle is pf applied for resolving conﬂicts only after saturating the set of deontic assumptions under monotonicity. The second condition models the fact that the conﬂicting prima-facie obligation O ( X /Y ) is overruled by the more speciﬁc prima-facie obligation O ( E / F ) by pf pf stating that F E ⇒ A, i.e., that what is enjoined by O ( E / F ) implies A. While pf this implies that E and X are inconsistent, one may wonder whether it is a too strong condition. In fact, as a consequence of our fundamental design decision to ﬁrst saturate the set of prima-facie obligations under monotonicity, and then ruling out conﬂicts, the obvious alternative of only demanding X and E to be inconsistent would lead to conﬂicting obligations rather quickly. Conceptually, this is due to the fact that the more speciﬁc obligation O ( E / F ) only suspends the part of the obligation O ( X /Y ) which pf pf is in conﬂict with E, but does not cancel the obligation completely. So in particular, if this part is unrelated to A, then the part of O ( X /Y ) which conﬂicts with O( A/ B) pf will remain unsuspended, and hence we should not be able to derive the latter. For example, given the list L ={O (¬ p/s), O ( p ∧ q/s), O (¬q ∧ r /t ), O (¬r /t )}, pf pf pf pf we would end up with the problematic situation of deriving both O(q/s ∧ t ),using O ( p ∧ q/s) as the main obligation, and O(¬q/s ∧ t ),using O (¬q ∧ r /t ) as the pf pf main obligation. The stronger condition used above prevents this situation. O (C / D) O (C / D) pf pf To turn these considerations into sequent rules (the rules O , O in L R Deﬁnition 1 below), we convert every (meta-)conjunction and universal quantiﬁer in this characterization into different premises, while (meta-)disjunctions and existential quantiﬁers yield a split into different rules. To write the rules in an economic way, we use the following notation. Notation 1 If P is a set of premisses, and S ={S ,..., S } is a set of sets of premisses 1 n we write P ∪ [S] P ∪ S P ∪ S 1 n C for the set of rules ,..., C C e.g., we write { X ⇒ Y }∪ [{{Σ ⇒ Π }, {Ω ⇒ Θ, A | A ∈ F }}] i i Γ ⇒ Δ for the set containing the two rules { X ⇒ Y }∪{Σ ⇒ Π } { X ⇒ Y }∪{Ω ⇒ Θ, A | A ∈ F } i i and . Γ ⇒ Δ Γ ⇒ Δ Note that we use set-theoretic notation for the sets of premisses, e.g., the rule above left has the two premisses X ⇒ Y and Σ ⇒ Π and the conclusion C. 123 362 B. Lellmann et al. Since the rules now also will mention underivability, we further need to add a judgment for this to some of the sequents, written as (F, L) , with the intended G cut MD meaning that the sequent is not derivable from the facts F and the prima-facie deontic statements L in the system G cut, in the sense deﬁned below (Deﬁnition 2). Thus MD O (C / D) pf we will obtain a set of rules O introducing a formula of the form O( A/ B) on the right hand side of the sequent. For technical reasons we will also add rules O (C / D) pf O introducing such a formula on the left hand side—these essentially follow from absorbing inferences using the axiom (D) into the previous rule. We will show below, in the discussion of the full system, that their addition indeed is merely a technical convenience (see Lemma 1). Remark 5 The formulae we want to infer might have nested deontic operators, setting the system apart from, e.g., the known systems of Input/Output logic (Makinson and van der Torre 2000). Indeed, they should capture key prescriptions like “under the condition of having to perform sacriﬁce α under the conditions β, you ought to do γ ”. However, to ensure decidability of the system we do not permit nested obligations in the deontic assumptions. Deﬁnition 1 Let L ={O ( A / B ),..., O ( A / B )} be a ﬁnite set of non-nested 1 1 n n pf pf prima-facie obligation formulae and let F be a set of propositional sequents. The rules of ga (for global assumptions from L)are giveninFig. 5.A proto-derivation with conclusion Γ ⇒ Δ in the system G from assumptions (F, L) is a ﬁnite labelled tree, MD where each internal node is labelled with a sequent, each leaf is labelled with an initial sequent, a sequent from F,oran underivability statement (F, L) Σ ⇒ Π, G cut MD such that the label of every internal node is obtained from the labels of its children using the rules of G or ga . The notion of a proto-derivation in the system G cut MD MD is deﬁned analogously, but also permitting applications of the cut rule Γ ⇒ Δ,AA,Σ ⇒ Π cut Γ, Σ ⇒ Δ, Π . The depth of a proto-derivation is the depth of the underlying tree, i.e., the maximal length of a branch in the tree plus one. For future reference we divide the premisses of the rules in Fig. 5 into different blocks in the following way: the ﬁrst two premisses, i.e., { B ⇒ D} and {C ⇒ A} together form the standard block, stating that the prima-facie obligation O (C / D) pf potentially can be used to derive the conclusion O( A/ B). The following block ⎧ ⎡ ⎤ ⎫ (F, L) B ⇒ Y ⎨ ⎬ G cut MD ⎣ ⎦ (F, L) Y ⇒ D | O ( X /Y ) ∈ L G cut pf MD ⎩ ⎭ (F, L) X , A ⇒ G cut MD is called the not-excepted block, and states that the prima-facie obligation O (C / D) pf is not overruled by another one which is at least as speciﬁc. The remaining premisses together form the no-active-conﬂict block, which states that there is no other conﬂicting 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 363 Fig. 5 The rules of ga with O (C / D) ∈ L L pf prima-facie obligation which is not overruled by a more speciﬁc one. For every formula O ( X /Y ) the choices are divided into the no-conﬂict block, stating that there is no pf conﬂict between the prima-facie obligation O ( X /Y ) and the desired conclusion pf O( A/ B), and consisting of the underivability statements (F, L) B ⇒ Y and (F, L) X , A ⇒, G cut G cut MD MD and the override block, consisting of the remaining possible premisses { B ⇒ F}∪{ F ⇒ Y }∪{ E , A ⇒}| O ( E , F ) ∈ L pf and stating that the prima-facie obligation O ( X /Y ) is overruled by another one pf O (C / D) pf which is at least as speciﬁc. The terminology for the rule O is analogous. Deﬁnition 2 A proto-derivation in G (in G cut)from (F, L) is valid if for each of MD MD the underivability statements (F, L) Σ ⇒ Π, occurring as one of the leafs of G cut MD that derivation, there is no valid proto-derivation of Σ ⇒ Π in G cut from (F, L). MD In case there is such a valid proto-derivation we also write (F, L) Γ ⇒ Δ and MD (F, L) Γ ⇒ Δ respectively. G cut MD Note that underivability statements are always evaluated in the system with the cut rule. Since the deﬁnition of a valid proto-derivation involves the notion of a valid proto-derivation itself, it is not immediately clear that this notion is well-deﬁned. We will show in the discussion of the full system below (Corollary 1) that this is indeed the case. In particular, this along with the decidability result follows from the crucial cut elimination theorem, stating the redundancy of the cut rule: 123 364 B. Lellmann et al. O (C / D) pf Fig. 6 The rule O from Example 3 Proposition 2 (Ciabattoni et al. 2018) For every F, L and sequent Γ ⇒ Δ we have (F, L) Γ ⇒ Δ if and only if (F, L) Γ ⇒ Δ. G cut G MD MD Since this proposition is a special case of the more general result for the full system in Theorem 2 below we omit the proof. Example 3 Consider the prima-facie obligations given by L ={O (agn/), pf O (¬agn/sdr)} (with agn, sdr and tch as in Example 1) and the set F =∅ pf of facts. Taking the formula O (agn/) as the formula O (C / D) in the general pf pf scheme of Fig. 5, we obtain the rules in Fig. 6. In particular, the sequent ⇒ O(agn/tch) would be derivable using, e.g., an instance of the rule B ⇒ agn ⇒ A (F, L) agn, A ⇒ (F, L) B ⇒ sdr G cut G cut MD MD B ⇒ ⇒ agn ⇒ A (F, L) B ⇒ sdr G cut MD ⇒ O( A/ B) Similarly, taking the formula O (C / D) to be O (¬agn/sdr) we obtain, e.g. pf pf B ⇒ sdr ¬agn ⇒ A (F, L) ⇒ sdr (F, L) ¬agn, A ⇒ G cut G cut MD MD B ⇒ sdr sdr ⇒ ¬agn ⇒ AB ⇒ sdr sdr ⇒ sdr ¬agn ⇒ A ⇒ O( A/ B) which serves to derive the sequent ⇒ O(¬agn/tch ∧sdr). Finally, using ga with O (¬agn/sdr) for the formula O (C / D) yields a derivation of O(agn/tch ∧ pf pf sdr) ⇒ and thus ⇒¬O(agn/tch ∧sdr). Note that even for just two prima-facie obligations we obtain many (often redundant) rules. 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 365 4 Extending bMDL with new deontic operators The preliminary analysis of M¯ımam ¯ s¯a reasoning purely in terms of obligations is rather simplistic, since it considers other deontic concepts such as prohibitions as deﬁned notions. It turned out, indeed, that obligations and prohibitions are treated markedly different in M¯ımam ¯ s¯a : on a “meta-logical” level, obeying a Vedic obligation gives positive results and disrespecting it implies just the lack of these results; conversely, the observance of a Vedic prohibition gives no result and the violation of it leads to a sanction (the accumulation of negative karma). Hence it is not enough to model prohibitions as negative obligations. In addition the difference between prescriptions and prohibitions is not only on the results of obeying or disrespecting them; one of the most important differences is the idea at the base of those two deontic concepts, i.e. “activation” in case of injunctions and “inhibition” for prohibitions. To conﬁrm such a distinction, let us consider the debate in M¯ımam ¯ s¯a commentaries on the injunction not to eat kalajañja (probably a variety of garlic). If the command represents a prohibition, it means that, independently from the agents’ desires and motivations, the agents have the duty not to consume this product: in principle, even eating it by accident would constitute a violation. On the other hand, if the command is a negative obligation, the agents have the duty to choose to refrain from eating kalañja; hence, theoretically, if the agents decide to eat that vegetable, they are not compliant with the obligation, even if an external contingency prevents them from realizing their intentions. For these reasons, prescriptions and prohibitions should be considered as genuine deontic concepts: they cannot be deprived of their deontic content and reduced to instructions for obtaining desirable results or avoiding sanctions. Such an interpretation—that could be formally represented by a Kanger- Andersonian reduction—would be closer to the instrumental reading of commands given by the late author Mandana Misra ´ (c. 8th century CE), which was in between . . the M¯ımam ¯ s¯a and the Vedanta schools of Indian philosophy. In this section we continue and reﬁne the analysis of M¯ımam ¯ s¯a reasoning in Cia- battoni et al. (2015) by extending the -free fragment of the logic bMDL with new operators for prohibitions and recommendations. We call the resulting logic MD+.As for the obligation operator O, axioms and rules for the new operators are extracted from the M¯ımam ¯ s¯a nyaya ¯ s, that have been in the meanwhile found in the texts, translated from Sanskrit, interpreted and abstracted. Prohibitions are modeled in MD+ using the operator F ( A/ B), to be read as “ A is forbidden under the conditions B”. As in the case of obligations, prohibitions are better expressed by a dyadic operator. They can apply unconditionally to the person throughout her life (purusartha ¯ ), as in the Vedic command “one should not perform violence on any living being”, or be relative to a particular ritual context (kratvartha), as for the example “one should not utter the ‘ye yajamahe’ ¯ mantra during the after- sacriﬁces” discussed in Jaimini’s P¯ urva M¯ ımams ¯ aS ¯ utr ¯ a (henceforth PMS). A similar phenomenon happens in Talmudic logic, were distinct operators are needed (Abraham et al. 2011). This is an ongoing project, which is carried out together with Sanskritists. 123 366 B. Lellmann et al. A ﬁrst natural property for prohibitions is expressed by the axiom D ¬(F ( A/ B) ∧ F (¬ A/ B)) motivated by the consideration that, because of the “meaningfulness of Vedic com- mands” (in PMS 1.2.23) stating that no injunction can be meaningless or inapplicable, there is always a way to obey all needed commands and avoid sanctions; this would be impossible having the prohibition of an action and its negation under the same conditions. The downward monotonicity rule (Mon ) C → AB ↔ D F ( A/ B) → F (C / D) is justiﬁed by argumentations as the one in Medhatithi’ ¯ s Manubhas ¯ ya, where the prohibition to commit suicide (F (suicide/)) is derived from the more comprehensive prohibition to commit violence (F (violence/)), since there it is explicitly assumed that suicide → kill and kill → violence. Finally, the axiom D OF ¬(O( A/ B) ∧ F ( A/ B)) arises again from the nyaya ¯ about meaningfulness of commands, and from what is known as “ought implies can principle”, extracted from the discussion on the concept of adhikara ¯ in Jaimini’s texts. According to this principle, a person who is prescribed to perform an action is assumed to have not only physical and economical capacities, but also the “practical” possibility to complete the action without undesirable conse- quences, like damages or sanctions. Therefore, performing a prescribed act can never lead to a sanction, hence it is impossible that the same act is prohibited. Recommendations Besides the distinction between obligations and prohibitions, a more reﬁned analysis should take into account the different notions of prescriptions used by M¯ımam ¯ s¯a authors. Traditionally, rituals prescribed by the Vedas are distin- guished into ﬁxed (nitya), occasional (naimittika), and elective (k¯ amya); ﬁxed ritual actions should be performed, in order to obtain the positive result of good karma, regu- larly throughout the whole life, occasional ones have similar properties, but should be carried out in special occasions, like the birth of a child, whilst the third kind includes rituals to be executed only in order to obtain a speciﬁc result. It has been noticed (Freschi et al. 2019) that, while the characteristics of the operator O in bMDL are well suited to describe ﬁxed and occasional prescriptions, the elective rituals represent rec- ommendations or instruction for achieving a result in a “Vedic” way, more than proper obligations. We call these weaker obligations recommendations and express them with the operator R(./.). We model R(./.) using the dyadic version of the modal logic MP, see, e.g., Chellas (1980), in line with the analysis in Freschi et al. (2019) where also the axioms for R(./.) are motivated. Note that it might be possible for something to be obligatory and recommended at the same time, as, e.g., in the case of the Agnihotra sacriﬁce. Indeed, the sequence of actions constituting the Agnihotra ritual represents the content both of a ﬁxed sacriﬁce (corresponding to obligations) and of an elective 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 367 Fig. 7 The Hilbert-style axiomatisations for prohibitions and weak obligations in MD+ one (corresponding to recommendations). In other words, if the agents perform such a ritual perfectly, according to the stricter rules governing elective sacriﬁces, they are compliant both with the recommendation (hence obtaining the desired result) and with the obligation (fulﬁlling their duties). The axiom ¬R(⊥/ A) guarantees that there are no self-contradictory recommendations, which represents a minimal condition for any Vedic instruction. Notice that it is weaker than D and D , as, in contrast with obligations and prohibitions, it is possible to have two rec- ommended rituals for getting the same result which cannot be performed at the same time. In those cases (e.g., in the case of the prescriptions of k¯ ariri sacriﬁce and twelve- nights sacriﬁce for obtaining the rain, see, e.g., Freschi et al. 2019)M¯ımam ¯ s¯a authors assume that one of the two sacriﬁces is enough to get the intended result, but both recommendations remain in force. The rule A →CB ↔ D R( A/ B) → R(C / D) is justiﬁed by the following abstraction of the nyaya ¯ sinthe Tantrarahasya IV.4.3.3 (see Freschi 2012): if the accomplishment of X presupposes the accomplishment of Y, the obligation to perform X prescribes also Y. Already mentioned in Ciabattoni et al. (2015) regarding obligations, this principle is suitable also for recommendations because, as noticed in Freschi et al. (2019), it is more about how M¯ımam ¯ s¯a authors consider the relations among facts than about a speciﬁc kind of prescription. The axioms and rules of all the operators in a Hilbert-style system are given in Fig. 7, and the corresponding sequent rules, obtained using the method in Lellmann and Pattinson (2013), are given in Fig. 8. Note that the resulting sequent calculus admits cut-elimination by construction and hence the resulting logic is consistent. Permissions MD+ does not include an explicit operator for permissions: they are instead treated exclusively as explicit exceptions to obligations or to prohibitions, and hence considered only on the prima-facie level. This formalization is motivated by M¯ımam ¯ s¯a authors’ interpretation, which assumes that “there cannot be a prescription prescribing a person to do something she is already inclined to do” (novelty nyaya ¯ in 123 368 B. Lellmann et al. Fig. 8 The sequent rules of G for the logic with prohibitions and recommendations MD+ Fig. 9 The global assumption rules for recommendations, based on the prima-facie recommendation R (C / D) ∈ L pf Jaimini’s PMS 1.2.19). Hence permissions, having the same linguistic form as pre- scriptions but conveying something that is naturally desired by anyone, are interpreted as exceptions to more general commands. For instance, the statement (in Sabara on PMS 10.7.28) “the ﬁve ﬁve-nailed animals can be eaten” is interpreted, at the prima- facie level, as the prohibition to eat meat plus the permission to eat ﬁve species of ﬁve-nailed animals (i.e., some species of wild rodents, wild boars, lizards, hares, and turtles). 5 Defeasible reasoning in M¯ımam ¯ s¯a We introduce a sequent calculus to reason in presence of speciﬁcity in the extended logic. In order to incorporate into our framework prohibitions, permissions as excep- tions to prescriptions or prohibitions, and recommendations, we extend the list of deontic assumptions or prima-facie (srauta) ´ prescriptions to also include prima- facie prohibitions, prima-facie obligation-permissions (exceptions to obligations), prima-facie prohibition-permissions (exceptions to prohibitions) and prima-facie rec- O F ommendations, denoted by the operators F (./.), P (./.), P (./.) and R (./.) pf pf pf pf respectively. Hence, the list L of prima-facie deontic statements now contains ﬁnitely many (non-nested) formulae of these forms. In particular, if L only contains prima-facie obligation formulae, we recover the simpliﬁed situation of Sect. 3.The construction of the global assumption rules then follows the same principle as before, incorporating speciﬁcity. For the recommendations we need to make sure that we do not derive R( A/ B) with A equivalent to ⊥. In particular, following Freschi et al. (2019)weuse theM¯ımam ¯ s¯a reasoning that the Vedas do not recommend anything which is self-contradictory, to rule out prima-facie recommendations R( A/ D) where it follows from the facts that A implies ⊥. Hence we only need one global assumption rule of the form given in Fig. 9. Remark 6 Due to the presence in MD+ of axiom D , the cases for the obligations OF and prohibitions are somewhat more complex than cases involving recommendations, 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 369 as prima-facie obligations and prohibitions can overrule each others according to the speciﬁcity principle. For the sake of an economical presentation we employ Notation 1 from Sect. 3.1. The rationale for the construction of the rules then is as follows: –Due to D (resp. D ), more speciﬁc conﬂicting obligations (resp. prohibitions) O F overrule less speciﬁc obligations (resp. prohibitions). – Due to the interaction rule D , more speciﬁc conﬂicting obligations overrule OF less speciﬁc prohibitions and vice versa. – Due to the interpretation of permissions as explicit exceptions to obligations or prohibitions, more speciﬁc obligation-permissions overrule less speciﬁc obliga- tions, but have no relevance for prohibitions, and analogously for prohibition- permissions. Right rules Following this, for obligations we obtain the following characterisation. An obligation O( A/ B) follows from a set L of srauta ´ deontic statements if there is a srauta ´ obligation O (C / D) in L such that: pf – The assumption is applicable, because the condition B is at least as speciﬁc as D, i.e., (F, L) B ⇒ D. – A is entailed by C, i.e., (F, L) C ⇒ A. – There is no more speciﬁc conﬂicting srauta ´ obligation or obligation-permission O O (P ), i.e., for every O ( X /Y ) ∈ L or P ( X /Y ) ∈ L we have ((F, L) B ⇒ Y pf pf pf or (F, L) Y ⇒ D or (F, L) X , A ⇒ ). – There is no more speciﬁc conﬂicting srauta ´ prohibition, i.e., for every F ( X /Y ) ∈ pf L we have ((F, L) B ⇒ Y or (F, L) Y ⇒ D or (F, L) A ⇒ X). – Every conﬂicting srauta ´ obligation is overruled by a more speciﬁc obligation, pro- hibition, or obligation-permission (P ) i.e., for every srauta ´ obligation O ( X /Y ) pf pf with (F, L) B ⇒ Y and (F, L) X , A ⇒ there is a srauta ´ obligation O ( E / F ) with ((F, L) B ⇒ F and (F, L) F ⇒ Y and (F, L) E ⇒ A) pf or there is a srauta ´ prohibition F ( E / F ) with ((F, L) B ⇒ F and (F, L) pf F ⇒ Y and (F, L) ⇒ A, E)orthereisa srauta ´ permission P ( E / F ) with pf ((F, L) B ⇒ F and (F, L) F ⇒ Y and (F, L) E ⇒ A). – Every conﬂicting srauta ´ prohibition is overruled by a more speciﬁc obligation, prohibition or prohibition-permission, i.e., for every srauta ´ prohibition F ( X /Y ) pf with (F, L) B ⇒ Y and (F, L) A ⇒ X there is a srauta ´ obligation O ( E / F ) pf with ((F, L) B ⇒ F and (F, L) F ⇒ Y and (F, L) E ⇒ A)orthereis a´srauta prohibition F ( E / F ) with ((F, L) B ⇒ F and (F, L) F ⇒ Y and pf (F, L) ⇒ A, E)orthereisa srauta ´ permission P ( E / F ) with ((F, L) B ⇒ pf F and (F, L) F ⇒ Y and (F, L) E ⇒ A). – Every conﬂicting srauta ´ obligation-permission is overruled by a more speciﬁc obligation, i.e., for every srauta ´ obligation-permission P ( X /Y ) with (F, L) pf B ⇒ Y and (F, L) A, X ⇒ there is a srauta ´ obligation O ( E / F ) with pf ((F, L) B ⇒ F and (F, L) F ⇒ Y and (F, L) E ⇒ A). The notion of being conﬂicting here is different, depending on the two conﬂicting statements. In particular, two obligations O ( A/ B) and O (C / D) are conﬂicting if pf pf 123 370 B. Lellmann et al. Fig. 10 The assumption right rule for obligations in presence of prohibitions and permissions what is obligatory, i.e., A and C, cannot be true at the same time. This is equivalent to stating that we can derive ¬( A ∧ C ), or equivalently the sequent A, C ⇒ from the facts. In contrast, an obligation O ( A/ B) conﬂicts with a prohibition F (C / D) if pf pf following the obligation would necessarily violate the prohibition, or in other words if the implication A → C follows from the facts, i.e., the sequent A ⇒ C is derivable from the facts. Two prohibitions F ( A/ B) and F (C / D) then conﬂict if it is not pf pf possible to follow both. This means that the formula A ∨ C resp. the sequent ⇒ A, C follows from the facts. Finally, a prohibition-permission P ( A/ B) conﬂicts with a pf prohibition F (C / D) if the permitted A implies the forbidden C, i.e., if A → C resp. pf the sequent A ⇒ C is derivable. Incorporating this rationale into the construction of the assumption right rule for O (C / D) pf obligations leads to the rules O for every formula O (C / D) ∈ L shown in pf Fig. 10. 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 371 Similarly, using the above rationale to construct the assumption right rule for pro- F (C / D) pf hibitions yields the rule F given in Fig. 11. Left rules In order to obtain a cut-free system again we need to absorb cuts between the principal formulae of these two rules and the remaining rules into the rule set. In particular, saturating the rule set under cuts between the assumption right rule for obligations and the rule D and the interaction rule D respectively yields the rules O OF O (C / D) O (C / D) pf pf O and F shown in Figs. 12 and 13. Using these rules it is possible L L to derive an obligation and prohibition respectively on the left, from a prima-facie obligation O (C / D) ∈ L. Similarly, cuts between the assumption right rule for pf prohibitions and the rule D and the interaction rule D respectively yield the rules F OF F (C / D) F (C / D) pf pf F and O shown in Figs. 14 and 15, respectively. Note that in the case L L where L contains only prima-facie obligation formulae we obtain exactly the rules of ga in Fig. 5 of the previous section. Again, for each of these rules we divide the premisses into the standard block consisting of the ﬁrst two premisses, the not-excepted block, consisting of the under- ivability statements stating that there is no conﬂicting and at least as speciﬁc prima facie deontic statement, and the no-active-conﬂict block, consisting of the last three sets of premisses and stating that every conﬂicting prima-facie deontic statement is overruled by a more speciﬁc one. For every prima-facie deontic statement, the corre- sponding possible premisses in the no-active-conﬂict block again are divided into the no-conﬂict block consisting of the underivability premisses, and the override block, consisting of the premisses stating that the deontic statement is overridden. Similarly to the simpliﬁed case without prohibitions and permissions, we use the following notation. Deﬁnition 3 We write (F, L) Γ ⇒ Δ if there is a valid proto-derivation of G cut MD+ Γ ⇒ Δ from F in the system G extended with the following global assumption MD+ rules for L: op1 ∈{O, F }, op2 ∈{O , F }, op2(C / D) pf pf ga := op1 | . op2(C / D) ∈ L, s ∈{L, R} O (C / D) O (C / D) F (C / D) pf pf pf The following lemma shows that the rules O , F , F and L L L F (C / D) pf O in the presence of the cut rule can be seen as a mere technical convenience, because they do not change the set of derivable sequents. However, in order to be able to perform automated reasoning in our system, we also would like to eliminate the cut rule itself, and the resulting system would not be complete without these rules. Lemma 1 (Redundancy of the left rules) If there is a valid proto-derivation of Γ ⇒ Δ in G cut from (F, L), then there is a valid proto-derivation of Γ ⇒ Δ from (F, L) MD+ in the system without the rules in Figs. 12, 13, 14 and 15. Proof We show how to replace every application of one of these rules by cuts and an O (C / D) F (C / D) pf pf application of O and F respectively. So consider ﬁrst an application R R O (C / D) pf of the rule O as in Fig. 12. From every premiss of the form Γ, A ⇒ Δ and 123 372 B. Lellmann et al. Fig. 11 The assumption right rule for prohibitions Σ ⇒ A,Π using weakening and the implication rules we obtain the corresponding premiss Γ ⇒ A →⊥,Δ and Σ, A →⊥⇒ Π, respectively. Further, from every underivability statement (F, L) Γ ⇒ A,Δ we obtain the corresponding G cut MD+ statement (F, L) Γ, A →⊥⇒ Δ, since if for the latter there were a valid G cut MD+ proto-derivation, we could extend it to one of the former via: Γ, A ⇒ A, ⊥,Δ Γ ⇒ A, A →⊥,Δ Γ, A ⇒ Δ cut Γ ⇒ A,Δ Analogously, from every underivability statement (F, L) Σ, A ⇒ Π we G cut MD+ obtain the corresponding statement (F, L) Σ ⇒ A →⊥,Π. Hence we G cut MD+ O (C / D) pf have all the premisses necessary to apply the rule O with conclusion ⇒ O( A →⊥/ B). From this we obtain the conclusion of the original application of the 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 373 Fig. 12 The assumption left rule for obligations in presence of prohibitions and permissions O (C / D) pf rule O using cut on the conclusion of the rule D as follows: A →⊥, A ⇒ B ⇒ B ⇒ O( A →⊥/ B) O( A/ B), O( A →⊥/ B) ⇒ O( A/ B) ⇒ The premisses are clearly derivable. O (C / D) pf In a similar way we obtain the conclusion of an application of F from an O (C / D) pf application of O and a cut with the conclusion of the rule D . The reasoning OF for the remaining rules is analogous. The central technical result about the system stating elimination of the cut rule then follows a reasonably standard pattern of a cut elimination proof, but slightly adjusted 123 374 B. Lellmann et al. Fig. 13 The derived left rule for prohibitions in presence of prohibitions and permissions to also accommodate for the underivability statements. The proof is detailed in the “Appendix”. In the following we write (F, L) for the cut-free system, i.e., the MD+ calculus (F, L) with the cut rule removed. Note that again in the cut-free G cut MD+ calculus G the non-derivability statements range over the system with the cut rule. MD+ Theorem 2 (Cut elimination) If (F, L) Γ ⇒ Δ, then (F, L) Γ ⇒ Δ. G cut G MD+ MD+ Proof See the “Appendix”. From the cut elimination theorem we then obtain equivalence of the systems with and without the cut rule: Proposition 3 For every F, L we have (F, L) Γ ⇒ Δ if and only if (F, L) Γ ⇒ Δ. G cut G MD+ MD+ 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 375 Fig. 14 The assumption left rule for prohibitions Proof The “only if” direction is the statement of the cut elimination theorem. The proof for the “if” direction is straightforward, since every rule in G is also a rule in MD G cut, and since the underivability statements range over the same system for valid MD proto-derivations in both systems. 5.1 Consequences of cut elimination The Cut Elimination Theorem has a number of important consequences, in particular the fact that the notion of a valid proto-derivation is well-deﬁned, consistency of the system, and a decidability and complexity result. The latter shows that despite the somewhat complicated shape of the assumption rules, reasoning in the calculus does not have a higher complexity than reasoning in standard modal logics such as K or in intuitionistic logic. The fact that valid proto-derivations are well-deﬁned can be seen by considering the following alternative stratiﬁed deﬁnition. 123 376 B. Lellmann et al. Fig. 15 The new assumption left rule for obligations Deﬁnition 4 A proto-derivation of rank n in G (in G cut) from (F, L) with MD+ MD+ conclusion Γ ⇒ Δ is a proto-derivation in G (in G cut)from (F, L) with MD+ MD+ conclusion Γ ⇒ Δ such that – every formula occurring in the proto-derivation has modal nesting depth at most n – every formula occurring in an underivability statement in the proto-derivation has modal nesting depth at most n − 1. If n is a natural number, then a proto-derivation is n-valid if it is of rank n and for every k < n, for none of the underivability statements occurring in it there is a k-valid proto-derivation in G cut from (F, L). MD+ Since the modal nesting depth of the formulae in the underivability statements in the rules of ga is stricly lower than that of the formulae in the conclusion, the question whether a proto-derivation is n-valid only depends on k-validity for k < n. Hence this deﬁnition is inductive and not circular. Using the Cut Elimination Theorem we obtain that it is equivalent to unrestricted validity of proto-derivations as follows. 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 377 Theorem 3 For every sequent Γ ⇒ Δ with modal nesting depth at most n there is a valid proto-derivation in G cut from (F, L) with conclusion Γ ⇒ Δ if and only if MD+ there is a n-valid proto-derivation in G cut from (F, L) with conclusion Γ ⇒ Δ. MD+ Proof By induction on n. So suppose the statement holds for every k < n. If there is a valid proto-derivation of Γ ⇒ Δ from (F, L) in G cut with conclusion Γ ⇒ Δ, then by Cut Elimination MD+ (Theorem 2) there is a valid proto-derivation of Γ ⇒ Δ from (F, L) in G . Since MD+ none of the rules of G or ga increases the modal nesting depth from conclu- MD+ sion to premiss(es), the maximal modal nesting depth of formulae occurring in this proto-derivation is n. Further, since the modal nesting depth of the formulae in the underivability statements in the rules ga is strictly smaller than that of the conclu- sion, every formula occurring in an underivability statement in this proto-derivation has modal nesting depth at most n−1. Hence the proto-derivation is of rank n. Since the modal nesting depth of the formulae in the underivability statements is at most n − 1, by induction hypothesis we obtain that for these there is no k valid proto-derivation for any k ≤ n − 1. Hence the proto-derivation is n-valid. Conversely, if we have a n-valid proto- derivation, then again by induction hypothesis we obtain that for none of the underivability statements occurring in it there is a valid proto-derivation. Since a proto-derivation of rank n in particular is a proto-derivation, we obtain a valid proto- derivation for the same conclusion. Well-deﬁnedness of the notion of a valid proto-derivation then follows immediately from the previous theorem together with the fact that n-validity is well-deﬁned: Corollary 1 (Well-deﬁnedness) The notion of a valid proto-derivation is well-deﬁned. As a second consequence of Cut Elimination we obtain that the rules ga are compatible with the logic MD+ as given in Figs. 1 and 7 in the sense that they do not yield any conﬂicting obligations or prohibitions: Theorem 4 (Consistency) For any L and F not containing the empty sequent, the consequences of L under F are consistent over MD+, i.e., (F, L) ⇒⊥. G cut MD+ Hence in particular – there are no A, B with (F, L) ⇒ O( A/ B) ∧ O(¬ A/ B); G cut MD+ – there are no A, B with (F, L) ⇒ F ( A/ B) ∧ F (¬ A/ B); G cut MD+ – there are no A, B with (F, L) ⇒ O( A/ B) ∧ F ( A/ B); G cut MD+ – there is no B with (F, L) ⇒ R(⊥/ B). G cut MD+ Proof By inspection it is clear that all the rules in the calculus G ga have the sub- MD+ formula property relative to L in the sense that every formula occurring in a premise of a rule, including the underivability statements, is a subformula of a formula occurring in its conclusion or in L. Since the empty sequent is not in F, and apart from W there is no rule introducing ⊥ on the right hand side of a sequent, we cannot derive ⇒⊥. The second statement follows from this using derivability of O( A/ B)∧O(¬ A/ B) ⇒ and the analogous sequents for the statements involving F and R together with the cut rule. 123 378 B. Lellmann et al. Fig. 16 The system G3 without the assumption rules MD+ The third major consequence of the Cut Elimination Theorem is that it permits the restriction of proof search to proto-derivations in the system without the cut rule. Using the (extended) subformula property of the rules of the cut-free system this yields a decision procedure for the logic. To make this precise, the derivability problem is given by the following: M¯ımam ¯ s¯a derivability using speciﬁcity Input: Finite lists F, L of propositional facts and prima-facie deontic statements, and a sequent Γ ⇒ Δ Question: Do we have (F, L) Γ ⇒ Δ? G cut MD+ We will show a decidability and complexity result via a natural implementation of backwards proof search on an alternating Turing machine (see Chandra et al. 1981 for details). As usual, for this we ﬁrst eliminate the structural rules from the system. Deﬁnition 5 The system G3 is the system in Fig. 16, obtained from G by MD+ MD+ restricting initial sequents to atomic formulae, dropping the weakening and contraction rules, and absorbing weakening into the conclusion of the logical rules. Similarly, for a list L of prima-facie deontic statements, the rules ga are the rules from ga O (C / D) pf with weakening absorbed into the conclusion (only!). E.g., the rule O has O (C / D) pf exactly the same premisses as the rule O from Fig. 10, but the conclusion Γ ⇒ O( A/ B), Δ.A valid proto-derivation in G3 from (F, L) is deﬁned as for MD+ G with the exception that leaves may also be labelled with sequents Γ, Σ ⇒ MD+ Π, Δ, where Σ ⇒ Π ∈ F and Γ ⇒ Δ is an arbitrary sequent. In particular, the underivability statements also range over G cut. MD+ The following properties of the calculus are shown by standard methods. Lemma 2 (Generalised initial sequents) The generalised initial sequent rule Γ, A ⇒ A,Δ is admissible in G3 . MD+ Proof By induction on the complexity of the formula A, using the rules Mon , Mon , O F Mon in case the outermost connective is one of O, F , R. 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 379 ∗ ∗ Lemma 3 (Invertibility of the propositional rules) The rules → and → are depth- R L preserving invertible in G3 , i.e., whenever there is a valid proto-derivation of their MD+ conclusion in G3 with depth n from (F, L), then for each of the premisses there is MD+ a valid proto-derivation with depth n from (F, L) as well. Proof By induction on the depth of the valid proto-derivation. Lemma 4 (Admissibility of weakening and contraction) The weakening and the contraction rules are depth-preserving admissible, i.e., whenever there are valid proto- derivations in G3 of the premisses of these rules with depth n from (F, L), then MD+ there are valid proto-derivations of their conclusions with the same depth from (F, L) as well. Proof By induction on the depth of the valid proto-derivation, using Lemma 3 in case the contracted formula is a principal formula in a propositional rule. Lemma 5 Let F, L be ﬁnite lists of propositional facts and prima-facie deontic state- ments, respectively, and let Γ ⇒ Δ be a sequent. Then (F, L) Γ ⇒ Δ if and MD+ only if (F, L) Γ ⇒ Δ. G3 MD+ Proof Since the underivability statement range over the same system, we only need to show how to convert proto-derivations from one system to the other. For the “only if” direction, we replace applications of the weakening and contraction rules with invocations of Lemma 4, and simulate the generalised intitial sequents of G using MD+ Lemma 2. For the “if” direction, we make the absorbed weakening explicit using the rules W , W . L R Using the previous lemma, to solve the M¯ımam ¯ s¯a derivability problem, it is then enough to perform backwards proof search in the system G3 with the rules ga . MD+ Recall that for the assumption rules from Figs. 10, 11, 12, 13, 14 and 15 we divide the schematic premisses into blocks:the standard block contains the ﬁrst two premisses; the not-excepted block contains the schematic premisses of the sets in the second and third line, i.e., all those underivability statements stating that the prima-facie deontic statement is not overridden by a more speciﬁc one; the no-active-conﬂict block contains the schematic premisses of the remaining sets. The premisses of the no-active-conﬂict blocks for each formula from L are further divided into the conﬂict block consisting of the ﬁrst two premisses in the [.] construct and the override block consisting of the remaining ones (which again depend on additional formulae from L). Theorem 5 (Decidability and complexity) The M¯ ımam ¯ s¯ a derivability problem using speciﬁcity is decidable in polynomial space. Proof The implementation of the decision procedure on an alternating Turing machine is shown as Algorithm 1. Intuitively, the algorithm makes existential guesses for the last applied rule, then makes universal choices to verify that every premiss is derivable. Claim 1 Algorithm 1 terminates in polynomial time. Suppose that n is the size of the input, i.e., the sum of the number of symbols in F, L and Γ ⇒ Δ.Let the complexity of a sequent be the number of occurrences of propo- sitional or modal connectives in it. Every application of a propositional rule removes 123 380 B. Lellmann et al. Algorithm 1: Decision procedure for the M¯ımam ¯ s¯a derivability problem Input:atuple (F, L) of ﬁnite sets of propositional facts and prima-facie deontic statements and a sequent Γ ⇒ Δ Output:Is (F, L) Γ ⇒ Δ? G cut MD+ 1 if ⊥∈ Γ or Γ ∩ Δ =∅ then 2 halt and accept; 3 if there is Σ ⇒ Π ∈ F with Σ ⊆ Γ and Π ⊆ Δ then 4 halt and accept; 5 existentially guess a rule scheme R (propositional, modal or assumption) from G3 or ga and a MD+ matching (tuple of) principal formula(e) from Γ ⇒ Δ; 6 else if R is a propositional rule scheme then 7 universally choose one of its premisses Σ ⇒ Π; 8 check recursively whether (F, L) Σ ⇒ Π, output the answer and halt; G3 cut MD+ 9 else if R is a modal rule scheme then 10 universally choose one of its premisses Σ ⇒ Π; 11 check recursively whether (F, L) Σ ⇒ Π, output the answer and halt; G3 cut MD+ 12 else /* Then R is an assumption schema */ 13 universally choose a block B of premisses; 14 if B is the standard block then 15 Universally choose a premiss Σ ⇒ Π in B; 16 Recursively check whether (F, L) Σ ⇒ Π, output the answer and halt; G3 cut MD+ 17 else if B is the non-excepted block then 18 universally choose a formula from L and existentially guess a premiss (F, L) Σ ⇒ Π from the block of premisses for this formula; G3 cut MD+ 19 Recursively check whether (F, L) Σ ⇒ Π, ﬂip the answer and halt; G3 cut MD+ 20 else /* Then B is the no-active-conflict block */ 21 Universally choose a formula from L and existentially guess a block C of premisses for this formula; 22 if C is the conﬂict block then 23 existentially guess a premiss (F, L) Σ ⇒ Π from C; G3 cut MD+ 24 Recursively check whether (F, L) Σ ⇒ Π, ﬂip the answer and halt; G3 cut MD+ 25 else /* Then C is the override block */ 26 existentially guess a formula from L and universally choose a premiss Σ ⇒ Π from the corresponding set of premisses; 27 Recursively check whether (F, L) Σ ⇒ Π, output the answer and halt; G3 cut MD+ 28 end 29 end 30 end 31 halt and reject; a propositional connective by replacing a propositional formula with its immediate subformulae, and hence reduces the complexity of the sequents. Hence the number of such applications is bounded by the number of subformulae of the conclusion, F or L, and thus bounded by n. Moreover, from the shape of the modal and assumption rules together with the fact that the assumptions in L do not contain nested modal operators it follows that each of the recursive calls in lines 11, 16, 19, 24 and 27 is on a sequent of strictly lower maximal nesting depth of the modal operators. Hence the 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 381 maximal modal nesting depth of a sequent is bounded by the size n of the input as well, and the maximal number of formulae in a sequent is thus bounded by 2n.Due to the fact that there are only ﬁnitely many different rule schemes, all the existential and universal choices can be encoded by a suitable combination of a rule scheme, principal formula(e), or sequents of size bounded by 2n, consisting only of subformulae of the conclusion. This yields witnesses of size polynomial in n for each of the nondeter- ministic steps. Moreover, since each recursive call either reduces the complexity or decreases the maximal modal nesting depth, a run of the algorithm makes at most O(n ) recursive calls, after which it either accepts with lines 2 or 4 or rejects with line 31. Thus, the algorithm terminates after at most O(n ) steps. Claim 2 Algorithm 1 accepts an input (F, L), Γ ⇒ Δ if and only if (F, L) Γ ⇒ Δ. G cut MD+ We show the claim by induction on the maximal modal nesting depth of Γ ⇒ Δ. If (F, L) Γ ⇒ Δ, then there is a valid proto-derivation for Γ ⇒ Δ in G cut MD+ G cut. Hence by Cut Elimination (Theorem 2) and Lemma 5 there is a valid proto- MD+ derivation for Γ ⇒ Δ in G3 . By induction hypothesis we know that the algorithm MD+ rejects all the underivability statements occurring in this proto-derivation. Hence the existential and universal choices corresponding to the rules of the proto-derivation together with recursive calls of the algorithm witness that the algorithm accepts the input. Conversely, from an accepting run of the algorithm we obtain ﬁrst a cut-free proto-derivation of Γ ⇒ Δ in G3 by applying the rules corresponding to the MD+ existential choices of the algorithm. Then by induction hypothesis we obtain that none of the underivability statements occurring in this proto-derivation are derivable in G cut from (F, L), and hence the proto-derivation is valid. Now Lemma 5 yields MD+ a valid proto-derivation in G and hence in G cut. MD+ MD+ The two claims together show that Algorithm 1 decides the M¯ımam ¯ s¯a derivability problem in alternating polynomial time, which is equivalent to polynomial space (Chandra et al. 1981). A Prolog implementation of the decision procedure is available under http://subsell. logic.at/bprover/deonticProver/version1.2/. 6 Applications: deciding between diﬀerent interpretations Possible applications of the introduced system are provided by the problem of val- idating the interpretation or formalisation of a normative text, and in particular the related problem of deciding between different interpretations or formalisations. Both of these problems are of course common to many areas also outside of Indian Philoso- phy, including Legal Representation, see, e.g., Bartolini et al. (2018), Libal and Steen (2020). The main setting here is the following. Suppose that we are given a natural language text, e.g., a passage of a M¯ımam ¯ s¯a text or a speciﬁc law or regulation, that we would like to formalize. Because of the ambiguity inherent in natural language as well as, e.g., certain difﬁculties of interpretation speciﬁc to Sanskrit we are almost guar- anteed to obtain not a single formalisation, but a number of different competing ones. Hence we are faced with the task of deciding which of these is the most appropriate. 123 382 B. Lellmann et al. Fig. 17 The procedure for deciding between different interpretations One way of doing so is to consider the consequences of the different interpretations under an assumed system of background reasoning, in our case the logic MD+, and to compare them with respect to certain criteria. We will see a speciﬁc possible criterium used by M¯ımam ¯ s¯a authors below, but in general such criteria would involve a basic sanity check in the form of consistency, or checking whether certain statements, which intuitively should hold, are derivable (compare, e.g., the quality assurance procedures in Libal and Steen 2020). Whenever the principles of the assumed background rea- soning match the guiding principles of our system MD+, the decision procedure given above can be used to check the consequences of the different interpretations, and hence aid in comparing them. The general procedure is illustrated in Fig. 17.Inviewofthe fact that in this article our main application is to M¯ımam ¯ s¯a reasoning, in the ﬁgure the stage containing the different competing ﬁrst formalisations is labelled the sr ´ auta level, but it is worth noting again that the procedure itself in principle can be applied to any collection of normative statements, including regulations and laws, as long as the formal language and the assumed system of background reasoning match the ones considered here. Example 4 Suppose we encounter the following statements: 1. One should refrain from smoking in the presence of a baby. 2. Smoking in a bar incurs a sanction. 3. One may smoke if there is sufﬁcient ventilation. Further suppose that we have already established that the ﬁrst of these should be read as an obligation, e.g., in a moral sense, whereas due to the mention of a sanction the second one should be read as a prohibition in the legal sense. The corresponding formalisation would be given by: 1. O (¬smoke/baby) pf 2. F (smoke/bar) pf 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 383 For the third statement, however, it is not clear whether the “may” constitutes an exception to the moral kind of obligation of the ﬁrst statement or to the legal kind of prohibition of the second. I.e., in the formal language of MD+ we could formalise this O F statement either as P (smoke/vent) or as P (smoke/vent). pf pf A possible way to decide between these two interpretations is to consider the con- sequences together with the formalisations of the ﬁrst two statements. In particular, under the interpretation as P (smoke/vent) and assuming our system of back- pf ground reasoning, we obtain that O(¬smoke/baby ∧bar ∧vent) is not derivable, whereas F (smoke/baby ∧ bar ∧ vent) is, i.e., while it still incurs a legal sanc- tion, from a moral point of view one need not refrain from smoking in the presence of a baby in a well-ventilated bar. In contrast, under the alternative interpretation as P (smoke/vent) we would derive O(¬smoke/baby ∧bar ∧vent), but would pf not derive F (smoke/baby ∧ bar ∧ vent). I.e., in the same situation smoking would not be illegal, but still morally objectionable. Checking these two possible sets of consequences either against our moral intuitions or against information on the legal aspects of smoking would provide us with a method for deciding which of the two possible interpretations of the permission statement would be more plausible. Similarly to the previous example, we can also use the general procedure of Fig. 17 to decide about assumptions on the level of facts as follows. Example 5 Suppose that we have a text stipulating that unjustiﬁed violence incurs certain sanctions, i.e., is explicitly forbidden. Hence F (violence/). pf Further, suppose that we are interested in the status of suicide. Adding the seemingly plausible fact that suicide is unjustiﬁed violence, i.e., suicide → violence to the factual assumptions would, in absence of any other prima-facie deontic statements, yield derivability of the formula F (suicide/). Thus, if we ﬁnd evidence that (attempted) suicide is not forbidden, possibly because it does not incur any sanctions, we should conclude that the authors do not consider suicide to be a form of unjustiﬁed violence, and hence remove the corresponding factual assumption. Note that in a certain sense the factual assumptions hence could also be used in a coarse (and perhaps oversimpliﬁed) representation of constitutive norms, see, e.g., Boella and van der Torre (2004). 6.1 The evaluation criterium of vikalpa Coming back to M¯ımam ¯ s¯a reasoning, it is interesting to note that the outlined proce- dure seems to be the method employed, in an informal manner, by various M¯ımam ¯ s¯a authors to decide between different interpretations of deontic statements found in the Vedas. One particular decision criterium employed in this context is that of minimising instances of the so-called vikalpa principle. This principle was already stated explicitly in the founding text of the M¯ımam ¯ s¯a school, the P¯ urva M¯ ımam ¯ s¯aSutr ¯ a of Jaimini, . . with the following English translation (and reformulation). If a prescription enjoins X and a prohibition forbids one to perform the same act X under the same conditions, and no other interpretation is possible, the act X should be considered optional. 123 384 B. Lellmann et al. The problem with such optional acts lies in the fact that for the M¯ımam ¯ s¯a authors if the act is optional, the deontic statement prescribing or prohibiting it would be superﬂuous in the sense of not being applicable. However, from their point of view the Vedas would not give superﬂuous information. Hence they see vikalpa as a very last resort, and strive to develop an interpretation of the Vedas which makes use of this device as little as possible. In our formal framework the discussion around the vikalpa principle has two aspects. The ﬁrst one is that we should be able to derive it in our system. Abstracting from the particular action X in the quote above, and concentrating on obligations only, this means that, for a set L ={O (a/b), O (c/d)} and facts F ={a, c ⇒} establishing pf pf that a and c are not jointly possible, neither of the formulae O(a/b ∧ d) and O(c/b ∧ d) should be derivable, because it should be considered optional whether a or c is performed. However, while it is optional which of the two is performed, it is still obligatory to perform one of them, hence the formula O(a ∨ c/b ∧ d) should be derivable. Remark 7 It is worth noting that more than two millennia after its formulation by Jai- mini this principle was also formulated in modern deontic logic and in nonmonotonic reasoning: In the former, it is known, e.g., under the name of disjunctive response, where from the two conﬂicting assumptions O(a/c) and O(b/c) we are able to derive at least the obligation of the disjunction O(a ∨ b/c), see Goble (2013); in the area of nonmonotonic reasoning it roughly corresponds to the phenomenon of ﬂoating con- clusions for skeptical semantics, where in our example neither of the contents a and b of the two conﬂicting assumptions would be in all the extensions, but the formula a ∨ b is, see Makinson and Schlechta (1991). Generalising the above example to sets of obligations, and adding that all the enjoined acts should be possible by themselves and that the result should not be blocked by any obligation, prohibition, or permission outside the set, we can derive the vikalpa principle in our system: Theorem 6 Let X ={O ( A / B ),..., O ( A / B )}⊆ L be a set such that 1 1 n n pf pf – (F, L) A ⇒ for every i ≤ n. G cut i MD+ – (F, L) A , C ⇒ for every O (C / D) ∈ L X with G cut i pf MD+ i ≤n (F, L) B ⇒ D. G cut i MD+ i ≤n – (F, L) A ⇒ C for every F (C / D) ∈ L X with (F, L) G cut i pf G cut MD+ i ≤n MD+ B ⇒ D. i ≤n – (F, L) A , C ⇒ for every P (C / D) ∈ L X with G cut i MD+ i ≤n pf (F, L) B ⇒ D. G cut i MD+ i ≤n Then (F, L) ⇒ O( A / B ). G cut i i MD+ i ≤n i ≤n O ( A / B ) 1 1 pf Proof We show that we have all the premises to apply the rule O . From the propositional rules we obtain (F, L) A ⇒ A and G cut 1 i MD+ i ≤n (F, L) B ⇒ B . Moreover, for every j ≤ n we obtain G cut i 1 MD+ i ≤n (F, L) A , A ⇒ , since otherwise in particular we would have G cut j i MD+ i ≤n 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 385 (F, L) A , A ⇒ , and hence (F, L) A ⇒ . Furthermore, by G cut j j G cut j MD+ MD+ assumption, for every O (C / D) ∈ L X we have either (F, L) B ⇒ pf G cut i MD+ i ≤n D or (F, L) C , A ⇒ . The analogous statement holds for every prima- G cut i MD+ i ≤n facie deontic statement of the form F (C / D) or P (C / D). Now applying the rule pf pf O ( A / B ) pf 1 1 O yields ⇒ O( A / B ). i i R i ≤n i ≤n It should be noted that for the statement of the theorem it is not relevant whether the A from the set X are jointly possible or not, only that their disjunction A i i i ≤m is not blocked by any C from outside that set. In particular, it also applies to the case where the A are not jointly possible. Thus, our system as described indeed satisﬁes the disjunctive response resp. vikalpa. The corresponding statement for prohibitions F ( A / B ) O ( A / B ) pf 1 1 pf 1 1 is shown completely analogously, using the rule F instead of O : R R Theorem 7 Let X ={F ( A / B ),..., F ( A / B )}⊆ L be a set such that pf 1 1 pf n n – (F, L) ⇒ A for every i ≤ n. G cut i MD+ – (F, L) C ⇒ A for every O (C / D) ∈ L X with (F, L) G cut i pf G cut MD+ i ≤n MD+ B ⇒ D. i ≤n – (F, L) ⇒ A , C for every F (C / D) ∈ L X with (F, L) G cut i pf G cut MD+ i ≤n MD+ B ⇒ D. i ≤n – (F, L) C ⇒ A for every P (C / D) ∈ L X with (F, L) G cut i G cut MD+ i ≤n MD+ pf B ⇒ D. i ≤n Then (F, L) ⇒ O( A / B ). G cut i i MD+ i ≤n i ≤n The previous theorems show that in our system we can use the fundamental prin- ciple of vikalpa to obtain derived deontic statements from conﬂicting prima-facie statements. But how can we evaluate different interpretations with respect to minimis- ing the number of applications of this principle? At this point it is important to note that what should be minimised is not the number of applications of the vikalpa prin- ciple to different derived formulae, but to prima-facie deontic statements, since only superﬂuousness of the latter is problematic. The general idea then is that a prima-facie deontic statement O (a/b) or F (a/b) is involved in an application of the vikalpa pf pf principle with another prima-facie deontic statement in a context given by (L, F) exactly when the corresponding formula O(a/b) or F (a/b) is not derivable from (L, F). Hence given such a context we can use the decision procedure of Algorithm 1 to identify exactly those prima-facie deontic statements involved in applications of the problematic principle. Apart from providing us with the number of possibly problem- atic prima-facie formulae, this method has the additional beneﬁt that it explicitly gives these problematic formulae, hence yielding clues as to which parts of the interpretation could be changed in order to avoid applications of the vikalpa principle. Note that this amounts to a form of inconsistency checking using the formalisation of a text similar to that used in Libal and Norotná (2020) for ﬁnding and correcting inconsistencies in legal texts This general idea for identifying applications of the vikalpa principle is made formally precise in the following two propositions. Proposition 4 Let O ( A/ B) ∈ L. Then (F, L) ⇒ O( A/ B) holds if and pf G cut MD+ only if at least one of the following holds: 123 386 B. Lellmann et al. – There is a O (C / D) ∈ L with (F, L) A, C ⇒ and (F, L) ⇒ pf G cut G cut MD+ MD+ B ↔ D; or – There is a P (C / D) ∈ L with (F, L) A, C ⇒ and (F, L) ⇒ G cut G cut MD+ MD+ pf B ↔ D; or – There is a F (C / D) ∈ L with (F, L) A ⇒ C and (F, L) ⇒ pf G cut G cut MD+ MD+ B ↔ D. Proof Using cut elimination and a close inspection of the rules, the only way in which the sequent ⇒ O( A/ B) could be derived is via an instance of the assumption rule O ( E / F ) pf O for some O ( E / B) ∈ L. From this the “if” part follows directly, since in pf case one of the three conditions hold, not all of the underivability statements in the not-excepted block hold. For the “only if” direction, suppose that (F, L) ⇒ G cut MD+ O( A/ B) holds. Then in particular, the sequent ⇒ O( A/ B) is not derivable via the O ( A/ B) pf speciﬁc rule O . Since the premisses of this rule in the standard block are B ⇒ B and A ⇒ A which are initial sequents, some of the premisses in the not-excepted block or in the no-active-conﬂict block must not hold. However, since the formula O ( A/ B) is pf in L and overrules any conﬂicting formula from L (since both B ⇒ B and B ⇒ Y are derivable for any conﬂicting obligation O ( X /Y ), prohibition F ( X /Y ), or permission pf P ( X /Y ) for which B ⇒ Y is derivable), all of the premisses in the no-active-conﬂict block do hold. Hence some of the premisses in the not-excepted block hold, which means that there is a formula O ( X /Y ), F ( X /Y ) or P ( X /Y ) from L satisfying pf pf pf the conditions given in the statement of the proposition. The analogous proposition for prohibitions is proved in the same way: Proposition 5 Let F ( A/ B) ∈ L. Then (F, L) ⇒ F ( A/ B) holds if and pf G cut MD+ only if at least one of the following holds: – There is a F (C / D) ∈ L with (F, L) ⇒ A, C and (F, L) ⇒ G cut G cut pf MD+ MD+ B ↔ D; or – There is a P (C / D) ∈ L with (F, L) C ⇒ A and (F, L) ⇒ G cut G cut pf MD+ MD+ B ↔ D; or – There is a O (C / D) ∈ L with (F, L) C ⇒ A and (F, L) ⇒ pf G cut G cut MD+ MD+ B ↔ D. Using these propositions we can now formally evaluate an interpretation given by a list L of prima-facie deontic statements and a set F of propositional facts with respect to the criterion of minimising the number of instances of vikalpa among the prima- facie deontic statements as follows: For every prima-facie statement O ( A/ B) ∈ pf L and for every F (C / D) ∈ L check whether (F, L) ⇒ O( A/ B) and pf G cut MD+ (F, L) ⇒ F (C / D) respectively, and return all those formulae for which G cut MD+ this holds. 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 387 The implementation available under http://subsell.logic.at/bprover/deonticProver/ version1.2/ includes this check together with the possibility of automatically generat- ing alternative formalisations by systematically reinterpreting the deontic operators, e.g., by rewriting prohibitions as negative obligations. The following M¯ımam ¯ s¯a example arises from the consideration that recommen- dations and obligations are expressed by the same Sanskrit word: vidhi. Therefore in absence of further discriminating elements the classiﬁcation of a command could be based on the principle of avoiding vikalpa as much as possible, as shown in the following example. Example 6 Consider the following simpliﬁed interpretation of part of the debate about the Sat¯ı sacriﬁce , discussed in depth in Brick (2010): (i) “When a woman’s husband has died, she should perform the Sat¯ı sacriﬁce by ascending the funeral pyre after him.” O (satı/widow). pf (ii) “Every rite which is violence itself is forbidden, therefore the Sat¯ı sacriﬁce for widows is forbidden” F (satı/widow). pf None of these injunctions is derivable in the logic, therefore, under this interpretation, the Sat¯ı sacriﬁce should be considered optional. However some M¯ımam ¯ s¯a authors propose to interpret the injunction (i) as the recommendation R (satı/widow), pf conditioned by a general woman’s desire of positive karma for her husband and herself. This explanation should be preferred as not giving rise to cases of vikalpa: the sacriﬁce remains forbidden, but a woman can chose to perform it for obtaining a desired result; i.e., both F (satı/widow) and R(satı/widow) are derivable in the logic. Another example of how the mechanism can be used for choosing between con- ﬂicting interpretations is given by the discussion about permissions: even if they do not appear as operators in MD+, reading a prima-facie permission as an exception F O to a prohibition (P ) or to an obligation (P ) affects the derivability of the other pf pf prescriptions. Example 7 Consider the following situation which abstracts part of the discussion in Kumarila’ ¯ s Tantravarttika ¯ on 1.3.3-4: (i) “During a particular sacriﬁce it is forbidden to eat” F (eat/sacr). pf (ii) “In the second part of this sacriﬁce it is also obligatory (rewarded with good karma) not to eat” O (¬eat/sacr_IIpart). pf (ii) “In the second part of this sacriﬁce it is also permitted to eat”. If the permission is considered as an exception to the obligation and formalized as P (eat/sacr_IIpart), then it blocks the derivation of O(¬eat/sacr_IIpart) pf in the logic. Hence, to ensure that the maximum number of srauta ´ injunctions are derivable in the logic (i.e. the instances of vikalpa are minimized) the permission should be interpreted as an exception to the ﬁrst prohibition—since, intuitively, sacr_IIpart → sacr—and formalized as P (eat/sacr_IIpart). Under pf this interpretation, eating is forbidden only in the other parts of the sacriﬁce (F (eat/sacr) is derivable and F (eat/sacr_IIpart) is not), but it remains pf pf obligatory not to eat in the second part (O(¬eat/sacr_IIpart) is derivable). Sat¯ı is an old custom where a widow immolates herself on her husband’s funeral pyre. 123 388 B. Lellmann et al. 7 Conclusion Focusing on the speciﬁcity principle, we have explored connections between the M¯ımam ¯ s¯a school of Indian philosophy and symbolic deontic logic. We have ﬁrst extended the basic logic of M¯ımam ¯ s¯a in Ciabattoni et al. (2015) with new operators for prohibitions and recommendations whose properties have been extracted from the M¯ımam ¯ s¯a texts. The paper’s main result is a sequent-based system to reason in this logic using speciﬁcity/gunapradhana; ¯ some of its properties have been investigated and its potential use as a tool for M¯ımam ¯ s¯a philosophy as well as to aid formalisation tasks, e.g., in Legal Representation, have been explored. Future research directions include the extension of the system to deal with further M¯ımam ¯ s¯a rules (nyaya ¯ s) to avoid contradictions; we are planning to work on those just discovered in Kumarila’ ¯ s Tantravarttika ¯ on 3.3.14 (balabala-adhikaran ¯ a) which comprise a prioritisation of rules based on an existing hierarchy of sources (e.g. a Vedic prescription defeats a contradictory prescription in the ‘traditional texts based on the Vedas’), and on the criteria of invalidating as few injunctions as possible. From the technical side, we plan to investigate the system’s semantics, to abstract and generalize it to work for base deontic logics other than MD+ and to further explore its use in other ﬁelds, in particular Legal Representation and Reasoning, in detail. Apart from the technical content, this paper illustrates some of the vast potential for cross-fertilisation between M¯ımam ¯ s¯a and deontic logic. This enterprise is the subject of ongoing work in collaboration with Sanskritists and experts of Indian philosophy. Funding Open access funding provided by TU Wien (TUW). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. A Appendix Theorem 8 (Cut elimination) If (F, L) Γ ⇒ Δ, then (F, L) Γ ⇒ Δ. G cut G MD+ MD+ Proof We show how to eliminate topmost applications of the multicut rule n m Γ ⇒ Δ, A A ,Σ ⇒ Π mcut Γ, Σ ⇒ Δ, Π from a proto-derivation, preserving validity (here A is the multiset containing n copies of A). Since cut is a case of mcut and mcut is derivable using Con , Con and cut, L R this sufﬁces. The proof is by double induction on the complexity of the cut formula A and the sum of the depths of the derivations of the two premises of the application 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 389 of mcut (see Troelstra and Schwichtenberg (2000), Sect. 4.1.9 for the classical case without underivability statements). If the complexity of the cut formula is 0, then it is a propositional variable, and hence not principal in a modal or propositional rule or a rule from ga . Thus, as usual, we permute mcut into the premises of the last applied rules using the inner induction on the depths of the derivations, until it is absorbed by an application of weakening, or reaches the leaves of the proto-derivation. In this case the premises of the multicut are initial sequents or elements of F. If at least one of these is an initial sequent, the multicut is eliminated as usual, if both sequents are elements of F we use that F is closed under contraction and cuts and replace the multicut with the corresponding element of F. So assume that the complexity of the cut formula is n + 1. Again, using the inner induction on the depth of the proto-derivation we permute the multicut into the premise(s) of the last applied rules, until it is in an initial sequent or it is princi- pal in the last rules of the derivations of both premises of the multicut. In case the cut formula is propositional we use the standard transformation, see Troelstra and Schwichtenberg (2000). The only interesting case is where the cut formula is a deontic formula and neither of the two premisses of the multicut is an initial sequent. If the last applied rules both are among P , P , P , D , D , D , Mon , Mon , Mon , then the transformation is O F R O F OF O F R essentially as for the system G , see Lellmann and Pattinson (2013) for the general MD transformations. E.g., if the last applied rules were Mon and D , the multicut has O O the following form: A, E ⇒ B ⇒ FF ⇒ B C ⇒ AD ⇒ BB ⇒ D Mon O O O(C / D) ⇒ O( A/ B) O( A/ B), O( E / F ) ⇒ mcut O(C / D), O( E / F ) ⇒ Using the induction hypothesis on the complexity of the cut formula we obtain valid proto-derivations of the conclusions of C ⇒ AA, E ⇒ D ⇒ BB ⇒ F F ⇒ BB ⇒ D mcut mcut mcut C , E ⇒ D ⇒ F F ⇒ D Now an application of the rule D yields the sequent Γ, O(C / D), Σ , O( E / F ) ⇒ Δ, Π. In case both principal formulae of the application of D are cut formulae, we proceed similarly, only using the rule P in the last step. The other cases of the modal rules are similar. In the most interesting cases at least one of the premises of the cut was derived using a rule from ga . For each operator op ∈{O, F , R} there are three major groups of op (C / D) op (C / D) cases: (i) op or op versus a rule not from ga where the multicut R L op (C / D) op (C / D) has non-empty conclusion; (ii) op or op versus a rule not from ga R L op (C / D) op (G/ H ) where the multicut has an empty conclusion; or (iii) op versus op . R L We consider all the different cases for op = O. The cases for the operators F and R are analogous, and much simpler in the case of R. 123 390 B. Lellmann et al. Case (i) The prime example of this case is the case where the two last applied rules O (C / D) pf were O and Mon . Then the two derivations end in an instance of a rule from { B ⇒ D} ∪ {C ⇒ A} ⎧ ⎡ ⎤ ⎫ ⎪ ⎪ ⎪ (F, L) B ⇒ Y ⎪ G cut ⎪ MD+ ⎪ ⎨ ⎢ ⎥ ⎬ ⎢ ⎥ ∪ ∪ (F, L) Y ⇒ D | O ( X /Y ) ∈ L or P ( X /Y ) ∈ L ⎢ ⎥ G cut pf MD+ pf ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ X , A ⇒ ∪ (F, L) G cut MD+ ⎧ ⎡ ⎤ ⎫ ⎪ ⎪ ⎪ (F, L) B ⇒ Y ⎪ G cut ⎪ MD+ ⎪ ⎨ ⎢ ⎥ ⎬ ⎢ ⎥ ∪ ∪ (F, L) Y ⇒ D | F ( X /Y ) ∈ L ⎢ ⎥ G cut pf MD+ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ∪ (F, L) A ⇒ X G cut MD+ ⎧ ⎡ ⎤ ⎫ ⎪ ⎪ (F, L) B ⇒ Y , ⎪ ⎪ G cut ⎪ MD+ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ∪ (F, L) X , A ⇒ ⎪ ⎢ ⎥ ⎪ G cut ⎪ ⎪ MD+ ⎨ ⎢ ⎥ ⎬ ⎢ ⎥ O ( E / F ) ∈ L { B ⇒ F } ∪ { F ⇒ Y } ∪ ⎢ pf ⎥ | O ( X /Y ) ∈ L pf ∪ | ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ∪ { E ⇒ A} or P ( E / F ) ∈ L ⎪ ⎢ ⎥ ⎪ ⎪ pf ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ { B ⇒ F } ∪ { F ⇒ Y } ⎪ ⎪ ⎪ ⎪ ∪ | F ( E / F ) ∈ L ⎩ ⎭ pf ∪ {⇒ E , A} ⎧ ⎡ ⎤ ⎫ ⎪ ⎪ (F, L) B ⇒ Y , ⎪ ⎪ G cut ⎪ MD+ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ∪ (F, L) A ⇒ X ⎪ ⎢ G cut ⎥ ⎪ ⎪ MD+ ⎪ ⎨ ⎢ ⎥ ⎬ ⎢ ⎥ O ( E / F ) ∈ L ∪ { B ⇒ F } ∪ { F ⇒ Y } pf | F ( X /Y ) ∈ L ⎢ ⎥ pf ∪ | ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ or P ( E / F ) ∈ L ⎪ ⎢ ∪ { E ⇒ A} ⎥ ⎪ ⎪ pf ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ { B ⇒ F } ∪ { F ⇒ Y } ⎪ ⎪ ⎪ ⎩ ∪ | F ( E / F ) ∈ L ⎭ pf ∪ {⇒ A, E } ⎧ ⎫ ⎡ ⎤ {{(F, L) B ⇒ Y }} G cut ⎪ ⎪ MD+ ⎪ ⎪ ⎨ ⎬ ⎢ ⎥ {{(F, L) X , A ⇒}} G cut ⎢ MD+ ⎥ ∪ | P ( X /Y ) ∈ L ⎣ ⎦ pf ⎪ { B ⇒ F}∪{ F ⇒ Y } ⎪ ⎪ ⎪ ⎩ | O ( E / F ) ∈ L ⎭ pf ∪{ E ⇒ A} O (C / D) pf ⇒ O( A/ B) R (1) and A ⇒ GB ⇒ HH ⇒ B Mon O( A/ B) ⇒ O(G/ H ) respectively. By induction hypothesis on the complexity of the cut formula we obtain valid proto-derivations of H ⇒ D and C ⇒ G, as well as the sequents H ⇒ F and F ⇒ Y and E ⇒ G whenever the corresponding sequents occur in the O (C / D) pf application of O . Further, for every underivability statement (F, L) G cut MD+ B ⇒ Y together with derivability of B ⇒ H we obtain the underivability statement (F, L) H ⇒ Y by contradiction: assuming there is a valid proto-derivation G cut MD+ of H ⇒ Y in G ga cut from F we could apply cut to this and B ⇒ H to obtain MD F B ⇒ Y , in contradiction to (F, L) B ⇒ Y . Similarly, for G ga cut G cut MD MD+ every underivability statement (F, L) X , A ⇒ using derivability of A ⇒ G G cut MD+ we obtain the underivability statement (F, L) X , G ⇒ ; analogously for the G cut MD+ underivability statements (F, L) A ⇒ X using derivability of A ⇒ G we G cut MD+ 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 391 O (C / D) pf get (F, L) G ⇒ X. Hence we can apply the rule O to obtain a proto- G cut MD+ R derivation of ⇒ O(G/ H ). By the reasoning above, all the underivability statements hold, hence the proto-derivation is valid. O (C / D) pf The cases where the two last applied rules were O and D with only one O (C / D) pf of the principal formulae a cut formula or Mon and O are similar, in each O (C / D) pf case ﬁnishing with an application of O . O (C / D) pf Similarly, in the case where the last applied rules were O and D OF O(C / D) we reason as above, but ﬁnishing with an application of F . F (C / D) pf The case where the last applied rules were Mon and O is also similar, F (C / D) pf ﬁnishing with an application of O . O (C / D) pf Case (ii) The prime example for this case is when the last rules were O and P . We claim that this case actually cannot occur. For otherwise the derivations end in an instance of (1) and A ⇒ O( A/ B) ⇒ . However, then for X := C and Y := D we have valid proto-derivations for all three of B ⇒ Y and Y ⇒ D and X , A ⇒ . The ﬁrst one is the ﬁrst premise of the application O (C / D) pf of O , the second one is easily derivable since Y = D, and the last one follows from the premise of P using W . But then the proto-derivation of ⇒ O( A/ B) cannot O L have been valid since for some of the underivability statements in the not-excepted O (C / D) pf block of thepremissesofthe rule O there is a valid proto-derivation. O (C / D) pf The case where the last rules were O and D with both principal formulae of the latter cut formulae is analogous to the previous case. Case (iii) Assume that both last applied rules are from ga . The prime example of O (C / D) O (G/ H ) pf pf this is where the last rules were O and O . Again, we claim that this R L cannot happen. For suppose it did, then the derivations would end in (1) and 123 392 B. Lellmann et al. { B ⇒ H } ∪ {G, A ⇒ } ⎧ ⎡ ⎤ ⎫ ⎨ (F, L) B ⇒ Y ⎬ G cut MD+ ⎣ ⎦ ∪ ∪ (F, L) Y ⇒ H | O ( X /Y ) ∈ L or P ( X /Y ) ∈ L G cut pf MD+ pf ⎩ ⎭ ∪ (F, L) X ⇒ A G cut MD+ ⎡ ⎤ ⎨ (F, L) B ⇒ Y ⎬ G cut MD+ ⎣ ⎦ ∪ ∪ (F, L) Y ⇒ H | F ( X /Y ) ∈ L G cut pf MD+ ⎩ ⎭ ∪ (F, L) ⇒ A, X G cut MD+ ⎧ ⎡ ⎤ ⎫ (F, L) B ⇒ Y , G cut ⎪ MD+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ∪ (F, L) X ⇒ A ⎪ G cut ⎪ MD+ ⎪ ⎢ ⎥ ⎪ ⎨ ⎬ ⎢ ⎥ { } { } O ( E / F ) ∈ L B ⇒ F ∪ F ⇒ Y pf ⎢ ⎥ ∪ ∪ | | O ( X /Y ) ∈ L O pf ⎢ ⎥ or P ( E / F ) ∈ L ⎪ ∪ { A, E ⇒ } ⎪ ⎪ ⎢ pf ⎥ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ { B ⇒ F } ∪ { F ⇒ Y } ⎪ ⎪ ⎪ ⎩ ⎭ ∪ | F ( E / F ) ∈ L pf ∪ { A ⇒ E } ⎧ ⎡ ⎤ ⎫ (F, L) B ⇒ Y , ⎪ G cut ⎪ MD+ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ∪ (F, L) ⇒ A, X ⎥ ⎪ ⎪ G cut ⎪ MD+ ⎪ ⎢ ⎥ ⎪ ⎨ ⎬ ⎢ ⎥ { B ⇒ F } ∪ { F ⇒ Y } O ( E / F ) ∈ L pf ⎢ ⎥ ∪ ∪ | | F ( X /Y ) ∈ L ⎢ ⎥ ⎪ ∪ { A, E ⇒ } or P ( E / F ) ∈ L ⎪ ⎪ ⎢ pf ⎥ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ { B ⇒ F } ∪ { F ⇒ Y } ⎪ ⎪ ⎩ ⎭ ∪ | F ( E / F ) ∈ L pf ∪ { A ⇒ E } ⎧ ⎡ ⎤ ⎫ {{(F, L) B ⇒ Y }} ⎪ G cut ⎪ MD+ ⎪ ⎪ ⎨ ⎬ ⎢ ⎥ {{(F, L) X ⇒ A}} G cut MD+ O ⎢ ⎥ ∪ | P ( X /Y ) ∈ L ⎣ ⎦ pf ⎪ { B ⇒ F}∪{ F ⇒ Y } ⎪ ⎪ ⎪ ⎩ | O ( E / F ) ∈ L ⎭ pf ∪{ E , A ⇒} O (G/ H ) pf O( A/ B) ⇒ But then in particular the no-active conﬂict block of the application of the rule O (C / D) pf O has either (a) one of the premises (F, L) B ⇒ H and G cut R MD+ (F, L) G, A ⇒ ; or (b) all of the three premises G cut MD+ B ⇒ FF ⇒ HE ⇒ A for some O ( E / F ) or P ( E / F ) from L; or (c) all of the three premises pf pf B ⇒ FF ⇒ H ⇒ A, E for some F ( E / F ) from L. However, the ﬁrst case of (a) gives a contradiction with the pf O (G/ H ) pf premise B ⇒ H of the application of O using validity of the proto-derivation. O (G/ H ) pf The second case gives a contradiction with the premise G, A ⇒ of O ,again using validity of the proto-derivation. Case (b) gives a contradiction because the not- O (G/ H ) pf excepted block of the application of O contains one of the premises (F, L) B ⇒ F (F, L) F ⇒ H (F, L) E ⇒ A G cut G cut G cut MD+ MD+ MD+ and the proto-derivation is valid. Case (c) is similar, but with (F, L) ⇒ A, E G cut MD+ instead of the last underivability statement. Hence this case also cannot occur. O (C / D) F (G/ H ) pf pf The case of O versus O is analogous. R L 123 M¯ımam ¯ s¯a deontic reasoning using speciﬁcity: a proof… 393 References Abraham M, Gabbay DM, Schild U (2011) Obligations and prohibitions in Talmudic deontic logic. 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Artificial Intelligence and Law – Springer Journals
Published: Nov 9, 2020
Keywords: Deontic logic; Non-monotonic inference; Specificity; Mīmāṃsā; Sequent systems; Legal representation
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