Mīmāṃsā Deontic Logic: Proof Theory and Applications

Abstract

Abstract. Starting with the deontic principles in Mīmām . sā texts we introduce a new deontic logic. We use general proof-theoretic methods to obtain a cut-free sequent calculus for this logic, resulting in decidability, complexity results and neighbourhood semantics. The latter is used to analyse a well known example of conflicting obligations from the Vedas.

Duty and Sacrifice: A Logical Analysis of the Mīmāṃsā Theory of Vedic Injunctions

“…These axioms represent the 'deontic reversal' mentioned in Section 2.1.6: on the one hand, fixed sacrifices represent duties in a strong sense (or obligations, O), while their subsidiaries have to be performed only as much as possible (amp); on the other hand, elective sacrifices represent duties in a weak sense (or recommendations, R), while their subsidiaries, once the main action is undertaken, have to be carried out precisely as prescribed (eap). Furthermore, we need to add axioms for our operators O and R. In the case of O, a possible solution consists in extracting the intended axioms and rules from a preliminary system to represent deontic reasoning in M¯ımāṁ sa, developed in Ciabattoni et al 2015; such system, called bMDL (basic M¯ımāṁ sa Deontic Logic), is based on a propositional language with both alethic and deontic modalities. The result of this 'extraction procedure' is the following list of principles (see the Appendix for the detailed proof and for an explanation of the sense in which bMDL can be said to be a system inspired by the M¯ımāṁ sa school):…”

“…Dyadic deontic operators are exploited in Ciabattoni et al 2015 to formalize an argument discussed by Mīmāṁ sā authors, which can be called theśyena dilemma. Such dilemma pertains to the performance of theśyena sacrifice, a malefic sacrifice meant for harming one's enemy.…”

“…Mīmāṁ sā authors all agree that it should not be performed because it violates the prohibition to harm living beings. Ciabattoni et al 2015 argue that the seeming conflict between the prohibition to harm any living being and the injunction to perform theśyena under the desire to harm an enemy can be formally solved through a dyadic operator. Indeed, it is possible to build a formal system, like bMDL, in which the dilemma is represented by a conjunction of formulas O ¬φ ∧ O ψ φ, where φ can be read 'you harm some living being' (which is a necessary consequence of the performance of theśyena), ψ 'you desire to harm your enemy' and represents any tautology.…”

“…In order to axiomatize the properties of the dyadic operator O, we choose to rely again on the preliminary system bMDL developed in Ciabattoni et al 2015. In the Appendix to this article it is shown that we can extract from bMDL the dyadic versions of the principles (7), (8) and (9), 27 together with the following rule:…”

“…We conclude this section with a remark on useful variations of S 0 d. We argued that one of the reasons to move from monadic deontic operators to dyadic ones is the possibility of treating deontic dilemmas. For instance, as illustrated in the analysis of theśyena dilemma in Ciabattoni et al 2015, one can block problematic inferences from a generic triggering condition to a more specific one; however, if the only principles relating triggering conditions are those formalized by 15and 16, there is some risk that too many inferences are blocked. For instance, it seems plausible to assume that sometimes the addition of a triggering condition preserves the injunction to perform an action.…”