A Very Brief Introduction to the Logic of the Mīmāṃsā.
Mīmāṃsā is an orthodox (roughly, pro-Vedic) school of Hindu philosophy. Recently, several philosophers (see e.g. Ciabattoni, et. al. 2013, Srinivasan and Parthasarathi 2012 & 2017, and Srinivasan 2014) have advanced formal deontic logics on the basis of the theory of inference derived from Mīmāṃsā philosophy in the work of Jamini (principally, the Pūrva Mīmāṃsā Sutras) and his commentators.
Ciabattoni et. al. define “basic Mīmāṃsā Deontic Logic” as an extension of S4 with the following axioms:
(◻(p→q) ∧ Opr)→Oqr
◻(q→~p)→~(Opr ∧ Oqr)
(◻(q↔r) ∧ Opq)→Opr
Where the operator “Oyz” is a dyadic deontic modal meaning “y is obligatory given z” or expressed as a conditional imperative: “If z, y!”
These three axioms are derived from three principles found in Mīmāṃsā texts:
If the accomplishment of X presupposes the accomplishment of Y, the obligation to perform X prescribes also Y.
Given that purposes Y and Z exclude each other, if one should use item X for the purpose Y, then it cannot be the case that one should use it at the same time for the purpose Z. (a.k.a. the principle of the half-hen)
If conditions X and Y are always equivalent, given the duty to perform Z under the condition X, the same duty applies under Y.
Now, these principles are derived and abstracted by the authors from the texts to express additional axioms for a deontic, mixed modal logic extended from standard S4. The authors also point out that they use classical rather than intuitionistic S4 because the Mīmāṃsā rely on several principles that imply the legitimacy of reductio ad absurdum proofs. Further, S4 is chosen over S5 because while the Mīmāṃsā is not precise with their modal terminology, it is strongly implied that “necessity” refers to epistemic certainty. Because S4 is the weakest canonical epistemic modal system, that’s what the authors choose.
The analysis of Mīmāṃsā deontic logic also has a trivalent satisfaction semantics. (cf. Vranas 2008, 2011, & 2016) Since the conditional imperative is basic, the satisfaction values an imperative can take are satisfied, violated, and avoided. Anyone who’s taken an intro logic course and gotten mad about the truth table for the material conditional (myself included) will find some comfort in this. Conditional imperatives are satisfied when their antecedents are true and the imperative is performed, violated when their antecedents are true and the imperative refused, and avoided otherwise. Unconditional imperatives are formed by making the antecedent tautological: “Ox⊤”.
Srinivasan and Parthasarathi 2012 have a more elaborate system of conditional and unconditional imperatives based on Mīmāṃsā texts, and they introduce several logical operators to make sense of ordered imperatives, grounded imperatives, and goal-directed imperatives. Examples of these are, respectively, “Put on your socks and shoes,” “If it’s snowing put on a jacket,” and “Take my card to pay for dinner.” Each of these imperatives enjoins a kind of structure between the two parts—either a pair of imperatives or an imperative and proposition—and connects the imperative(s) via a kind of “tense” to some earlier or later state of affairs.
All this is to say that I think the Mīmāṃsā logic is an interesting deontic system worthy of investigation. I hope to do more investigation of my own in the near future.
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