The Modal Catuṣkoṭi: Formalizing Emptiness in Classical Modal Logic

Abstract

Nāgārjuna’s arguments in the Mūlamadhyamakakārikā rely extensively on modus tollens, yet this inference is invalid in the paraconsistent frameworks typically used to formalize the catuṣkoṭi. We develop a modal extension of Belnap-Dunn semantics to address this, expressed as: ¬□(v(P) = t) ∧ ¬□(v(P) = f) ∧ ¬□(v(P) = b) ∧ ¬□(v(P) = n). This formula applies self-reflexively, resolves longstanding interpretive debates regarding verses within Nāgārjuna’s Mūlamadhyamakakārikā and Vigrahavyāvartanī such as his “no thesis” defense, and bridges interpretive gaps between traditionally designated Eastern and Western logical traditions, providing a Western analytical translation of Indian dialectic into symbolic modal logic that is system-agnostic across K, T, S4, S5, non-normal logics, and neighborhood semantics. The modal operators function classically while the four-valued structure is modalized, preserving modus tollens at the level of modal reasoning while accommodating paraconsistent valuation at the object level. Self-application produces a non-terminating regress that is virtuous rather than vicious: each application is self-contained, demonstrating the unavailability of stable ground rather than constituting a failure to find it.