Can contradictions be meaningful? How can one assert 'P soku not-P' or 'P and yet not-P' without sacrificing intelligibility? Expanding on previous attempts, mainly by Dilworth and Heisig, to demystify the soku connective, a formal system is presented here for the logic of soku. Through a formal distinction between internal and external negation, grammatical features of the soku connective are shown to be logically irrelevant, and the principle of non-contradiction is preserved. Disparities with traditional logic are noted, with a focus on negation rather than 'soku'. The formal examination of the logic of soku is intended to present the logic in a way acceptable to more analytically minded philosophers and thereby enhance East-West and Japanese-Anglo-American interaction and criticism.