Axiomatic Extensions of Hohle's Monoidal Logic

We introduce an axiomatic extension of H¨ Monoidal Logic called Semi‐divisible Monoidal Logic, and prove that it is complete by showing that semi‐divisibility is preserved in MacNeille completion. Moreover, we introduce Strong semi‐ divisible Monoidal Logic and conjecture that a predicate formula is derivable in Strong Semi‐divisible Monadic logic if, and only if its double negation ¬¬ is derivable in Eukasiewicz logic.