Abstract

In this paper we intend to study implications in their most general form, generalizing different classes of implications including the HeytingHeyting, A. implication, sub-structural implications and weak strict implicationsStrict implication. Following the topological interpretation of the intuitionistic logicIntuitionistic logic, we will introduce non-commutative spacetimes to provide a more dynamicDynamics and subjective interpretation of an intuitionistic proposition. These combinations of space and time are natural sources for well-behaved implications and we will show that their spatio-temporal implications represent any other reasonable abstract implication. Then to provide a faithful well-behaved syntax for abstract implications, we will develop a logical system for the non-commutative spacetimes for which we will present both topological and KripkeKripke, S. semantics. These logics unify sub-structural and sub-intuitionist logics by embracing them as their special fragments.