“Subintuitionistic Logics,” Notre Dame Journal of Formal Logic 35 (1994) 116–129.
Once the Kripke semantics for normal modal logics were introduced, a whole family of modal logics other than the Lewis systems S1 to S5 were discovered. These logics were obtained by changing the semantics in natural ways. The same can be said of the Kripke-style semantics for relevant logics: a whole range of logics other than the standard systems R, E and T were unearthed once a semantics was given. In a similar way, weakening the structural rules of the Gentzen formulation of classical logic gives rise to other “substructural” logics such as linear logic. This process of “strategic weakening” is becoming popular today, with the discovery of applications of these logics to areas such as linguistics and the theory of computation. This paper examines what the process of weakening does to the Kripke-style semantics of intuitionistic logic, introducing the family of subintuitionistic logics.