Simulation logic

Simulation relations have been discovered in many areas: Computer Science, philosophical and modal logic, and set theory. However, the simulation condition is strictly a first-order logic statement. We extend modal logic with modalities and axioms, the latter's modeling conditions are the simulation conditions. The modalities are normal, i.e., commute with either conjunctions or disjunctions and preserve either Truth or Falsity (respectively). The simulations are considered arrows in a category where the objects are descriptive, general frames. One can augment the simulation modalities by axioms for requiring the underlying modeling simulations to be bisimulations or to be p-morphisms. The modal systems presented are multi-sorted and both sound and complete with respect to their algebraic and Kripke semantics.