Reflexive intermediate logics

I ask the general question, "Which logics can prove their own completeness?" Specifically, we will call a logic reflexive if a second-order metatheory of arithmetic created from the logic is sufficient to prove the completeness of the original logic.; I define a lattice of intermediate propositional logics with the finite model property, and prove that the reflexive logics within that lattice are exactly those that are in the principal filter generated by testability logic, i.e. intuitionistic logic plus the scheme ¬4∨¬¬4 . I show that this result holds regardless of whether Tarskian or Kripke semantics is used in the definition of completeness. I also show that the operation of creating a second-order metatheory is injective, thereby insuring that we are actually considering each logic independently.; I define a lattice of intermediate first-order logics, each generated by a finite list of axiom schema, and prove that the reflexive logics within that lattice are exactly those whose metatheories prove the principle of testability and the principle of Double Negation Shift (DNS). This result holds regardless of whether Tarskian or Kripke semantics is used in the definition of completeness. I then demonstrate the relationship between DNS and two interesting concepts related to first-order logics, coconsistency and Godel translations.; Several open questions are mentioned at the conclusions of chapters, and throughout Chapter 7.