Intuitionistic Rules : Admissible Rules of Intermediate Logics

Abstract

In this thesis, we investigate the admissible rules of intermediate logics. On the one hand, one can characterize the admissibility of rules in certain logic, and on the other hand, one can characterize logics through their admissible rules. We take both approaches, and reach new results in both directions.

The first approach can be subdivided into several, more specific questions. First, we investigate the semantics of admissible rules. We show that exact models provide sound and complete semantics for the admissible rules of intermediate logics with the finite model property. Moreover, we give a correspondence between constraints imposed upon order-defined models and the validity of certain rules in said models, including the disjunction property, a weakening thereof, and several variants of the Visser rules. In closing, we prove that finite models can not provide sound and complete semantics for logics of width greater than two that admit one particular variant of the Visser rules. This result encompasses IPC and the logics of bounded branching.

Second, we investigate the decidability of the admissible rules of IPC. The novelty here does not lie in the answer, which has been known since the eighties, but in the presentation of the proof. We proceed semantically, introducing a generalization of exact models. Moreover, we effectively characterize projective formulae in the logics of bounded branching and IPC as being those formulae that are closed under the so-called de Jongh rules.

Third, we provide a basis for the admissible rules of the logics of bounded branching and the logics of height at most two. In the former logics, the proof proceeds via the above-mentioned characterization of projective formulae. In the latter logics, the proof proceeds via the observation that the totality of formulae on a finite number of variables is finite. Both of the proofs are effective in nature, and both spring from the observation that projectivity can be expressed by means of the closure under certain rules.

The second approach is studied in two forms. First, we investigate the unification type of the intermediate logics mentioned above. We show how the information about their admissible rules allows one to prove their unification type to be finitary. We discuss the notion of an admissible approximation, which can roughly be interpreted as the left-adjoint to the inclusion of derivability into admissibility. The logics at hand all enjoy such admissible approximations that are disjunctions of projective formulae.

Second, we characterize IPC and each of the logics of bounded branching as being the greatest intermediate logic that admits a particular version of the Visser rules. Analogously, Medvedev’s logic is described as the greatest intermediate logic above Kreisel-Putnam logic that enjoys the disjunction property. The key observation lies in translating the existence of a counter model into a syntactic statement. In this translation, we make essential use of our previously obtained knowledge on the admissible rules of the logics at hand. Moreover, our method allows us to construct refutation systems for all logics mentioned in this paragraph.