Basic Relevant Theories for Combinators at Levels I and II

Abstract

The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of all B+T theories is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader.