Combinator logics (Q1826366)

Combinatory logic deals with combinators and reduction, illative combinatory logic is an extension of this theory within which intuitionistic predicate logic can be developed in a natural way. Combinator logic, developed here, is quite different to either. Substructural logics can generally be represented using sets of axioms each of which is the type of a combinator. Such axioms are considered here as possible additions to a positive relevance logic \textbf{B} with fusion. Within this logic it is shown that each combinator axiom (i.e. the type of each combinator) can be represented, using fusion, in a way that is directly related to the reduction property of the combinator. The author also uses a Routley-Meyer semantics for \textbf{B} to characterise each combinator axiom. Each semantic characterisation, when expressed in a certain way, also has the combinator's reduction pattern! Next, the author extends \textbf{B} with propositional constants named after the combinators, which, in the fusion notation, mimic their reduction rules. Again the reduction rules are found in the semantics when this is extended to cover the combinator propositional constants.