Schönfinkel-type Operators for Classical Logic

Abstract

We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator (Nallor) that is the dual of Schonfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic (Fol). Having scrutinized the proof, we pinpoint the theorems of Fol that are needed in the proof. Using the insights gained from the proof, we show that there are four binary operators that each can serve as the only undefined logical constant for Fol. Finally, we show that from every n-ary connective that yields a functionally complete singleton set of connectives two Schonfinkel-type operators are definable, and all the latter are so definable.