Nelson's logic ℒ

Besides the better-known Nelson logic ($\mathcal{N}3$) and paraconsistent Nelson logic ($\mathcal{N}4$), in 1959 David Nelson introduced, with motivations of realizability and constructibility, a logic called $\mathcal{S}$. The logic $\mathcal{S}$ was originally presented by means of a calculus (crucially lacking the contraction rule) with infinitely many rule schemata and no semantics (other than the intended interpretation into Arithmetic). We look here at the propositional fragment of $\mathcal{S}$, showing that it is algebraizable (in fact, implicative), in the sense of Blok and Pigozzi, with respect to a variety of three-potent involutive residuated lattices. We thus introduce the first known algebraic semantics for $\mathcal{S}$ as well as a finite Hilbert-style calculus equivalent to Nelson’s presentation; this also allows us to clarify the relation between $\mathcal{S}$ and the other two Nelson logics $\mathcal{N}3$ and $\mathcal{N}4$.