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 |$\mathcal{N}4$|⁠.

中文翻译：

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尼尔森的逻辑𝒮

除了更著名的Nelson逻辑（⁠| $ \ mathcal {N} 3 $ |⁠）和超一致的Nelson逻辑（⁠| $ \ mathcal {N} 4 $ |⁠）之外，1959年David Nelson引入了可实现性的动机。和可构造性，称为| $ \ mathcal {S} $ |⁠的逻辑。逻辑| $ \ mathcal {S} $ | 最初是通过演算（极其缺乏收缩规则）来表示的，该演算具有无数的规则图式且没有语义（除了对算术的预期解释之外）。我们在这里查看| $ \ mathcal {S} $ |⁠的命题片段，表明它在Blok和Pigozzi的意义上相对于各种三势对合残差格是可代数的（实际上是含蕴的）。因此，我们介绍了| $ \ mathcal {S} $ |的第一个已知的代数语义。以及等效于Nelson的希尔伯特式有限演算；这也使我们可以弄清| $ \ mathcal {S} $ |之间的关系。和另外两个纳尔逊逻辑| $ \ mathcal {N} 3 $ | 和| $ \ mathcal {N} 4 $ |⁠。