An Almost Classical Logic for Logic Programming and Nonmonotonic Reasoning

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The model theory of a first-order logic called N⁴ is introduced. N⁴ does not eliminate double negations, as classical logic does, but instead reduces fourfold negations. N⁴ is very close to classical logic: N⁴ has two truth values; implications in N⁴ are material, like in classical logic; and negation distributes over compound formulas in N⁴ as it does in classical logic. Results suggest that the semantics of normal logic programs is conveniently formalized in N⁴: Classical logic Herbrand interpretations generalize straightforwardly to N⁴; the classical minimal Herbrand model of a positive logic program coincides with its unique minimal N⁴ Herbrand model; the stable models of a normal logic program and its so-called complete minimal N⁴ Herbrand models coincide.