An Almost Classical Logic for Logic Programming and Nonmonotonic Reasoning

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