On the Semantics of Classical First-order Logic with Constructive Double Negation

Constructive negation in intuitionistic logic (called strong negation [7]) can be used to directly represent negative assertions, and for which its semantics [8, 1] is defined in Kripke models by two satisfaction relations (|= P and |= N ). However, the interpretation and satisfaction based on the conventional semantics do not fit in with the definition of negation in knowledge representation when considering a double negation of the form ∼¬, for strong negation ∼ and classical negation ¬ (which we call constructive double negation). The problem is caused by the fact that the semantics makes the axiom ∼¬A ↔ A valid. By way of solution, this paper proposes an alternative semantics for constructive double negation ∼¬A by capturing the constructive meaning of the combinations of the two negations. In the semantics, we consider the constructive double negation ∼¬A as partial to the classical double negation ¬¬A and as exclusive to the classical negation ¬A. Technically, we introduce infinite satisfaction relations to interpret the partiality that is sequentially created by each constructive double negation (of the forms ∼¬A,∼¬∼¬A, . . . ).