# NII Technical Report (NII-2005-010E)

 Title On the Semantics of Classical First-Order Logic with Constructive Double Negation Authors Ken Kaneiwa Abstract Constructive negation in intuitionistic logic (called strong negation) can be used to directly represent negative assertion, and for which its semantics is defined in Kripke models by two satisfaction relations. 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 \$\sim \lnot\$, for strong negation \$\sim\$ and classical negation \$\lnot\$ (which we call constructive double negation). The problem is caused by the fact that the semantics makes the axiom \$\sim \lnot A \leftrightarrow A\$ valid. By way of solution, this paper proposes an alternative semantics for constructive double negation \$\sim \lnot A\$ by capturing the constructive meaning of the combinations of the two negations. In the semantics, we consider the constructive double negation \$\sim \lnot A\$ as partial to the classical double negation \$\lnot \lnot A\$ and as exclusive to the classical negation \$\lnot A\$. Technically, we introduce infinite satisfaction relations to interpret the partiality that is sequentially created by each constructive double negation (of the forms \$\sim \lnot A, \sim \lnot \sim \lnot A,\ldots\$). Language English Published Dec 28, 2004 Pages 18p PDF File 04-010E.pdf

ISSN:1346-5597
NII Technical Reports
National Institute of Informatics