Completeness of LCOI+PLCOI under an Alternative Implication Function

Determine whether there exists a binary function \(\mathcal{R}\) such that the completeness theorem holds for the logic LCOI+PLCOI with continuous-valued semantics.

Background

The paper introduces continuous-valued semantics for the logic LCOI+PLCOI, whose formulas receive truth values in the interval $[0,1]$. To establish soundness, the authors select the implication function $\mathcal{R}_o$ and prove that all axioms are $\lambda$-tautologies and that the resulting system is sound.

For this particular implication function, the paper explicitly states that completeness fails. The unresolved issue is whether some other binary implication function $\mathcal{R}:[0,1]^2\to[0,1]$ can yield a complete continuous-valued semantics for LCOI+PLCOI. The problem concerns the alignment between semantic consequence and formal derivability in the proposed three-negation logic.