CE is not a conservative extension of E
The logic \({\mathbf {CE}}\) results from adding Boolean negation, \(\neg\), to the Anderson-Belnap logic \({\mathbf E}\) of entailment. Semantically, this is treated classically, so that \(\neg A\) is true at a point, or `possible world', iff \(A\) is not true at that point. Boolean negation contrasts with the de Morgan negation, \(\sim\), of \({\mathbf E}\) and other relevant logics, which requires a more complex semantical rule. As the title indicates, this paper demonstrates that \({\mathbf {CE}}\) is not a conservative extension of \({\mathbf E}\). The proof is based on the Routley-Meyer ternary relational semantics for relevant logics, wherein \({\mathbf {CE}}\) can be proved sound and complete, and to contain a formula in the vocabulary of \({\mathbf E}\) that is not provable in \({\mathbf E}\). The result is interesting since the addition of Boolean negation to the logic \({\mathbf R}\), which is stronger than \({\mathbf E}\), is conservative, and so is its addition to relevant logics weaker than \({\mathbf E}\).
scientific article; zbMATH DE number 1491206