New constants in the superintuitionistic logic L2 (Q2440003)

The paper studies the logic L2 (the intermediate logic of the frames of depth at most 2, also known as \(\mathrm{BD}_2\)) endowed with a finite number of new (logical) constants. The obtained results are generalizations of the results from [\textit{A. D. Yashin}, Algebra Logic 50, No. 2, 171--186 (2011; Zbl 1271.03043); translation from Algebra Logika 50, No. 2, 246--267 (2011)], where the logic L2 with a single additional constant had been researched. In particular, the description of all Novikov-complete extensions (i.e., maximal conservative extensions of L2) is given, as well as decidability of all these extensions is established. The problem of deciding by an additional axiom \(A(\overline{\phi})\) (in the extended language) whether the extension \(\mathrm{L2} + A(\overline{\phi})\) is conservative is also proven to be decidable. A \(\overline{\phi}\)-logic \(\mathcal{L}\) does not admit any explicit relations if \(\mathcal{L}\) is conservative over \(L\) and, for every additional constant \(\phi_i\) from \(\mathcal{L}\) and every explicit relation \(\phi_i \leftrightarrow B\), the \(\overline{\phi}\)-logic \(\mathcal{L} + \phi_i \leftrightarrow B\) is not conservative over \(L\). A straightforward method of finding whether or not a given complete logic admits new constants has been introduced.