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New constants in the superintuitionistic logic L2. (English. Russian original) Zbl 1331.03027

Math. Notes 94, No. 6, 938-950 (2013); translation from Mat. Zametki 94, No. 6, 918-932 (2013).
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 [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.

MSC:

03B55 Intermediate logics

Citations:

Zbl 1271.03043
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References:

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