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A short note on essentially \(\Sigma_1\) sentences. (English) Zbl 1270.03039
A modal formula \(F\) is essentially \(\Sigma_1\) with respect to theory \(T\) if, under any arithmetical interpretations \(\ast\) into \(T,\) \(F^\ast\) is a \(\Sigma_1\) formula. A. Visser proved in [Ann. Pure App. Logic 73, No. 1, 109–142 (1995; Zbl 0828.03008)] that a formula \(F\) of the provability logic GL (Gödel-Löb) is essentially \(\Sigma_1\) with respect to PA if and only if \(F\) is provably equivalent in GL to a disjunction of formulas of the form \(\square B\). In [Stud. Fuzziness Soft Comput. 24, 246–254 (1999; Zbl 0923.03025)], D. de Jongh and D. Pianigiani extended the result to system \(R\) of Guaspari-Solovay. E. Goris and J. J. Joosten classified in [Log. J. IGPL 16, No. 4, 371–412 (2008; Zbl 1162.03033); ibid. 20, No. 1, 1–21 (2012; Zbl 1252.03140)] all essentially \(\Sigma_1\) sentences of ILM (interpretability logic with Montagna principle) with respect to interpretations in essentially reflexive recursively enumerable arithmetical theories. In the paper under review, the authors show that a characterization of this kind can be obtained also for formulas of the interpretability logic ILP, with respect to any finitely axiomatizable \(\Sigma_1\)-sound extension of \(\mathrm{I}\Delta_0 + \mathrm{Supexp}.\) It is proved that the same characterization does not extend to \(\mathrm I\Delta_0+\mathrm{Exp}\) and a conjecture is formulated about essentially \(\Sigma_1\) ILP-formulas with respect to \(\mathrm I\Delta_0+\mathrm{Exp}.\)

MSC:
03B45 Modal logic (including the logic of norms)
03F45 Provability logics and related algebras (e.g., diagonalizable algebras)
03F30 First-order arithmetic and fragments
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