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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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