Abstract
Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ1 (i.e., it is Σ1 under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form \({\square{B}}\) . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ1 formulas of languages including witness comparisons using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of IΔ 0 + Supexp and we address a similar problem for IΔ 0 + Exp.
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The present paper is dedicated to Dick De Jongh
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Montagna, F., Pianigiani, D. A Short Note on Essentially Σ1 Sentences. Log. Univers. 7, 103–111 (2013). https://doi.org/10.1007/s11787-012-0070-9
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DOI: https://doi.org/10.1007/s11787-012-0070-9