# zbMATH — the first resource for mathematics

Solution of a problem of David Guaspari. (English) Zbl 0923.03025
Orłowska, Ewa (ed.), Logic at work. Essays dedicated to the memory of Helena Rasiowa. Heidelberg: Physica-Verlag. Stud. Fuzziness Soft Comput. 24, 246-254 (1999).
If $$L$$ is a logic with a collection of arithmetical interpretations, then we call a formula $$A$$ of $$L$$ essentially $$\Sigma_1$$ with respect to theory $$T$$ if, under any arithmetical interpretation $$*$$ of $$L$$ into $$T$$, $$A^*$$ is a $$\Sigma_1$$-formula. D. Guaspari [J. Symb. Log. 48, 777-789 (1983; Zbl 0547.03035)] stated as a question: whether in the Guaspari-Solovay system $$R$$, which extends the system $$GL$$ (Gödel-Löb) with symbols for witness comparisons, the essentially $$\Sigma_1$$ formulae w.r.t. an r.e. theory extending $$I\Sigma_1$$ are the ones which are, provably in $$R$$, equivalent to a (possibly empty) disjunction of conjunctions of $$\square$$-formulae and witness comparison formulae. A. Visser answered Guaspari’s question for the system $$GL$$ positively. In the paper under review, the authors prove first the conjecture for the simple case of $$GL$$, because this clarifies the method (very different from Visser’s) used in the more complicated case of $$R$$. By using interpretability logic $$ILM$$ and Kripke models, the authors solve the Guaspari problem for the system $$R$$.
For the entire collection see [Zbl 0910.00016].

##### MSC:
 03B45 Modal logic (including the logic of norms) 03F30 First-order arithmetic and fragments