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The logic of \(\Pi_ 1\)-conservativity continued. (English) Zbl 0790.03018

Summary: It is shown that the propositional modal logic IRM (interpretability logic with Montagna’s principle and with witness comparisons in the style of Guaspari and Solovay’s logic \(R\) [D. Guaspari and R. Solovay, Ann. Math. Logic 16, 81-99 (1979; Zbl 0426.03062)]) is sound and complete as the logic of \(\Pi_ 1\)-conservativity over each \(\Sigma_ 1\)-sound axiomatized theory containing \(I\Sigma_ 1\). The exact statement of the result uses the notion of standard proof predicate. This paper is an immediate continuation of our earlier paper [Arch. Math. Logic 30, 113-123 (1990; Zbl 0713.03007)]. Knowledge of that paper is presupposed. We define a modal logic, called IRM, which includes both ILM and \(R\), and prove an arithmetical completeness theorem in the style of Guaspari and Solovay [loc. cit.], thus showing that IRM is the logic of \(\Pi_ 1\)-conservativity with witness comparisons. The reader is recommended to have C. Smoryński’s book [Self-reference and modal logic (1985; Zbl 0596.03001)] at his/her disposal.

MSC:

03B45 Modal logic (including the logic of norms)
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References:

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