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On theses without iterated modalities of modal logics between \(\mathbf{C1}\) and \(\mathbf{S5}\). II. (English) Zbl 1423.03063

Summary: This is the second, out of two papers, in which we identify all logics between \(\mathbf{C1}\) and \(\mathbf{S5}\) having the same theses without iterated modalities. All these logics can be divided into certain groups. Each such group depends only on which of the following formulas are theses of all logics from this group: \((\mathsf N)\), \((\mathsf T)\), \((\mathsf D)\), \(\ulcorner (\mathsf T) \vee\square \mathsf q\urcorner\), and for any \(n>0\) a formula \(\ulcorner (\mathsf T) \vee (\mathsf{alt}_{\mathrm n})\urcorner\), where \((\mathsf T)\) has not the atom ‘\(q\)’, and \((\mathsf T)\) and \((\mathsf{alt}_{\mathrm n})\) have no common atom. We generalize J. L. Pollock’s result from [J. Symb. Log. 32, 356–365 (1967; Zbl 0149.24404)], where he proved that all modal logics between \(\mathbf{S1}\) and \(\mathbf{S5}\) have the same theses which does not involve iterated modalities (i.e., the same first-degree theses).
For Part I see [the author, ibid. 46, No. 1–2, 111–133 (2017; Zbl 1423.03062)].

MSC:

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

[1] J. L. Pollack, Basic Modal Logic, The Journal of Symbolic Logic 32 (3) (1967), pp. 355-365. DOI: 10.2307/2270778 Department of Logic Nicolaus Copernicus University in Toru´n ul. Moniuszki 16, 87-100 Toru´n, Poland e-mail: Andrzej.Pietruszczak@umk.pl
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