Propositional Epistemic Logics with Quantification Over Agents of Knowledge (An Alternative Approach)

Abstract

In the previous paper with a similar title (see Shtakser in Stud Log 106(2):311–344, 2018), we presented a family of propositional epistemic logics whose languages are extended by two ingredients: (a) by quantification over modal (epistemic) operators or over agents of knowledge and (b) by predicate symbols that take modal (epistemic) operators (or agents) as arguments. We denoted this family by $${\\mathcal {P}\\mathcal {E}\\mathcal {L}}_{(QK)}$$PEL(QK). The family $${\\mathcal {P}\\mathcal {E}\\mathcal {L}}_{(QK)}$$PEL(QK) is defined on the basis of a decidable higher-order generalization of the loosely guarded fragment (HO-LGF) of first-order logic. And since HO-LGF is decidable, we obtain the decidability of logics of $${\\mathcal {P}\\mathcal {E}\\mathcal {L}}_{(QK)}$$PEL(QK). In this paper we construct an alternative family of decidable propositional epistemic logics whose languages include ingredients (a) and (b). Denote this family by $${\\mathcal {P}\\mathcal {E}\\mathcal {L}}^{alt}_{(QK)}$$PEL(QK)alt. Now we will use another decidable fragment of first-order logic: the two variable fragment of first-order logic with two equivalence relations (FO$$^2$$2+2E) [the decidability of FO$$^2$$2+2E was proved in Kieroński and Otto (J Symb Log 77(3):729–765, 2012)]. The families $${\\mathcal {P}\\mathcal {E}\\mathcal {L}}^{alt}_{(QK)}$$PEL(QK)alt and $${\\mathcal {P}\\mathcal {E}\\mathcal {L}}_{(QK)}$$PEL(QK) differ in the expressive power. In particular, we exhibit classes of epistemic sentences considered in works on first-order modal logic demonstrating this difference.