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Epistemic Logics with Quantification Over Epistemic Operators: Decidability and Expressiveness

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Abstract

The optimal balance between decidability and expressiveness is a big problem of logical systems, in particular, of quantified epistemic logics (QELs). On the one hand, decidability is a very significant characteristic of logics that allows us to use such logics in the framework of artificial intelligence. On the other hand, QELs have important expressive capabilities that should not be lost when we construct decidable fragments of these logics. QELs are known to be much more expressive than first-order logics. One important example of their extra expressive power is that they allow us to distinguish between de dicto and de re readings of epistemic sentences. It is clear that such capabilities should be preserved as much as possible in decidable fragments. In this paper, we consider extensions of QELs that include quantification over modalities. Denote this extensions by \(\hbox {Q}_{\Box }\)Ls. \(\hbox {Q}_{\Box }\)Ls allows us to make more subtle distinctions between de dicto and de re readings of epistemic sentences, and we also should keep these new features as much as possible in decidable fragments. It is known that there are not much interesting decidable QELs. The situation with \(\hbox {Q}_{\Box }\)Ls is the same. But in recent years (after 2018), we have obtained a variety of decidable \(\hbox {Q}_{\Box }\)Ls constructed in different ways. We distinguish between (1) the approach in which for every undecidable \(\hbox {Q}_{\Box }\)L and for every variant of its decidable fragment, a specific proof is constructed, and (2) the approach in which a class of decidable \(\hbox {Q}_{\Box }\)Ls is obtained using general tools and a uniform method for all \(\hbox {Q}_{\Box }\)Ls of this class. In this paper, we compare the results of these approaches.

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Notes

  1. It is known that transitive relations and, in particular, equivalence relations are not expressible in FO\(^2\).

  2. We will look at these fragments in detail in Sects. 5.1 and 5.2.

  3. Note that the paper [32] has been accepted, but the paper [33] is only submitted.

  4. Quine’s original sentence is ‘Ralph believes that someone is a spy’. But the difference between ‘know’ and ‘believe’ is inessential for this paper.

  5. Propositional atoms are viewed as nullary predicates.

  6. Since Ralph saw Ortcutt in a brown hat and at the beach in different circumstances and at different times, we can always choose sets of worlds (situations) \(W_1\) and \(W_2\) such that \(W_1\cap W_2 = \emptyset \).

  7. See the reading (S\(\forall \)1) and also its formalization (S\(\forall \)1)\(^{TML^2}\) in Sect. 6.1.

  8. Recall that the monodic fragment of a modal logic is a restriction allowing at most one free variable within the scope of a modality.

  9. Note that in the monodic fragment of PTML, we can express only unary predicates.

  10. In contrast, the monodic fragment of PTML can be extended with constants (as indices of modalities) and equality without loss of decidability (see [23, Sect. 3]).

  11. The notion ‘first-order correspondence language’ is defined in [3, pp. 83, 127].

  12. Super predicate symbols; are predicate symbols that take as arguments both individual variables and predicate variables (see [6, Sect. 4]).

  13. In (f1), the superscript \(^{(w)}\) indicates that name \(P^{(w)}\) is associated with some world (in this case with the current world).

  14. The paper [15] establishes the 2-Nexptime upper complexity bound for satisfiability of FO\(^2\)+2E. But in [31], we use a fragment of FO\(^2\)+2E the satisfiability problem of which is Nexptime-complete (see [31, Sect. 3]).

  15. U corresponds to the relation R of Definition 3.1.

  16. Such a quantifier \(\forall \overline{Y}\) exists in \(\varphi \), since formulas of HO-LGF\(^{\Box }\) do not contain second-order predicate variables free (see Definition 5.4).

  17. Obviously, this formula belongs to HO-LGF\(^{\Box }\).

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  1. Odessa National Academy of Communications, Odessa, Ukraine

    Gennady Shtakser

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Shtakser, G. Epistemic Logics with Quantification Over Epistemic Operators: Decidability and Expressiveness. Log. Univers. 17, 297–330 (2023). https://doi.org/10.1007/s11787-023-00330-2

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