Abstract
Combining relevant and classical modal logic is an approach to overcoming the logical omniscience problem and related issues that goes back at least to Levesque’s well known work in the 1980s. The present authors have recently introduced a variant of Levesque’s framework where explicit beliefs concerning conditional propositions can be formalized. However, our framework did not offer a formalization of implicit belief in addition to explicit belief. In this paper we provide such a formalization. Our main technical result is a modular completeness theorem.
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Notes
- 1.
Such agents can be seen as reasoning according to Harman’s clutter avoidance principle [7] in that they do not clutter their minds with trivial but unrelated consequences of the given information.
- 2.
\(\Box _L\) can be seen as a sort of provability operator; see Lemma 7.
- 3.
This condition has to do with the canonical model construction (see Sect. 3), since in the canonical model \(Q_I^c\) will not be upward monotone.
- 4.
We can define \(\mathfrak {M}\) similarly as in the \(+\)-construction used in the proof of Lemma 7, with the proviso that we do not add a new possible world w since v itself is seen as the only possible world in the model.
- 5.
Note that \(\mathsf {(L12)}\) is problematic since \(\varphi \) in general is not an implication, so we cannot use item (1) and \(\mathsf {(BR)}\).
- 6.
The presence of the bounds 0, 1 is necessary for the following reason. The bound-free versions of Conditions (5–6) are sufficient for Lemma 3, but these simpler versions do not hold in the canonical model. For instance, \(\emptyset \) is a perfectly legitimate prime \(\textsf{L}\)-theory, and \(Rw\emptyset t\) obviously holds for all \(w \in W^c\) and \(t \in S^c\). Hence, \(Rwst \Rightarrow w \subseteq t\) fails. (The argument that \(Rwst \Rightarrow s \subseteq w\) fails is similar, exploiting the possibility that \(t = \mathfrak {L}\).) In this situation we can either add extensional truth constants to the language, and so rule out \(\emptyset \) and \(\mathfrak {L}\) as legitimate \(\textsf{L}\)-theories, or work with \(\emptyset \) and \(\mathfrak {L}\) as special kinds of states in the model while modifying the frame conditions (5–6) so that they refer to these special states. We chose the second option.
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Acknowledgement
We thank the anonymous reviewers for comments. This work was supported by the Czech Science Foundation grant no. GA22-01137S.
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Sedlár, I., Vigiani, P. (2023). Relevant Reasoning and Implicit Beliefs. In: Hansen, H.H., Scedrov, A., de Queiroz, R.J. (eds) Logic, Language, Information, and Computation. WoLLIC 2023. Lecture Notes in Computer Science, vol 13923. Springer, Cham. https://doi.org/10.1007/978-3-031-39784-4_21
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