Abstract
We present a logic which allows to reason about the relationship between an agent’s beliefs and the information that the agent obtains by his senses. Our logic reconstructs standard doxastic logic by primitives denoting that an agent receives from his senses the datum that some propositional variable is true or false. We give a sound and complete axiomatization and prove that the logic is decidable. We also discuss extensions of the basic logic by principles of introspection and a variant in terms of knowledge instead of belief. We finally extend the logic by events in the style of dynamic epistemic logics.
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Notes
- 1.
These are the L-PB models \(V \cup \{ {p}^{0}\}\), \(V \setminus \{{p}^{0}\}\), \((V \cup \{{\mathsf{datum}}^{\top }(i,p)\})\setminus \{{\mathsf{datum}}^{\perp }(i,p)\}\), \((V \cup \{{\mathsf{datum}}^{\perp }(i,p)\})\setminus \{{\mathsf{datum}}^{\top }(i,p)\}\), \(V \setminus \{{\mathsf{datum}}^{\top }(i,p)\}\), and \(V \setminus \{{\mathsf{datum}}^{\perp }(i,p)\}\).
- 2.
See van Ditmarsch et al. (2009) for a similar notion of forgetting.
- 3.
Precisely, \(\mathsf{MBel}_{C}^{k}p\) can be defined in the three steps as follows. First, we define \(\mathsf{EBel}_{C}^{}p\) as an abbreviation of \(\bigwedge _{i\in C}\mathsf{B}_{i}\). Then, we inductively define \(\mathsf{EBel}_{C}^{k}p\) for every \(k \in \mathbb{N}\): \(\mathsf{EBel}_{C}^{0}p\ \stackrel{\mathtt{def}}{=}\ p\) and \(\mathsf{EBel}_{C}^{k}p\ \stackrel{\mathtt{def}}{=}\ \mathsf{EBel}_{C}^{}\mathsf{EPerc}_{C}^{k-1}p\) for k > 0. Finally, we define \(\mathsf{MBel}_{C}^{n}p\) as an abbreviation of \(\bigwedge _{1\leq k\leq n}\mathsf{EBel}_{C}^{k}p\).
- 4.
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Herzig, A., Lorini, E. (2014). A Modal Logic of Perceptual Belief. In: Lihoreau, F., Rebuschi, M. (eds) Epistemology, Context, and Formalism. Synthese Library, vol 369. Springer, Cham. https://doi.org/10.1007/978-3-319-02943-6_12
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