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Formal Model for Explicit Aware Knowledge

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Awareness in Logic and Epistemology

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 52))

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Abstract

This chapter presents a formal model that constitutes the logical correlation to the conceptual schema for Explicit Aware Knowledge (EAK-Schema). Using neighbourhood semantics this model depicts the informational attitudes of a non-omniscient epistemic agent. It represents the awareness of and awareness that of the agent and the different combinations of them that result in Explicit Aware Knowledge, implicit knowledge and other forms of knowledge. The model also includes the formalisation of epistemic actions and discusses the main properties of the informational attitudes and the corresponding epistemic actions that change these attitudes.

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Notes

  1. 1.

    Most of what is stated in this chapter has been published in Fernández-Fernández and Velázquez-Quesada (2020). A previous work, containing a first version of this model, is published in Fernández-Fernández and Velázquez-Quesada (2019).

  2. 2.

    For more details on this model construction, see Chellas (1980), Sect. 7.3.

  3. 3.

    Note that \(\mathop {\mathrm {A^t}}\) uses the ‘set’ semantic interpretation of the \(\mathop {\Box }\)-operator in neighbourhood models. (Recall: the ‘subset’ semantic interpretation makes \(\mathop {\Box }\varphi \) true at w in M not only when \(\llbracket \varphi \rrbracket ^{M}\) is in N(w), but also when any of its subsets is: \(\llbracket \mathop {\Box }\varphi \rrbracket ^{M} := \{ w \in W \mid \text {there is}\, U \in N(w) \,\text {such that}\, U \subseteq \llbracket \varphi \rrbracket ^{M} \}\).)

  4. 4.

    When awareness-that is understood as explicit knowledge it does not have this closure property (like in Konolige 1984; Grossi and Velázquez-Quesada 2015, for example), simply because it is represented by a set of formulas that is not required to have any closure property.

  5. 5.

    In a semantic setting, a modus ponens can be understood as a three-step process: conjunction introduction (from \(\llbracket \varphi \rrbracket ^{M}\) and \(\llbracket \varphi \rightarrow \psi \rrbracket ^{M}\) to \(\llbracket \varphi \wedge (\varphi \rightarrow \psi ) \rrbracket ^{M}\)), logical equivalence (from the latter to \(\llbracket \varphi \wedge \psi \rrbracket ^{M}\)) and conjunction elimination (from the now-latter to \(\llbracket \psi \rrbracket ^{M}\)).

  6. 6.

    In this, the present setting coincides with the one in Grossi and Velázquez-Quesada (2015).

  7. 7.

    Velázquez-Quesada (2013) (see also Balbiani et al. 2018) already uses this known relationship between neighbourhood models and relational models. Still, the technical details are slightly different, as the referred paper works on the finite-domain case and does not take the concept of awareness-of into account.

  8. 8.

    This differs from the same notion in Fagin and Halpern (1988), where the agent knows implicitly every validity.

  9. 9.

    In fact, the referred logic of local reasoning (Fagin and Halpern 1988) is equivalent to a neighbourhood semantics in which each neighbourhood is assumed to be non-empty and also closed under supersets, thus making the knowledge of the agent closed under conjunction elimination.

  10. 10.

    Define \(\mathrm {at}(\mathop {[+ \chi ]}\varphi ) := \mathrm {at}(\chi ) \cup \mathrm {at}(\varphi )\).

  11. 11.

    Defining a ‘diamond’ becoming aware-of modality in the standard way, \(\mathop {\langle +\chi \rangle }\varphi := \lnot \mathop {[+ \chi ]}\lnot \varphi \), implies \(\llbracket \mathop {\langle +\chi \rangle }\varphi \rrbracket ^{M} = \llbracket \varphi \rrbracket ^{M^{+\chi }}\): under this definition, \(\mathop {[+ \chi ]}\varphi \) and \(\mathop {\langle +\chi \rangle }\varphi \) are logically equivalent.

  12. 12.

    If there are no such atoms (if \(\mathrm {at}(\varphi ) \subseteq \mathrm {at}(\chi )\)), the formula’s right-hand side collapses to \(\top \): after becoming aware of \(\chi \), the agent will be aware of every \(\varphi \) built only from atoms in \(\chi \).

  13. 13.

    More precisely, the validity holds exactly for those formulas \(\varphi \) whose truth-set is not affected by the operation, that is, for every \(\varphi \in \mathcal {L}\) for which \(\llbracket \varphi \rrbracket ^{M} = \llbracket \varphi \rrbracket ^{M^{+\chi }}\).

  14. 14.

    Define \(\mathrm {at}(\mathop {[-\chi ]}\varphi ) := \mathrm {at}(\chi ) \cup \mathrm {at}(\varphi )\).

  15. 15.

    Just as before, defining \(\mathop {\langle -\chi \rangle }\varphi := \lnot \mathop {[-\chi ]}\lnot \varphi \) implies \(\Vdash \mathop {[-\chi ]}\varphi \leftrightarrow \mathop {\langle -\chi \rangle }\varphi \).

  16. 16.

    In particular, by becoming unaware of \(\chi \), the agent becomes unaware of all \(\chi \)’s subformulas.

  17. 17.

    Define \(\mathrm {at}(\mathop {[-\texttt {Q}]}\varphi ) := \texttt {Q}\cup \mathrm {at}(\varphi )\).

  18. 18.

    Again, \(\mathop {\langle -\texttt {Q}\rangle }\varphi := \lnot \mathop {[-\texttt {Q}]}\lnot \varphi \) implies \(\Vdash \mathop {[+ \texttt {Q}]}\varphi \leftrightarrow \mathop {\langle +\texttt {Q}\rangle }\varphi \).

  19. 19.

    Take \(\varphi := \bigwedge _{p \in \mathrm {at}(\chi )} \lnot \mathop {\mathrm {A^o}}p\), stating that the agent is unaware of all atoms in \(\chi \). Clearly, it holds when all atoms are removed, but fails when some of them remain.

  20. 20.

    Take \(\varphi := \bigvee _{p \in \mathrm {at}(\chi )} \mathop {\mathrm {A^o}}p\), stating that the agent is aware of at least one of \(\chi \)’s atoms: it holds when \(\chi \) has at least two atoms and only one of them is removed, but it fails when all atoms are discarded.

  21. 21.

    Other proposals including operations with the same spirit are the deductive inferences of Velázquez-Quesada (2009), Velázquez-Quesada (2013), Solaki (2017), Smets and Solaki (2018), Balbiani et al. (2018), and the belief-based inference of Velázquez-Quesada (2014).

  22. 22.

    Define .

  23. 23.

    E.g., the case of a public announcement (Plaza 1989; Gerbrandy and Groeneveld 1997), whose precondition is for the announced formula to be true.

  24. 24.

    Define \(\mathrm {at}(\mathop {[! \chi ]}\varphi ) := \mathrm {at}(\chi ) \cup \mathrm {at}(\varphi )\).

  25. 25.

    See the relational-model-based forgetting operations of van Ditmarsch et al. (2009), Fernández-Duque et al. (2015).

  26. 26.

    Define \(\mathrm {at}(\mathop {[\setminus \chi ]}\varphi ) := \mathrm {at}(\chi ) \cup \mathrm {at}(\varphi )\).

  27. 27.

    Still, the most important properties of the main concepts have been discussed.

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  1. Departamento de Filosofía, Universidad de Málaga, Málaga, Spain

    Claudia Fernández-Fernández

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  1. Claudia Fernández-Fernández
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Fernández-Fernández, C. (2021). Formal Model for Explicit Aware Knowledge. In: Awareness in Logic and Epistemology. Logic, Epistemology, and the Unity of Science, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-030-69606-1_6

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