Abstract
The possibility of better understanding belief and knowledge modalities through justifications is not a novel one, however, the machinery of justifications has never been employed to explore the nature of ignorance from a formal perspective. By including justification terms into a modal logic for belief a major project (among others) can be pursued: different cognitive attitudes can be formalized that imply ignorance, therefore highlighting even better the possible culprits of the emergence of the phenomenon of being ignorant. This would allow the possibility of developing strategies that could be employed in different scenarios to tackle ignorance, thus adapting interventions to the specific situations in which ignorance arises.
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Notes
- 1.
Throughout the paper, I will take propositions and facts as synonyms. In particular, a proposition will indicate a fact, i.e., that the world is in a particular way. At the same time, each fact could be represented with a proposition, i.e., the fact specifies a way in which the world is and the proposition describes such world.
- 2.
Obviously, other approaches are also possible, e.g., interpreting ignorance as a primitive notion [6]. Moreover, even following this simple approach of reducing ignorance to the lack of knowledge can produce different formalizations depending on how the authors interpret the phrase “lack of knowledge”, e.g., as not knowing that or not knowing whether.
- 3.
Being agnostic: \(\lnot B(\phi ) \wedge \lnot B(\lnot \phi )\), misbelieving: \(B(\phi )\wedge \lnot \phi \), and doubting: \(B(\phi )\wedge \phi \wedge \lnot K(\phi )\). Where \(B(\phi )\) should be interpreted as \(\phi \) is believed and \(K(\phi )\) as \(\phi \) is known.
- 4.
The phenomenon of doubting seems particularly obscure, since it is left unexplained why \(\phi \) is not known even though it is believed and it is true.
- 5.
It might be useful to keep track of the fact that \(\lnot K(\varphi ){:}{=}\lnot B(\varphi ) \vee \lnot \varphi \vee \lnot E_c(\varphi )\).
- 6.
See [16] for some examples.
- 7.
The origin of the notion of ignoring whether could be traced back to Hintikka [12], who, however, only discusses it briefly. In short, ignoring whether describes ignorance as a lack of knowledge both about a proposition \(\phi \) and its negation \(\lnot \phi \). On the other side, ignorance could also be interpreted as lack of knowledge that, where only one of the two propositions is taken into consideration.
- 8.
In order to define the attitudes that are indicated in this paper, I employed the Oxford Languages online dictionary [13]. Note that different interpretations of the attitudes might modify the way that they are formalized.
- 9.
First class: Mislead, negative belief perseverance, credulity, misbelief; second class: positive belief perseverance, doubt, and intuition.
- 10.
From now on, the attitudes are presented employing only \(B(\varphi )\). Obviously, the same considerations would be true employing, mutando mutandis, \(B(\lnot \varphi )\).
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Tagliaferri, M. (2023). Using Justified True Beliefs to Explore Formal Ignorance. In: Masci, P., Bernardeschi, C., Graziani, P., Koddenbrock, M., Palmieri, M. (eds) Software Engineering and Formal Methods. SEFM 2022 Collocated Workshops. SEFM 2022. Lecture Notes in Computer Science, vol 13765. Springer, Cham. https://doi.org/10.1007/978-3-031-26236-4_31
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