Abstract
When building artificial agents that have to make decisions, understanding what follows from what they know or believe is mandatory, but it is also important to understand what happens when those agents ignore some facts, where ignoring a fact is interpreted to stand for not knowing/not being aware of something. This becomes especially relevant when such agents ignore their ignorance, since this hinders their ability of seeking the information they are missing. Given this fact, it might prove useful to clarify in which circumstances ignorance is present and what might cause an agent to ignore that he/she is ignoring. This paper is an attempt at exploring those facts. In the paper, the relationship between ignorance and beliefs is analysed. In particular, three doxastic effects are discussed, showing that they can be seen as a cause of ignorance. The effects are formalized in a bi-modal formal language for knowledge and belief and it is shown how ignorance follows directly from those effects. Moreover, it is shown that negative introspection is the culprit of the passage between simply ignoring a fact and ignoring someone’s ignorance about that fact. Those results could prove useful when artificial agents are designed, since modellers would be aware of which conditions are mandatory to avoid deep forms of ignorance; this means that those artificial agents would be able to infer which information they are ignoring and they could employ this fact to seek it and fill the gaps in their knowledge/belief base.
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Notes
- 1.
- 2.
It is necessary to clarify that the term ignorance employed in this paper is given a specific meaning, i.e., to not know/not be aware of something.
- 3.
In this paper, the term doxastic will always refer to the act of believing.
- 4.
In the paper, ignorance will always be indicated with a specific order, which indicates the depth of the ignoring phenomenon. First-order ignorance means that a given fact is ignored; second-order ignorance means that it is ignored that a given fact is ignored, and so forth.
- 5.
The interested reader is referred to [17] for a standard presentation of Kripke structures.
- 6.
Note that an ignorance formula could represent instances of ignorance of any order, depending on how many occurrences of ignorance operators appear in the formula \(\phi \).
- 7.
Note that, given the semantic framework employed to interpret the two notions, those notions also distribute over implications.
- 8.
See, e.g., [2].
- 9.
The abbreviations that will be employed in the proofs of this paper will all be reported here. Ass. will stand for “assumption”; P. Taut. will stand for “propositional tautology”; Elim. will stand for “elimination rule”; Intr. will stand for “introduction rule”; Contrap. will stand for “contrapposition”; MP will stand for “Modus Ponens”; DM will stand for “DeMorgan rules”; DS will stand for “disjunctive syllogism”; Nec. will stand for “necessitation rule”; finally Distr. will stand for “distributivity rule”.
- 10.
See [6] for a discussion about different aspects that relate misbelieving and ignoring.
- 11.
It should be pointed out that Fine does not use the terms “first-order ignorance”, “second-order ignorance” and so on. However, to maintain coherence with the rest of the paper, those terms will be employed when the concepts expressed by Fine are aligned with the meanings attributed to those terms in this paper.
- 12.
Proofs of lemmas will not be provided. If the reader is interested, in [7] it is possible to find all the details concerning the lemmas which are introduced here. The only important detail is that Fine provides proofs in the axiomatic system S4, which is a system that defined languages weaker than the one employed in this paper. Therefore, every proof provided by Fine could be easily reproduced inside \(\mathcal {L}\).
- 13.
Fine proves such lemma while proving his main theorem. However, to make the proof easier to read, this lemma will be given separately. The proof of such lemma can be found in appendix A.
- 14.
This result is not novel to this paper, but is well known in the logical literature on formalizing ignorance.
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A Formal Proofs
A Formal Proofs
Proof
(Proposition 1)
The proof will be split into two parts.
Case 1: The proof will be given directly.
Case 2: The proof will be given directly.
Proof
(Proposition 2)
The proof will be given directly.
Proof
(Proposition 3)
The proof will be split into two parts, showing that each disjunct of the antecedent of the conditional implies the consequent of the conditional.
Case 1: The proof will be given directly.
Case 2: The proof will be given directly.
Proof
(Lemma 5) The proof is given by contradiction.
Since a contradiction has been reached, one of the assumptions must be rejected. Thus, \(I(I(\phi ))\rightarrow \lnot K(I(I(\phi )))\) holds in \(\mathcal {L}\). \(\square \)
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Aldini, A., Graziani, P., Tagliaferri, M. (2021). Reasoning About Ignorance and Beliefs. In: Cleophas, L., Massink, M. (eds) Software Engineering and Formal Methods. SEFM 2020 Collocated Workshops. SEFM 2020. Lecture Notes in Computer Science(), vol 12524. Springer, Cham. https://doi.org/10.1007/978-3-030-67220-1_17
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