Abstract
The Dunning–Kruger effect focuses our attention on the notion of invisibility of ignorance, i.e., the ignorance of ignorance. Such a phenomenon is not only important for everyday life, but also, above all, for some philosophical disciplines, such as epistemology of sciences. When someone tries to understand formally the phenomenon of ignorance of ignorance, they usually end up with a nested epistemic operator highly resistant to proper regimentation. In this paper, we argue that to understand adequately the ignorance of ignorance phenomenon we have to understand satisfactorily the concept of disbelief and, as we call it, the concept of “radical ignorance”. We propose also prerequisites that a notion of radical ignorance useful for the philosophy of science ought to fulfill, and we sketch a possible formalization of this notion. Finally, we propose some comments on the problem of propagation of ignorance proposed by Fine (Synthese, 2007. https://doi.org/10.1007/s11229-017-1406-z).
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Notes
In this paper, we will use the expressions “unknown unknowns”, “the ignorance of ignorance”, “square ignorance” and “second-order ignorance” as synonymous.
Note that, even though there seems to be a strict reference to “knowledge-how” rather than “knowledge-that” in the Dunning-Kruger effect, we will concentrate only on the phenomenon of radical ignorance applied to the latter case of knowledge, i.e., “knowledge-that.” We thank an anonymous referee for highlighting this point.
Thanks to an anonymous referee for the objection.
That is to say things that we know that should somehow exist but simultaneously we reckon to know nothing about, such as in the case of dark matter and dark energy.
Fine (2017).
Obviously, the “Fitch knowability paradox” doesn’t arise in our propositional logic approach. However, we would like to highlight that the fundamental premise of Fitch’s paradox is that all sentences are knowable. This principle applied to empirical sciences would presuppose that all what there is would be connected causally with us and we have no reason to believe, for instance, that there is nothing out of our past light cone, even taking into account the expansion of the universe. Therefore, sentences about what is out of our past light cone seem unknowable.
See for example Williamson (2000).
See, for instance, Fagin et al. (1995), which apply epistemic logic to computer science.
Williamson (2000, chap. V), attempts to show that KK is epistemologically and not only psychologically wrong. To make simple something complex, investigating a case of imprecise knowledge—for instance “at best of our knowledge, Hubble’s constant is 71.9 ± 2.7 km/s/Mpc” – Williamson argues that applying KK one reaches the conclusion that one knows that Hubble constant has not its true value. And obviously this is absurd. But one of the premises of his argumentation is that “one does know that the constant is not 100H”, where “100H” is any value surely wrong. Nevertheless, if ± 2.7 km/s/Mpc is the unavoidable error of our measurement, it is impossible that “one does know that the constant is not 100H” is true. Maybe “one does know that the constant is not 100H ± 2.7 km/s/Mpc” is true. And from this premise it is no longer possible to deduce Williamson’s paradox. See De Florio and Fano (2020).
In this case “K” refers to the distributivity over implication of the “K” operator and “4” is the KK principle.
If wiRkwj and wiRkwk then wjRkwk.
See, for instance, Hendricks and Symons (2015).
That is to say, every doxastically accessible world must be epistemically accessible as well.
This is a striking difference between the “B” operator and “K”: the latter is assumed to be factive (i.e. (1) holds).
The controversial transitivity of B is not essential to our argumentation.
In our fallibilist framework no sentence is completely justified, but a sentence can be strongly justified.
See also Fan et al. (2015).
Often epistemic logics work with the indication of the individual subject who knows, such as “a” in this case; since in our discussion this specification is not relevant, we will neglect from now on this index.
Hintikka (1962, p. 79) defines the notion of “epistemic implication”.
A third formalization is also interesting, but we reserve a different article for its analysis. It is as follows: (iii.) Agents are radically ignorant about “φ” if they do not know both “φ” and “~φ”, and either they believe to know “φ” but it is the case that “~φ”, or it is the case that “φ” but they believe to know “~φ”. Once again formally:
$$I_{S} \varphi =_{df} (( \sim K\varphi \wedge \sim K \sim \varphi ) \wedge ((BK\varphi \wedge \sim \varphi ) \vee (BK \sim \varphi \wedge \varphi )))$$Where ISφ should be read as “φ is strongly ignored”. Note that this formula is no longer redundant in the sense of (11).
This point was suggested by an anonymous referee.
Even though this issue is a very important one, we prefer do not analyze it in detail here, postponing its discussion in a future article. However, we would thank an anonymous referee for discussing with us the role of blind faith in scientific research.
Thanks to an anonymous referee for his/her comments in which s/he emphasizes these points.
And what about the above-mentioned epistemic inconsistency between “~K” and “~(K ~ K)”? This epistemic inconsistency holds in KT4-B4 + (14) + (14*) as well. But it must be interpreted differently in the light of Fine’s paper. Again, this means that we cannot know that we are ignorant about the ignorance of a given sentence “φ”, but this does not entail that we cannot know that we are ignorant of our ignorance in general.
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Acknowledgements
We are grateful to three anonymous referees for having read and commented on an earlier version of this paper. We would like to thank also our colleagues from Synergia Research Group at the University of Urbino for their helpful comments on the first draft of this paper. In particular, we would like to thank Mario Alai, Adriano Angelucci, Stefano Bonzio, and Mirko Tagliaferri for their helpful suggestions. This work was supported by the Italian Ministry of Education, University and Research through the PRIN 2017 project “The Manifest Image and the Scientific Image” prot. 2017ZNWW7F_004.
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Fano, V., Graziani, P. A working hypothesis for the logic of radical ignorance. Synthese 199, 601–616 (2021). https://doi.org/10.1007/s11229-020-02681-5
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DOI: https://doi.org/10.1007/s11229-020-02681-5