A working hypothesis for the logic of radical ignorance

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).

1 Introduction

Insofar as knowledge is supposed to guide someone’s choices, it’s safe to claim that what people know or ignore has a great effect on their lives. At the same time, people not only have knowledge and beliefs, but are also, in some cases, aware of the fact that they either know (knowing what is known or know knowns) or ignore (knowing what is not known or know unknowns) something. While the latter phenomenon is rare, i.e., people rarely know that they do not know something, a related, though different, phenomenon is quite common. The phenomenon we are referring to is the so called “ignorance of ignorance”¹ According to this phenomenon, people often fall prey to a cognitive bias that makes their ignorance invisible to them and thus, they end up being ignorant about their ignorance.

As Dunning states,

People are destined not to know where the solid land of their knowledge ends and the slippery shores of their ignorance begin. It is perhaps the cruelest irony, the one thing people are most likely to be ignorant of is the extent of their own ignorance – where it starts, where it ends, and all the space it fills in-between. (2011, p. 250)

While there are different types of knowledge and many ways to make it visible, there are also several types of ignorance and different ways in which it might escape the subject’s consciousness. For example, while many instances of ignorance fall into the category of unknown unknowns, where an agent is not only ignorant about something but also about her/his state of ignorance, other instances of ignorance fall into the category of ignorance in disguise, where an agent is not only ignorant about her/his ignorance, but also mistakes his/her misbeliefs for valid knowledge, i.e. the ignorance is disguised by misbeliefs accounted as knowledge. Radical ignorance is exactly a phenomenon of this last type. It is very difficult to explore radical ignorance; nevertheless, the so-called Dunning–Kruger effect (Kruger and Dunning 1999) is an example of how such a phenomenon might manifest itself in everyday life.

Specifically, for any given skill, some people have more expertise and some have less, some a good deal less. What about those people with low levels of expertise? Do they recognize it? According to the argument presented here, people with substantial deficits in their knowledge or expertise should not be able to recognize those deficits. Despite potentially making error after error, they should tend to think they are doing just fine. In short, those who are incompetent, for lack of a better term, should have little insight into their incompetence—an assertion that has come to be known as the Dunning–Kruger effect (Kruger and Dunning 1999). This is the form of meta-ignorance that is visible to people in everyday life. (Dunning 2011, p. 260)

From psychology to education studies, passing through philosophy and many other disciplines, a plenitude of deep analyses of knowledge and ignorance have been put forward (Peels and Blaauw 2016; Peels 2017; Fine 2017); many of these studies share a definition of ignorance in terms of lack or absence of knowledge. In line with such works, even logicians have favored such a definition and thus they have subordinated the notion of ignorance to that of knowledge in monomodal logics. However, the formal accounts of ignorance available in the literature fail to adequately express the important properties of ignorance experimentally highlighted by Dunning and Krueger’s research.² Thus, a proper formalization of a notion of ignorance that is important for the epistemology of science is still lacking.

This paper aims to direct attention to this last point. Radical ignorance is not only meaningful from a psychological point of view, but it also has general epistemological relevance. Indeed, throughout the history of science the phenomenon has manifested itself quite often. An example might be helpful here. Aristotle believed that bodies fall with a speed proportional to their weight (call this proposition β). Philoponus, an Aristotelian commentator of the sixth-century A.D., already knew that this is a false belief, i.e., that β is false. However, Aristotle also believed that his belief about β was justified. Given the standard definition of knowledge we possess (i.e., that of justified true belief), Aristotle believed to know β (he believed that he had a belief about β, that β is true and that he was justified in believing it), but he didn’t actually know it. It is evident that this epistemological situation is extremely common in the history of science.

Someone could object³ that radical ignorance, in contemporary times, is no longer a tenable position in the epistemology of sciences. Therefore, this opacity of radical ignorance would never be at work in reality. This objection is not convincing for at least one reason. This reason is that many scientists and philosophers of science today yet share the lack of consciousness of their fallibilism that the scientists of the past had, as shown, for instance, by Doppelt’s (2014) “best theory scientific realism” position.

Furthermore, even if one accepts this objection, it is very difficult to model a fallibilist thesis without incurring into inconsistencies. If one investigates in more detail examples such as that of Aristotle and the falling bodies, one sees immediately that Aristotle is not only ignoring his ignorance, but, at the same time, he is misbelieving that he is believing something false. Hence in the regimentation of radical ignorance one must take into account also beliefs. Let us consider the following example: Aristotle believed many sentences, call them α, β, γ …. He also believed that one of his beliefs is possibly false. This, as it is known, is an example of the Preface Paradox (Makinson 1965). Thus, we reach an inconsistency, which, as it is well known, it is not easy to remedy.

Historical examples like the previous one seems to point to the conclusion that radical ignorance is not only useful for explicating psychological situations, but also to shed light on epistemological limitations. Indeed, we think that twentieth-century science clearly shows that, although modern science has established a good method to produce knowledge, not only there are plenty of known unknowns,⁴ but surely there are also sentences that we consider knowledge which we will very likely discover to be false in the future. This means that we are systematically ignorant about our ignorance. To be clear: such a situation is not a fault; indeed, it seems to be somehow intrinsic to scientific knowledge itself. From this, it should follow quite naturally that an epistemic logic must be able to account both for the phenomenon of unknown unknowns and for the radical ignorance phenomenon.

Note that from a logical point of view, there is a form of higher order ignorance which is not problematic. For example: Aristotle didn’t know that other galaxies existed (call this proposition φ); neither did he know that he didn’t know φ. In epistemic logic (where K stands for knowledge) one could represent this cognitive situation in such a way:

$$\varphi \wedge \sim K\varphi \wedge \sim K \sim K\varphi$$

In the epistemic logic KT4, as we will see, “~K ~ Kφ” is not problematic, in line with our intuitions. The above formula is problematic only in the epistemic logic KT5 because “~K ~ Kφ” is contradictory with respect to “~Kφ”.

However, reconsidering the epistemological situation regarding Aristotle and the falling bodies, such a situation is much more difficult to represent, even employing languages such as KT4, i.e., we cannot say that Aristotle does not know that he ignores the falling bodies law. More precisely, normally ignorance is defined as not knowing whether “φ” or “~ φ” is true. From this kind of ignorance one can define the ignorance of ignorance, i.e. “IIφ”.⁵ But the problem is, as emphasized by Fine (2017), that “IIφ → IIIφ”, i.e. the ignorance of ignorance is opaque to our knowledge. Therefore, the aim of this paper is to try to understand what principles must be accepted by a logic of ignorance that wants to take also into account phenomena such as that of radical ignorance.

In what follows we try to establish a reasonable framework for the logic of knowledge and belief (Sect. 2), and we discuss ignorance, and in particular the radical ignorance phenomenon, in that framework (Sect. 3). In Sect. 4 we move on to investigate a possible integration into the classical framework for the logic of ignorance.⁶ We conclude the paper with some final remarks on Kit Fine’s problem about reiteration of ignorance (Sect. 5).

2 A classical framework

As the reader might already know, the logic of the operator “K” can be represented as a modal logic with possible worlds (scenarios) semantics. In particular, the possible worlds are those epistemically compatible with the knowledge possessed in the reference world. In this semantics, the interpretation of “Kφ” is “φ is true in all epistemically accessible worlds.” Moreover, the epistemic accessibility relation R_K is reflexive, since

$${\text{T}}.{\qquad}K\varphi \to \varphi$$

(1)

ought to be valid. At the same time, “R_K”, in our context, is not symmetric. Indeed, if w_j is epistemically accessible from w_i, then everything that is true in w_j is compatible with the knowledge in w_i; but in w_j there might be knowledge that make w_i incompatible with w_j. Finally, the transitivity of “R_K” is justified, since if w_j is accessible from w_i and w_k is accessible from w_j, nothing prevents w_k being accessible from w_i. One of the consequences of the transitivity of R_K is the validity of the formula “Kφ → KKφ”. Such a principle, sometimes dubbed “KK”, has been challenged and hotly debated in the literature.⁷ Indeed, it is quite common to think that it would be easy to find psychological counterexample to it. Consider, for instance, a student that had prepared during four months for a logic exam. Even though the student is very clever and naturally inclined toward the subject, one hour before the exam the student has a strong feeling of knowing nothing. Since studying for four months is more than enough for knowing the topics of the exam, we can conclude that the student does not know his/her knowing. Therefore, there is at least one case in which KK is false. Thus, assuming KK, as is done in this paper, requires a brief discussion on the notion of knowledge employed.

“Knowledge”, in this paper, is a normative term in the sense that such a word does not refer directly to something that we find in the world. On the contrary, we must establish what characterizes knowledge. The epistemic logic proposed by Hintikka (1962) is properly an attempt of determining which are the rules that hold, in general, for the notion of knowledge. It follows that only if someone fulfills these rules can s/he know something. Therefore, what psychologically happens to real subjects is not so relevant here. One may of course reply to such a perspective claiming that if the chosen definition of knowledge is too strong, then nobody would be a knower. But the reader ought to bear in mind that the notion of knowledge here employed is not a scientific (descriptive/explicative) concept. The way in which we are using the word “knowledge” does not distinguish actual knowers from fake ones; rather, its usage is convenient to establish an ideal toward which we must aim. Moreover, we explicitly admit that in different epistemic contexts we can use diverse—probably weaker—notions of knowledge. Indeed, that is precisely why we think that there must be different epistemic logics useful in diverse epistemic situations.⁸ Nonetheless, Hintikka’s (1962) intent was instead to establish which rules govern the ideal notion of knowledge; and it is precisely such a perspective that we are following in our work.⁹ For these reasons, psychological examples, as that of the scholar of logic, are not relevant for us.

Coming back to the regimentation of the “K” operator, we can say that the modal logic of knowledge is KT4¹⁰ (also known as S4). We note that if R_k is Euclidean,¹¹ in our system it would hold:

$$\sim K\varphi \to K \sim K\varphi$$

(2)

and the logical system would become KT5 or S5. In KT5 the knowers know if they don’t know something: in the world w_j accessible from w_i the knower cannot find any possible knowledge that precludes her/him to come back to w_i. Thus, if the accessibility relation is transitive and Euclidean, it would be symmetric as well. KT5 implies that we know when we don’t know something.

As already said, we take “knowledge” to be a normative notion, namely a concept useful in understanding the epistemology of science. Therefore, the question to which KT4 is the answer is ‘which rules must fulfill a set of sentences to be logically defensible?’ (Hintikka 1962, § 2.6). “Logical defensibility” is a necessary condition for a set of scientific sentences. The, by now common, experience in science of misbelieving suggests a notion of a so to speak “epistemological defensibility”, which should account for the phenomenon of being radically ignorant about our ignorance. One way to state our main thesis, then, is that a prerequisite of scientific knowledge is the presence of radical ignorance; but in KT5 this cannot happen, since each time we are ignorant about something, we know it. Therefore, we will assume KT4 as our standard model.

Since between KT4 and KT5 there are many intermediate systems, let us consider briefly some axioms that one might wish to add to KT4¹²:

$$\sim K \sim K\varphi \to K \sim K \sim \varphi$$

(3)

$$K(K\varphi \to K\psi ) \vee K(K\psi \to K\varphi )$$

(4)

$$\varphi \to (\sim{K}\sim{K}\varphi \to {K}\varphi )$$

(5)

(3) and (5) are clearly at odds with our notion of radical ignorance. Indeed, (3) expresses that being ignorant about ignorance entails knowing to be ignorant, whereas (5) says that if “φ” is true then being ignorant about the ignorance of “φ” entails the knowledge of “φ”. Therefore, they cannot be accepted in our system. (4) instead says that given every couple of sentences “φ” and “ψ”, either one knows that the knowledge of “φ” implies that of “ψ”, or vice versa. Even if (4) might seem prima facie compatible with our concept of ignorance, it is not; indeed, if one adds (4) to KT4 then (3) holds. Since (3) is incompatible with our notion of ignorance, we must reject (4) as well.

In order to create an adequate framework for talking about the phenomenon of radical ignorance, it seems to us, as we said, necessary to introduce the logical concept of belief as well, since disbelieving is an important component of radical ignorance. The desiderata of such a concept are the following. First, we interpret “Bφ” as “φ” is true in all doxastically accessible worlds. While the doxastically accessible worlds do not have to be the same as those which are epistemically accessible, the reverse,¹³ however, must hold true. This point deserves more attention.

It is quite obvious that if one knows “φ”, then, at the same time, s/he believes “φ”. Therefore:

$$K\varphi \to B\varphi$$

(6)

(6) means that if in all epistemically accessible worlds “φ” is true, then in all doxastically accessible worlds “φ” is true as well, but (6) doesn’t constrain the nature of a given accessibility relation.

Taking in consideration the interaction between the two accessibility relations, one have to investigate the following formulas:

$$K\varphi \to BK\varphi$$

(7)

$$B\varphi \to KB\varphi$$

(7*)

(7) undoubtedly holds in a scientific context; it determines that if a world is doxastically accessible it is epistemically accessible as well. This holds because if “Kφ” is true in all doxastic accessible world, then all doxastic worlds must be epistemically accessible. Since we have accepted the KK principle, that is if one knows, then one knows of knowing, it must hold also that if one knows, then one believes to know.

(7*) is much more problematic. According to such a formula, given a structure of worlds, if the relation is epistemically accessible, then it must be also doxastically accessible. The reason of doubtfulness of (7*) is that in the epistemically accessible worlds one must know all that s/he knows in the reference world; but in these worlds s/he could also know something more, and this new knowledge could go against his/her beliefs (Hintikka 1962, pp. 51–53). Take for instance the case of the cardinal Roberto Bellarmino: he knew Kepler’s Laws, yet he didn’t believe heliocentric system theory. For him a world in which he knows Kepler’s laws and the laws of gravity is epistemically accessible. And in this world he would know as well that the best description of the solar system would be the heliocentric one. But remember that Bellarmino didn’t believe in the heliocentric system theory; therefore for Bellarmino this world was epistemically accessible but not doxastically accessible.

To sum up, for the moment we refuse (7*), but we will come back to this issue later.

We move now to another peculiarity which differentiates B from K.

The doxastic relation of accessibility “R_B” is not reflexive, since “Bφ → φ” does not hold.¹⁴ Finally, it is reasonable to assume that the distributivity of “B” on the implication and the transitivity of the R_B relation hold¹⁵; as a consequence, the modal system of belief is B4.

Note that even B is not a psychological phenomenon. Indeed, in our perspective, B is what a scientist believes with a partial justification¹⁶ Here a much longer discussion would be necessary, but we are not framing the psychology of scientific research—though an interesting topic—but are looking for an epistemic formalization of the rules governing scientific knowledge.

3 Ignorance in the classical frameworks

In the previous section, we have presented a sketch of a (hopefully) not too controversial framework for knowledge and belief, useful for regimenting the notion of ignorance.

The first source for our approach to model the notion of ignorance will be Hintikka’s (1962) book. It might thus be appropriate to spend a few words explaining how he himself handles our topic and a couple of tentative modifications to his proposal; in particular, we will discuss in this paragraph the logics proposed by Hoek and Lomuscio (2004a, b) and Steinsvold (2008)¹⁷ before outlining (in §4) our formalization of radical ignorance.

Hintikka (1962, p. 3) distinguishes between “a does not know that φ” (“~K_aφ”) and “a does not know whether φ” (“~K_aφ ∧ ~ K_a ~ φ”).¹⁸ It seems that according to Hintikka, only the latter explicates the notion of ignorance; indeed, he (1962, p. 12) formalizes ignorance as “~Kφ ∧ ~ K~φ”. Such regimentation has become the standard one of the logical literatures on ignorance.¹⁹ Moreover, Hintikka (1962) distinguishes between “virtual”, “epistemic” and “doxastic” implication. Roughly speaking, virtual implication is captured by the standard material implication, whereas epistemic and doxastic implications between φ and ψ mean respectively that K(φ ∧~ψ) and B(φ ∧~ψ) are inconsistent. Hintikka (1962, p. 106) also points out that, although:

$$K\varphi \to KK\varphi$$

(8)

holds, “~Kφ” does not imply virtually “K ~ Kφ”. That might seem like good news for us, given our aim of framing a logic in which the concept of ignorance is not as tightly bounded as usual to that of knowledge. Indeed, that was precisely the reason for which in §2 we justified the dismissal of KT5 and other systems stronger than KT4: in KT5 “~Kφ → K ~ Kφ” holds, and partially similar formulas are derivable in systems intermediate between KT4 and KT5. At the same time, as Hintikka’s (1962, p. 106) emphasizes, in his system—similar to our framework—“~Kφ” implies epistemically “K ~ Kφ”.²⁰ From this it follows that:

$$K( \sim K\varphi \wedge \sim (K \sim K\varphi ))$$

(9)

is inconsistent; that is to say, it is impossible to know that one doesn’t know “φ” and doesn’t know his lack of knowledge of “φ”. Formula (9) is very important, since it emphasizes the impossibility of knowing the reiteration of absence of knowledge.

This propagation of absence of knowledge and ignorance, on one side, is good news for the notion of radical ignorance. On the other, instead, it is a problem, as already emphasized, since it seems that one cannot be aware of his/her ignorance, contrary to the well-known Socratic motto. We will see in the next section that there is an escape from this cul-de-sac.

Coming back to Hintikka, he instead considers the concept of radical ignorance to be an exclusively psychological phenomenon. Emphasizing what we previously said, since we want to flesh out what knowledge and belief must be, rather than representing the real situations in which they are involved, from our point of view the concept of radical ignorance is not just a psychological notion; indeed, it is an epistemologically relevant one. For such a reason, the epistemic inconsistency between “~Kφ” and “~(K ~ Kφ)” should be tamed, unless we find a different interpretation of this inconsistency.

Let us consider other two approaches to the logic of ignorance.

At a first sight, Hoek and Lomuscio’s (2004a, b) papers might seem to diverge from Hintikka’s book. They claim:

Note that by ignorance we do not mean the mere lack of knowledge, but something stronger. […] By state of ignorance about φ in the following we shall refer to a mental state in which the agent is unsure about the truth value of φ. So not only the agent does not know the truth value of φ but also that of ~ φ (Hoek and Lomuscio 2004a, p. 98, b, p. 2).

Hoek and Lomuscio’s papers are is, logically speaking, interesting, especially because in them they explicitly aim to define “ignorance as a first class citizen” (2004a, p. 99, b, p. 3). Moreover, they propose a logical system for ignorance (Ig) proving important meta-theorems for Ig. Despite their intention, their solution does not seem too far from the classical one. Indeed, they keep defining ignorance in terms of knowledge: an agent is ignorant about “φ” if he does not know “φ” and s/he does not know “~φ”, or using their words again: “formula Iφ is to be read as ‘the agent is ignorant about φ’ i.e. ‘he is not aware of whether or not φ is true’” (2004a, p. 99, b, p. 4). Such a definition is the same as that used by Hintikka (1962, p. 12), of which we spoke earlier.

From our point of view this system is too weak, since KK is not a valid principle of the system, which principle is essential to our epistemology. The authors are perfectly aware of this; therefore, they add to their system the following axiom:

$${\text{I}}4.{\qquad} \sim I\varphi \to \sim I \sim I\varphi ,$$

which in Hoek and Lomuscio’s system is equivalent to “IIφ → Iφ”. Then they go forward saying: “Can an agent be ignorant about one’s ignorance? Clearly, a negatively introspective agent cannot be. This would suggest the validity of the axiom:

$${\text{I}}5.{\qquad}I\varphi \to \sim II\varphi .\hbox{''}$$

(Hoek and Lomuscio 2004a, p. 105, b, p. 12)

To the contrary, our framework aims at keeping the possibility of radical ignorance. For this reason, surely the system Ig + I4. + I5. is not suitable for us. Therefore, we stay with Ig + I4. It must be stressed that a couple of axioms of this system are not very intuitive. But the main problem of their approach is that their definition of ignorance does not consider misbelieving, which seems an essential part of this concept. An advantage of Ig + I4. is that it is no longer possible to prove Fine’s propagation of ignorance, i.e. “IIφ → IIIφ”. But we will see that this problem is trivially solvable in KT4 as well.

Similarly, the Logic of Unknown Truths (LUT) and the subsequent logics of ignorance proposed by Steinsvold (2008) subordinates the concept of ignorance to that of knowledge. In these logics the black box “■” in fact stands for “φ ∧ ~ Kφ”; if the latter formula is true, and “φ → ~ K~φ” holds, then we can derive “~Kφ ∧ ~ K~φ”, which is again Hintikka’s definition of ignorance. Therefore, from our point of view, nothing new under the sun.

Therefore, let us go back to the above-mentioned framework, that is, KT4 and B4.

As underlined by Dunning (2011) and by our examples from history of science, one of the reasons for the invisibility of ignorance is the fact that ignorance is disguised by misbeliefs accounted as knowledge. Staying in the traditional framework, we propose the following two possible formalizations of radical ignorance:²¹

(i.) Agents are radically ignorant about “φ” iff they are ignorant about “φ”, but they believe they know “φ”. Formally:

$$I_{R} \varphi =_{df} (( \sim K\varphi \wedge \sim K \sim \varphi ) \wedge (BK\varphi \vee BK \sim \varphi )$$

(10)

Where I_Rφ should be read as “φ is radically ignored”.

However, we believe that neither “BKφ” nor “BK ~ φ” capture properly the concept of misbelief, because “to misbelieve” includes “to believe something that is wrong”, “to believe wrongly”, and “to hold an erroneous belief”. Therefore, we propose the following second formalization:

(ii.) Agents are radically ignorant about “φ” if they do not know both “φ” and “~φ”, and either they believe “φ” but it is the case that “~φ”, or it is the case that “φ” but they believe “~φ”; once again formally:

$$I_{R} \varphi =_{df} (( \sim K\varphi \wedge \sim K\sim\varphi ) \wedge ((B\varphi \wedge \sim \varphi ) \vee (B \sim \varphi \wedge \varphi ))$$

(11)

It should be noted that the two doxastic parts of the last formula (i.e. “(Bφ ∧ ~ φ)” and “(B ~ φ ∧ φ)”) are reminiscent of Moore’s paradox (Moore 1993). We wish to remember that these formulas are not per se contradictory; they become so only if we consider a first-person agent and we place before those formulas either a knowledge or a belief operator, i.e., these formulas are doxastically and epistemically inconsistent, using Hintikka’s (1962) terminology. This fact properly conveys the absurdity communicated by Moore’s paradox, still leaving open the logical possibility of such a strange doxastic attitude, typical of ignorance followed by misbelief.

It is also important to note that definition (11) is redundant. As correctly suggested to us by an anonymous referee, we can deduce “(~ Kφ ∧ ~ K~φ)” from “((Bφ ∧ ~ φ) ∨ (B ~ φ ∧ φ))” and the following formulas: “(Kφ → φ)”, “(Kφ → Bφ)” and “~(Bφ ∧ B ~ φ)”. This means that in (KT4-B4) we can prove that:

$$((B\varphi \wedge \sim \varphi ) \vee (B \sim \varphi \wedge \varphi )) \to ( \sim K\varphi \wedge \sim K \sim \varphi )$$

It is also provable that the vice versa does not hold.

Therefore, our definition of radical ignorance is equivalent to the definition of misbelief:

$$I_{R} \varphi =_{df} ((B\varphi \wedge \sim \varphi ) \vee (B \sim \varphi \wedge \varphi ))$$

(11*)

Even though the framework (KT4-B4) seems to capture one aspect of radical ignorance, actually it does not. Indeed, in this framework, formulas (10) and (11) are not derivable. For, by definition, in KT4-B4:

$$II\varphi =_{df} ( \sim K( \sim K\varphi \wedge \sim K \sim \varphi ) \wedge \sim K \sim ( \sim K\varphi \wedge \sim K \sim \varphi ))$$

(12)

Note that (12) is based on Hintikka’s standard definition of ignorance.

It is possible that either (10) or (11) hold, although the right-hand side of (12) does not; that seems to be a consequence of the fact that (12) is a condition of nested “~K”, whereas in (i.) and (ii.) this denied operator appears only once, and in our system KT4-B4 is not possible to prove reiteration of “~K”. This situation clarifies why the classical framework is not able to precisely regiment the notion of invisible ignorance we are looking for. It might be the case that the solution could come from a more careful usage of knowledge and belief. Such a consideration leads us to ask whether it is possible to investigate a formal logic for ignorance that includes misbelieving.

4 A sketchy investigation for new frameworks

Summing up what we have seen so far, we have understood that the formalizations of ignorance already present in the literature, even if they explicate, in a certain sense, the phenomenon of known unknown, fail to account for phenomena such as those that go under the label of Dunning–Kruger. These phenomena seem to deal with a more forceful ignorance, which we have called so far radical ignorance. As emphasized already, this kind of ignorance is not only a relevant psychological effect, but also an important epistemological requisite as well. Moreover, we have highlighted that it might be that the insufficiency in previous formalizations is a result of the fact that they are too hasty in reducing ignorance to knowledge and belief. Therefore, the general principles that a logic must maintain to account for a radical form of ignorance should consider not only the ignorance of ignorance phenomenon, but also the role that misbelief plays. Since misbelieving is an important part of radical ignorance, considering the relevance of epistemic habits in forming our knowledge should be important as well. All of this seems to point to a dynamic logic of ignorance and belief; however we will limit ourselves to consider a static aspect of radical ignorance, with the hope of dedicating a future work to a much more complete and systematic analysis. Our analysis here will be restricted, then, to examination of some principles that a logic for ignorance should embrace in order to account for those extreme cases in which radical ignorance is involved.

We express this kind of radical ignorance, which the other logical systems discussed cannot deal with, using the formula:

$$\left( C \right) \to II\varphi$$

(13)

Our job here will be to determine what the condition “C” is.

Remembering the bimodal framework KT4 and B4 mentioned earlier and the two possible conditions for ignoring ignorance presented in §3, let us consider again just clause (11*):

$$I_{R} \varphi =_{df} ((B\varphi \wedge \sim \varphi ) \vee (B \sim \varphi \wedge \varphi ))$$

Considering (11*) and (12) together, we suggest adding to the KT4-B4 bimodal framework the following axiom:

$$((B\varphi \wedge \sim \varphi ) \vee (B \sim \varphi \wedge \varphi )) \to (II\varphi )$$

(14)

Roughly speaking, (14) says that if we have a misbelief about “φ”, then we ignore our ignorance about “φ”, i.e., (14) says that radical ignorance entails second order ignorance.

Therefore, our proposal for C is “((Bφ ∧ ~ φ) ∨ (B ~ φ ∧ φ))”.

This is one of the most important claims in our paper.

Note that from (14) and the “blind faith” effect, i.e. “(KIφ ∧ Bφ)”, it is possible to deduce “φ”²²

Although blind faith can play a relevant role in some aspects of our lives, it is not a good thing for our system that belief can influence reality.²³ For this reason we have to add to our framework the following axiom:

$$\sim (KI\varphi \wedge B\varphi )$$

(14*)

For us, this is the best way to eliminate the problem.

However, we realize that one could move in another direction. We have previously refused (7*):

$$B\varphi \to KB\varphi$$

(7*)

because, as we have already argued, (7*) would require that all epistemically accessible worlds should be doxastically accessible as well. This is not reasonable. Even in scientific practice, we have a lot of beliefs that prevent us from accessing many possibilities, which, on the contrary, are epistemically accessible. In other words, the set of facts we know is much smaller than that of our rational beliefs. This implies that the set of our doxastically accessible worlds is much smaller than that of our epistemically accessible ones. On the other hand, if we accept (7*), we can easily prove that “blind faith” becomes impossible and therefore (14*) becomes superfluous.

In any case, the notion of belief employed in (14) is a bit different with respect to the standard one, since, if your belief is false, it brings you directly to second order ignorance.²⁴

Now a semiformal proof of the consistency of the system KT4 + B4 + 14 + 14* is in order.

Note also that (14*) can be written as:

$$\sim B\varphi \vee \sim KI\varphi$$

(14**)

Let us now consider (without loss of generality) a frame in which the doxastically accessible worlds are exactly the epistemically accessible ones. Let us suppose that the accessibility relation is reflexive and transitive. Let us imagine that φ is true in every world and “Kφ” and “Bφ” are true in the actual world. Then all axioms KT4 + B4 are true. The same holds for (6) “Kφ → Bφ” and (7) “Kφ → BKφ” as well. Moreover (14) is trivially true because the antecedent of the conditional is always false. And the same holds for the second term of the (14**) disjunction. Therefore, this very trivial model shows that our system is coherent.

We could reformulate (14) in the following way:

(iv.) If a has a wrong belief about “φ” and a is ignorant whether “φ” as well, this means not only that a is ignorant about “φ”, but also that, besides the scenarios in which a is ignorant whether “φ” there are epistemically accessible (but not doxastically accessible) scenarios where a knows “φ” and others where a does not know “φ”.

It seems, to us at least, that (iv.) expresses quite well what we had in mind when we discussed the phenomena of radical ignorance. Therefore, any formal representation of radical ignorance should include something like (14).

As a concluding remark, we would like to finish our investigation with a few comments on the logic of “square ignorance” investigated by Fine (2017). As already emphasized, Fine shows that in KT4, the formula

$$II\varphi \to III\varphi$$

(15)

holds. This means that:

It is good to know when one is ignorant. But sometimes one is ignorant of one’s ignorance. One might think that, in that case, it would be good to know of one’s second order ignorance. But it can be shown, under quite plausible assumptions, that it is impossible to know of one’s second order ignorance. Second order ignorance is inevitably third-order ignorance. Likewise, third-order ignorance is inevitably fourth order ignorance and so on, indefinitely, through all the orders of ignorance. When one is second order ignorant one enters a black hole from which there is no epistemic escape. (Fine 2017, p. 1)

Our comment on such an important point would be the following: from Fine’s deduction, it clearly derives that “KIIφ” is inconsistent; but since (15) is a theorem of KT4, then

$$K(II\varphi \to III\varphi )$$

(16)

is a theorem as well. This means that in the system here sketched, which we could dub KT4-B4 + (14) + (14*), we have no knowledge of our second order ignorance and, such ignorance is, as pointed out by Fine, clearly undetectable; at the same time we know which are the conditions that promote second order ignorance and that not only can it exist, but that it is a mute phenomenon in scientific knowledge. In other terms, we think that the point raised by Fine does not compromise the work developed here: on the contrary, even if we cannot know what our second order ignorance is, we can still know that if there is such ignorance, it is invisible. Moreover, if we are sure that this ignorance exists, it is sufficient to make a pessimistic meta-induction from history of science. In a certain sense, the propagation of second order ignorance highlighted by Fine is an essential peculiarity of it. But we know that radical ignorance behaves in this way from a logical point of view. Therefore, in a strongly idealized epistemology, one can accept something only when s/he is completely sure of it. Else radical ignorance is triggered. “Real” knowledge is clearly different; but this is another story. Therefore, the importance of the formalization of the radical ignorance, and then of the second order ignorance, remains unchanged.²⁵

5 Conclusion

Philosophers seem to be interested mostly in epistemic logics, in which an agent is highly idealized in the sense that their abilities are bounded only by logical and/or mathematical constraints. Therefore, it is reasonable that philosophical approaches to epistemic logic prescind from empirical evaluations of the cognitive limits and deficits of specific empirical subjects. But this does not imply that these approaches should also ignore phenomena such as those brought to light by Dunning and Krueger. Indeed, such discoveries should be an important stimulus for a better understanding of the structure and the limits of human beings’ epistemic status and, therefore, should be reflected upon during the attempt to formalize knowledge. The Dunning-Krueger effect directed our attention towards the notion of invisible ignorance and, since the formal system already present in the literature is not able to account for it, motivated our interest in its formalization. But the phenomenon of radical ignorance is very important also in epistemology of sciences, where the condition of misbelieving is very common. This is an even more important reason, with respect to the Dunning–Kruger effect, to have a good regimentation of radical ignorance. Although this paper does not present a ready formal system for radical ignorance, we think that having explored the reasonable modal prerequisites of such a notion and conducted an analysis of the previous contributions on this topic, we have provided a solid ground for future developments. Finally, we are confident that these reflections could also be useful for stimulating formal research on epistemic fallibilism (Rescher 1984) in philosophy of science and on ignoring machines in theoretical computer science, along the lines of the topics explored, for instance, in Aldini et al. (2016).