A Logical Modeling of Severe Ignorance

In the logical context, ignorance is traditionally defined recurring to epistemic logic. In particular, ignorance is essentially interpreted as “lack of knowledge”. This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of φ that stems from an unfamiliarity with its meaning. Contrarily to this trend, in this paper, we introduce and investigate a modal logic having a primitive epistemic operator I, modeling ignorance. Our modal logic is essentially constructed on the modal logics based on weak Kleene three-valued logic introduced by Segerberg (Theoria, 33(1):53–71, 1997). Such non-classical propositional basis allows to define a Kripke-style semantics with the following, very intuitive, interpretation: a formula φ is ignored by an agent if φ is neither true nor false in every world accessible to the agent. As a consequence of this choice, we obtain a type of content-theoretic notion of ignorance, which is essentially different from the traditional approach. We dub it severe ignorance. We axiomatize, prove completeness and decidability for the logic of reflexive (three-valued) Kripke frames, which we find the most suitable candidate for our novel proposal and, finally, compare our approach with the most traditional one.


Introduction
The study of ignorance is certainly as old as the study of knowledge; however the formal study of the logic of ignorance is still a young area of research. In the epistemological studies of ignorance the standard view is to define it as lack of knowledge (see for example the debate in [29][30][31][32]). In logic, it is not easy to reconstruct this tradition (see [14,17] and [23]). However, in our view an important step in the history is Hintikka's seminal work [26], where he distinguishes two notions of lack of knowledge relative to an agent, namely "a (an agent) does not know that ϕ" (ϕ ∧¬K a ϕ) and "a does not know whether ϕ" (¬K a ϕ ∧ ¬K a ¬ϕ). Such regimentations have become standard in the logical literature on ignorance [14,17,23]. Throughout this article, we will refer to the standard view by the expression "ignorance as lack/absence of knowledge". In particular, we will use the expression "whether view" to address the second notion of lack of knowledge, i.e. "a does not know whether ϕ". 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 [2,17,25,33,34] and the standard view in the literature describes ignorance in terms of lack, or absence, of knowledge. Therefore, it is not surprising that this is also the standard view in the logical treatment of ignorance. However, in more recent times, van der Hoek and Lomuscio [44] introduced a modal logic (Ig) where ignorance is modeled by a primitive modal operator, unrelated to (lack of) knowledge. The spirit behind Ig is expressing "ignorance as a first class citizen" [44, p.3]. However, despite their intention, their solution does not seem too far from "not knowing whether". Indeed, in their semantics for the operator I -for ignorance -an agent ignores ϕ if s/he has access to two (different) worlds, where ϕ is evaluated differently (true in one and false in the other). In their own words (again): "[the] formula Iϕ is to be read as the agent is ignorant about ϕ, i.e. s/he is not aware of whether or not ϕ is true ". The semantics of I reflects that of absence of knowledge, with the only difference that Ig "can not speak" about knowledge.
Similarly, the Logic of Unknown Truths (LUT) and the subsequent logics of ignorance proposed by Steinsvold [40] subordinate 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 also ¬Kϕ ∧ ¬K¬ϕ holds, which is again the "whether view" of ignorance.
Following the research trend opened in Fano and Graziani [15] (see also [1]), this article intends to discuss the fact that lack of knowledge is just one way to look at ignorance and, taking up van der Hoek and Lomuscio's challenge, to introduce a logic which addresses the purpose of defining "ignorance as a first class citizen". In this paper, after discussing the consequences of defining ignorance as lack of knowledge (in the epistemic logic S 4 ), we introduce and investigate a modal logic having a primitive epistemic operator I, modeling ignorance. In particular, the idea we have in mind is that of modelling a type of content-theoretic ignorance, so to say an ignorance of something that stems from an unfamiliarity with its meaning, i.e. a severe notion of ignorance that implies a lack of awareness 1 with respect to a subject-matter. In our view, this type of ignorance constantly affects the practice of science. For instance, consider the following situation: Max Planck, in approaching the black body radiation problem, knew that, in the theoretical predictions of the black body, there was a divergence for high frequencies, in contrast with experimental data. However, he did not simply ignore which physical phenomena constituted the cause, but, more importantly, he did not have any idea (was ignorant) of what could be a bundle of causes. In logical terms, it is not merely the case that Planck does not know the truth value of a physical statement (that could be the cause), but he does not know which kind of event could be a cause. In other terms, when thinking about severe ignorance, we have in mind situations where scientists are ignorant of the bundle of causes that might be at the root of a phenomenon. Contrarily, the "whether view" of ignorance appears related to the lack of knowledge of single agents, such as, for instance, a physical statement that is, perhaps, known in the community of trained physicists but, possibly, ignored by a non-physicist or a first-year student. To achieve the goal of modeling severe ignorance, we base the semantics of our (modal) logic on the presence of a third truth-value, whose behaviour is infectious, as severe ignorance ultimately is. Returning to the example about Planck's ignorance, the infectivity of his ignorance depends on the fact that every scientific issue whose content is theoretically connected to the explanation of the black body is ignored severely at the same way that the explanation is. The most natural examples of infectious logics are the so-called weak Kleene logics, which can be intuitively introduced via a matrix where truth-values {0, 1} are joined by a third truth-value 1/2 whose behaviour is infectious in the sense that a complex formula ϕ is evaluated to the third value 1/2 whenever any of its atomic formulas is evaluated to 1/2 (independently of the structure of ϕ). Our modal logic will be essentially constructed following the ideas of the modal logics based on (one of the) weak Kleene logics introduced by Prior [36] and Segerberg [39]. Our philosophical approach keeps fixed the classical account that ignorance, as well as knowledge, is an epistemic notion and, for this reason, the logical modeling we primarily purse is an epistemic (modal) logic, whose privileged semantics is a relational one (Kripke-style). As a byproduct of our analysis, we discover that the non-classical propositional basis chosen (Bochvar external logic) indeed already incorporates (some) connectives that can be interpreted as modalities, to be used (also) for the formal representation of severe ignorance. Therefore, we will highlight the coincidence between the Kripke-style interpretation of the modality for ignorance and that of one connective in the enriched language of Bochvar logic.
The paper is organized into four parts: in Section 2, we introduce the standard (logical) approach to ignorance as "lack of knowledge". In Section 3, we outline the key features Bochvar external logic which we will use in order to give a modal approach to severe ignorance. In Section 4, it is introduced the logic SI of severe ignorance; an axiomatization with relative completeness is proved in Section 4.2. We conclude the paper with Section 5 where we make some remarks on the validity of certain formulas relevant to capture a severe notion of ignorance, and compare the differences between the standard view and the proposed logic for severe ignorance.

Ignorance as Lack of Knowledge
As mentioned, the traditional logical approach to ignorance is based on the idea of defining ignorance as "lack of knowledge" (see [17] and [12]). This translates ignorance into a modal operator in the epistemic logic S 4 defined as follows: where K stands for the knowledge operator. It follows from Eq. 1 that, in the standard Kripke-style semantics for S 4 , the formula Iϕ is true in a world w (under a certain evaluation v, in symbols v(w, Iϕ) = 1) if and only if there exist two worlds w , w related to w, such that ϕ is true (false, respectively) in w (under v) and ϕ is false (true, resp.) in w (under v). In words, an agent ignores a formula ϕ, in a world w, if (and only if) s/he has access to two worlds each of which assigns a different truth value to ϕ. Roughly speaking, "do not knowing whether ϕ -ignoring ϕ according to the "whether view" -means seeing (at least two different) worlds where ϕ is assigned with different truth-values. We might say that this view models ignorance as a truth-theoretic notion: "ignoring whether ϕ" is translated as "being unsure" about the truth value of ϕ, due to the existing conflict of evaluation in the related worlds.
It is useful to underline that the semantics for the ignorance modality I is the same, as above, also in the logic Ig, introduced by Van der Hoek and Lomuscio (see [44,Definition 2.1]) with the aim of treating ignorance as a primitive notion, not subordinated to knowledge. Indeed, the main difference, with respect to the "whether view" approach, is that, in Ig, ignorance is not defined as "lack of knowledge" as Ig does not contain any primitive knowledge operator: it is modeled via the primitive operator I.
We wonder whether modeling ignorance as lack of knowledge (or via I in Ig) is the only way to logically address the notion of ignorance. Far from saying that it is not the correct way to analyze the concept, we simply claim that "lack of knowledge" is only one way to approach ignorance, whose features are exemplified by the logical laws in which I actually occurs. We recap the logical laws and the notable failures involving I in the following remarks.

Remark 1
It is immediate to check that the following formulas are logical truths in S 4 -where I is defined according to Eq. 1:

Remark 2
The following formulas do not hold, in general, in S 4 -where I is defined according to Eq. 1: where evaluation v is defined as follows: p ∈ w, p ∈ s and q ∈ w, q ∈ s. It then follows that v(w, Ip) = 1, v(w, Iq) = 1. On the other hand, v(w, p ∧ q) = v(s, p ∧ q) = 0, thus v(w, I(p ∧ q)) = 0.
Intuitively, to falsify a) it is sufficient to consider a model with two different related worlds, each of which makes one formula true and the other false, respectively. In this way, each formula is ignored but the conjunction is not, since is false in every world.
We are convinced that the set of formulas listed in Remarks 1 and 2 -although they might not constitute an exhaustive list -tells something relevant about the notion of ignorance that the supporters of the "whether view" had in mind (more detailed comments on this can be found in Section 5). Let us analyze, through some examples, the applicability (as well as limits of applicability) of this interpretation of ignorance.
Suppose that Magnus and Jan 2 are about to play a single chess match. It is plausible to think that a rational agent ignores (does not know) whether Magnus is going to win (although it is very unlikely to happen, he might lose or the match could end in a draw); similarly, s/he ignores whether Jan is going to win. On the other hand, our rational agent does not ignore whether Magnus and Jan is going to win, as s/he knows that the same chess match can not have two different winners. This shows a case in which the ignorance of two conjuncts does not translate in the ignorance of their conjunction, as it happens to be the case in S 4 (see Remark 2-a).
Observe, however, that ignorance as lack of knowledge behaves according to the principle that ignoring a conjunction implies ignoring both the conjuncts and the disjuncts (Remark 1), which shades some confusion between "and" and "or" when referring to notions that are ignored.
Nevertheless, we believe that, sometimes, lack of knowledge is understood in a way which is not exemplified by the behaviour of I in S 4 (and Ig). We try to clarify what we mean, giving some examples relative to the behaviour of I with respect to the conjunction.
Suppose that one of the authors of this paper has just concluded to examine a student, who aimed to pass his/her exam in modal logic. During the exam, s/he was asked to answer some questions (obviously, in a finite number!), each of which with the precise goal to verify whether s/he is ignorant -hopefully, is not ignorant -of the main topics which, together, form the program of the entire course. Unfortunately, due to her deficient answers, the examiner has collected enough evidence to conclude that s/he is ignorant of all the main topics, say ϕ 1 , . . . , ϕ n , characterizing the course. The rational examiner is so brought to conclude that the student is ignorant of the whole subject of the exam, which can be exemplified as the conjunction ϕ 1 ∧ · · · ∧ ϕ n , and thus can not pass the exam. In other words, s/he is ignorant of the program ϕ 1 ∧ · · · ∧ ϕ n of the exam. It seems reasonable to think that the above exemplified notion of ignorance is indeed lack of knowledge (the examiner is ultimately testing if the student "knows ϕ 1 , . . . , ϕ n ") and it seems reasonable also to think that the ignorance of each of the statements ϕ 1 , . . . , ϕ n implies the ignorance of the conjunction ϕ 1 ∧ · · · ∧ ϕ n (how could this not be the case?!). However, ignorance as lack of knowledge, modeled in S 4 , and closure with respect to conjunction can not stand together. Another weakness regarding the standard view (as discussed in recent literature, see [22,28]) is that the so called Factivity Principle (usually intended relative to knowledge as Kφ → φ), does not work in the standard view framework, i.e. it does not hold that if an agent ignores φ then φ is true. This fact is also highlighted in our Remark 2, where we prove that factivity of ignorance does not hold in S 4 (in contrast with the factivity of knowledge which clearly holds).
It is also possible to design other examples allowing us to stress that there are cases where ignorance is severe and does not coincide with lack of knowledge. Let us consider the discovery, happened at the beginning of November 2021, of the new Omicron variant of Coronavirus. The group of South-African scientists who isolated the variant communicated immediately their discovery, however it is reasonable to think that the sentence "Omicron is a variant of concern" was ignored by everyone at the time (and perhaps in the following days). This kind of ignorance is severe (in our sense), since it is natural that it spreads over sentences containing the previous. For instance, also any implication of the form "if the Omicron variant is of concern then there will be more deaths due to it" is genuinely to be ignored. This example seems to be convincing on the infectiousness of severe ignorance. More precisely, the lesson to learn from the above discussion is that the notion of ignorance is more subtle and problematic than it might appear at first look. Modeling it as "lack of knowledge" is surely a possibility, which has both qualities and flaws, depending on the context of applicability.
The aim of the present work is to propose a logical modeling of severe ignorance, a notion that differs from standard lack of knowledge ("whether view"). This change of perspective significantly impacts on the formulas holding/not holding in this new logic with respect to S 4 (see Section 5 for a comparison and further discussion). Indeed, when ignorance is conceived as severe, then the failure of certain formulas, such as (2) and (4) in Remark 1, comes with no surprise; similarly, it is not surprising that a formula like a) in Remark 2 holds in this new system. Intuitively, one could think that a way to address a severe ignorance is possible also in S 4 , by recurring to the so-called "second-order ignorance" [17], rendered by applying I twice to a formula. However, applying II does not validate the fact that ignoring two formulas implies ignoring their conjunction, as witnessed by the following.

Remark 3
The following formulas are not logical truths of S 4 : The same counterexample introduced in Remark 2 serves also for (1). Indeed, However, since v(w, I(p ∧ q)) = v(s, I(p ∧ q)) = 0 and there exists no world x ∈ W such that wRx and v(x, I(p ∧ q)) = 1, then v(w, II(p ∧ q)) = 0. A simple counterexample to (2) is given by the following. Consider a Kripke Evaluation is defined as follows: p, q ∈ {w, t, r} and q ∈ s, p ∈ s. It follows that v(s, I(p ∧ q)) = 1 and v(t, I(p ∧ q)) = 0, thus v(w, II(p ∧ q)) = 1. However, v(x, Iq) = 0, for every x ∈ W , thus v(w, IIq) = 0.
The possibility of nesting the modality I (i.e. having formulas such as IIϕ, IIIϕ, etc.), which is allowed in S 4 , as we just saw, presents also a remarkable disadvantage. Although Iϕ → IIϕ is not a theorem of S 4 , it is not difficult to check that the formula IIϕ → IIIϕ is a theorem. More in general, abbreviating with I n the n-times application of the modality I, in S 4 the formula I n ϕ → I n+1 ϕ holds, for n ≥ 2. This quite problematic phenomenon is usually referred to as the black hole of ignorance (see [17]).

Bochvar External Logic B e
Given a similarity type ν, the absolutely free algebra Fm of type ν over a countably infinite set X of generators will be called the formula algebra of type ν; its members will be called formulas. Members of X will be called (propositional) variables and referred to by the symbols p, q, . . . . We denote algebras by A, B, C . . . and the respective universes by A, B, C . . . We understand a logic (of type ν) as a pair L = Fm, L , where Fm is the formula algebra (of type ν), and L is a substitution-invariant consequence relation over Kleene's three-valued logics -introduced by Kleene in his Introduction to Metamathematics [27] -are traditionally divided into two families, depending on the meaning given to the connectives: strong Kleene logics -counting strong Kleene and the logic of paradox [35] -and weak Kleene logics, namely Bochvar logic [4] and paraconsistent weak Kleene -PWK in brief (sometimes referred to as Hallden's logic [24]).
The language L K e is significantly richer than L K and allows to define the socalled external formulas. 3 Intuitively, a formula α is external when it is evaluated to {0, 1} (which is the universe of a Boolean subalgebra of WK e ) from any homomorphism h : Fm → WK e . In other words, an external formula is one such that can not be evaluated to 1/2 (see [18, p. 208]).
Via J 2 , it is possible to define more connectives (which will be very useful for our analysis): Intuitively, connectives J 0 , J 1 , J 2 , J 3 allow to speak not only about a statement ϕ but also about its truthfulness, falseness and more. Bochvar  In words, B e is the logic preserving only the truth-value 1 ("true"). 4 The intuition behind external formulas is made precise by the following.

Definition 4 A variable p is open in a formula
Examples of external formulas are: A Hilbert-style axiomatization of B e has been introduced by Finn and Grigolia [18, p. 236]. In order to present it, let and α, β, γ denote external formulas. Axioms Deductive rule Observe that the axiomatization contains a set of axioms (A19-A29), which, together with the rule of modus ponens, yields a complete axiomatization for classical logic (relative to external formulas). Upon defining the notion of derivation ( B e ϕ) in the usual way, Finn and Grigolia proved weak completeness for B e . It is natural to wonder whether B e can be provided with a more synthetic Hilbertstyle axiomatization and/or with a different style axiomatization (natural deduction, Gentzen-style, etc.). Actually a stronger version of Theorem 6 can be proved (the details of the proof are displayed in the Appendix, where we also show that B e is algebraizable).

Theorem 7
B e ϕ if and only if |= B e ϕ.
Proof (⇒) By induction on the length of the derivation of ϕ (from ).

The Logic SI of Severe Ignorance
The logic of (severe) ignorance we are going to introduce consists of a modal logic, whose propositional basis is B e .
In the expanded language L I , we generalize the definition of covered variable (see Definition 4) as follows: a variable p is covered in a formula ϕ if it occurs in ϕ and all occurrences fall under the scope of J i , for some i ∈ {0, 1, 2, 3}, or under the scope of I. In other words, we are stipulating that formulas like Iϕ, Iψ, ... are external (for any ϕ, ψ ∈ F m I ), in the sense of Definition 5.
We introduce the logic SI as the one induced by the Hilbert-style axiomatization given by the following. The intuition behind the axiomatization is that ignoring ϕ implies that J 1 ϕ is true, i.e. ϕ takes the third value. Moreover, the rule [I] states that the formula Iϕ is derived from J 1 ϕ: intuitively, the ignorance of ϕ can be inferred from the assumption that J 1 ϕ is the case (i.e. semantically, J 1 takes the third value). Observe that this is different with respect to the rule of necessitation for standard modal logic (where ϕ can be inferred from any theorem ϕ).
By SI we intend the derivability relation of the deductive system defined by the above axioms and inference rules. We now introduce the intended Kripke-style semantics for the logic SI .

Semantics
The semantics of the logic of ignorance consists of a relational (Kripke-style) structure where formulas, in each world, are evaluated into WK e . We introduce these structures according to the current terminology adopted in many-valued modal logics (see, for instance, [11,19,20]).

Definition 9
A weak three-valued Kripke model M is a structure W, R, v such that: (1) W is a non-empty set (of possible worlds); (2) R is a binary relation over W (R ⊆ W × W ); (3) v is a map, called valuation, assigning to each world and each variable, an element in WK e (v : W × Fm I → WK e ).
Non-modal formulas will be interpreted as in B e , i.e. we assume that v is a homomorphism, in its second component, with respect to ¬, ∨, J 2 , 1, 0, 1/2. The reduct F = W, R of a model M is called frame. Notation: for ordered pairs of related elements, we equivalently write (w, s) ∈ R or wRs.
The semantical interpretation of the epistemic modality I in a weak three-valued Kripke model is given in the following.  The interpretation of the operator I is defined according to our intuition behind the notion of severe ignorance: a formula ϕ is ignored (in a world) when it is indeterminate (i.e. evaluated to 1/2) in all the related worlds.
Observe that, in accordance with the syntactic stipulations, we are establishing that is an external formula (in the sense that it can assume classical truth-value only). In other words, the recurse to the third truth-value is used only at the propositional level and does not affect modal formulas. Moreover, there is no special assumption behind the accessibility relation R in weak three-valued Kripke structures: it is simply interpreted as an epistemic accessibility relation. Accordingly, the rationale behind the interpretation of I is that a formula is being ignored in case it is neither true nor false -it is indeterminate -in every world s an agent has epistemic access to from w. Recall that the notion of ignorance we aim at modeling with this semantics is severe. To further clarify our goal imagine, for instance, the following situation. Charles Darwin was aware, in 1859, of the existence of a form of hereditariness; however he did not know exactly the functioning mechanism of such process. Moreover, in every scenario accessible to his mind in that period, the cause of hereditariness was not determined. So, if we think a formula ϕ exemplifying the mechanism of hereditariness, then, in 1859, it held that Darwin was (severely) ignorant of ϕ, because ϕ was indeterminated in every possibile scenario. Other mechanisms, although not entirely known, were not (severely) ignored by Darwin himself at that time. For instance, we can not say that he was ignorant of the so called "missing links". Although he could not find them, he had an idea of how to search them, thanks to the analysis of fossils.
Accordingly, it is false that a formula ϕ is being ignored (in a world w) when there is a (related) world where ϕ is either true or false.
We say that a formula ϕ is valid in a model M = W, R, v -we will write M |= ϕ -if, for every w ∈ W , v(w, ϕ) = 1. A comment on this choice is in order. The introduced semantics of I relies on the presence of the third truth-value 1/2 to be read as "indeterminate". In particular, severe ignorance, thought as a content-theoretic notion (in contrast with the truth-theoretic notion modeled by the standard view in S 4 ), is rendered thanks to the infectious behaviour of 1/2. For this reason, it is natural to take 1/2 as not designated, since the evaluation of a formula to 1/2 (in every related world) is a good reason for its ignorance.
We say that a formula ϕ is valid in a frame F = W, R (and write F |= ϕ) if it is valid in every model having F as frame. A frame (accordingly, a model) will be called reflexive if its accessibility relation is reflexive. From now on, we will write Kripke model instead of weak three-valued Kripke model. We define |= SI as the global modal logic on the class of all reflexive Kripke frames (see e.g. [11]), i.e. The above consideration is due to the peculiar behavior of the truth-value 1/2 in weak Kleene and gives already a gist of the severity of ignorance obtained via the introduced semantics of I. Indeed a (complex) formula ϕ is being ignored when a part of it (occurring open) is actually being ignored (as evaluated to 1/2 in every related world), independently of the logical form of ϕ (exceptions hold for external formulas).
The choice of defining the logic |= SI as that of all reflexive frames is mainly motivated by the fact that accessibility is interpreted in epistemic sense, thus is natural to think that every world is (epistemically) accessible to itself.
The following provides the behaviour of the epistemic modality I in |= SI .
Observe that the validity Iϕ → J 1 ϕ is strictly related with the reflexivity of the models. It is immediate to check that the formula is not valid in non-reflexive models (think, for instance, to a model with only one world with the empty relation). Indeed the formula characterizes the class of reflexive frames.

Proposition 13
Let F = W, R be a frame. Then F |= Iϕ → J 1 ϕ if and only if R is reflexive.
As noticed in [44], the essential feature of any notion of ignorance is captured by formulas (1) and (3) in Proposition 12. It is indeed very reasonable to think that an agent is ignorant about a formula if and only if is about its negation. Moreover, ignorance transfers from a conjunction to at least one constitutive part of it. Severe ignorance meets the minimal desiderata.
As a remarkable difference with S 4 (and Ig), in this new semantics for I, being ignorant of two (or more) formulas implies being ignorant also of their conjunction.
Not surprisingly, the converse (which holds in S 4 ) does not characterize severe ignorance (see Remark 14). It is indeed reasonable to think that being ignorant of a book (a conjunction of statements), does not mean to be ignorant of any single statements in the book, but, perhaps, some relevant parts of it. Moreover, (4) holds in virtue of the infectivity of the third truth-value.
We will discuss the significance of all the mentioned logical laws in Section 5. Some notable failures are collected in the following.

Remark 14
The following formulas are not valid in |= SI : In Section 5, we will argue that it is not a problem for the severe notion of ignorance not to have distribution of I over conjunction and implication (formulas (2) and (3)). On the other hand, since severe ignorance is here conceived as a content-theoretic notion, it is obvious to expect the failure of the factivity (5).

Completeness and Decidability
When no danger of confusion is occurring we will drop the subscripts SI and B e to .
The following result, whose form resembles a weakened version of the (classical) deduction theorem, can be proven for SI .
Proof (1). By induction on the length n of the derivation of ϕ from .

Remark 20
Observe that the content of Lemma 19 can not be simplified by deleting the second disjunct (as in [39,Lemma 4.9]) as, for instance, {J 1 ϕ, ϕ} is inconsistent but J 1 ϕ ¬ϕ.

Lemma 21
Let be a consistent set of formulas. The following are equivalent: (1) ϕ; (2) for every X ∈ X such that ⊆ X, ¬ϕ ∈ X and J 1 ϕ ∈ X.

Lemma 26
Let M = Y, R, v be a canonical model. Then, for every formula ϕ ∈ F m I and every X ∈ Y, the following hold: (1) v(X, ϕ) = 1 if and only if ϕ ∈ X; (2) v(X, ϕ) = 0 if and only if ¬ϕ ∈ X; Proof The claim is proved by induction on the complexity of ϕ. The basis follows from Definition 24. As for the inductive step, we show only the cases of ϕ = J 2 ψ and ϕ = Iψ, for some ψ ∈ F m I (the others are routine). As regards the former, suppose that ϕ = J 2 ψ, for some ψ ∈ F m I . For any valuation v (and any X ∈ Y), v(X, J 2 ψ) = 1/2 (in accordance with the fact that J 1 J 2 ϕ ∈ X), thus we only have to consider two cases: (a) v(X, J 2 ψ) = 1 iff v(X, ψ) = 1, thus, by induction hypothesis, ψ ∈ X and, by Lemma 18, Consider, first, the case v(X, ψ) = 0; by induction hypothesis, ¬ψ ∈ X and, since X is consistent, J 2 ψ ∈ X, thus by Lemma 18, ¬J 2 ψ ∈ X. In the second (sub)case, v(X, ψ) = 1/2, thus, by induction hypothesis, J 1 ψ ∈ X. Since X is maximal (and consistent) then ψ ∈ X, thus, by Lemma 18, J 2 ψ ∈ X, whence ¬J 2 ψ ∈ X. Consider now the case of ϕ = Iψ, for some ψ ∈ F m I . We only have to consider the following two cases. (i) v(X, Iψ) = 1 if and only if v(Y, ψ) = 1/2, for every Y ∈ Y such that XRY . By induction hypothesis, J 1 ψ ∈ Y thus, by Lemma 23, Iψ ∈ X.
We are now ready to prove (strong) completeness, i.e. that SI and |= SI are the same logic. (⇐) We reason by contraposition and suppose that SI ϕ; this implies that is consistent. Let X ⊆ X the set of maximal and consistent sets extending ( ⊆ Y , for every Y ∈ X ); observe that, by Lemma 21, X = ∅; in particular, there exists X ∈ X such that ⊆ X and ¬ϕ ∈ X or J 1 ϕ ∈ X. Consider now the canonical model M = X , R, v . By Lemma 26, M |= γ , for every γ ∈ . On the other hand, since ¬ϕ ∈ X or J 1 ϕ ∈ X, then v(X, ϕ) = 1, i.e. M |= γ , hence |= SI ϕ.
From now on, we will write SI to indicate both SI and |= SI (since they are equal). The completeness strategy applied insofar allows to prove decidability for SI (Theorem 30).
Let M = W, R, v be a model. We say that a model has cardinality n (with n ∈ N), if W has cardinality n (| W |= n).
We reason by contraposition and suppose that SI ϕ. Define the following relation on the set X : X ≡ Y if and only if, for all ψ ∈ Sub(ϕ), ψ ∈ X iff ψ ∈ Y . It is immediate to check that ≡ is an equivalence relation on X . Since ϕ ∈ Sub(ϕ), then clearly | X /≡ |≤ 2 n . Define the binary relation ρ on the set X /≡ (whose elements are denoted by [X], [Y ], [Z], . . . ) as follows: Consider the structure N = X /≡ , ρ, w , where w is defined as follows: for every [X] ∈ X /≡ . It is immediate to check that N is a model of SI. Moreover, let M = X , R, v be a canonical model: it is not difficult to prove that v(X, ψ) = w([X], ψ), for every X ∈ X and for every formula ψ ∈ Sub(ϕ) (the proof of this claim runs by induction on the length of the formula ψ). Since SI ϕ, then there exists X ∈ X and some v (in the canonical model M) such that v(X, ϕ) = 1 (this follows from the proof of Theorem 27). Then, by the previous claim, w ([X], ϕ) = 1 and the cardinality of the model N is at most 2 n .
As a direct consequence of Lemma 29 one gets:

Theorem 30
The logic SI is decidable.

Conclusion and Comparison with Other Approaches
We have introduced severe ignorance as a content-theoretic notion. In particular, we have focused on the logical modeling of such notion, assumed as primitive ("as a first class citizen"), i.e. disconnected from knowledge, via a modal logic based on a threevalued propositional logic. The intuition behind our proposal is that being ignorant of ϕ means that ϕ is indeterminate (is assigned with the third value 1/2) in all the n.d. stands for not defined worlds accessible to an agent. To the best of our knowledge, the unique existing system considering I as a primitive modality is the logic Ig, by Van der Hoek and Lomuscio [44]. However, as discussed in Section 2, in Ig the semantics of I coincides with the interpretation of ignorance as "lack of knowledge" in S 4 , although no (termdefinable) modality expressing knowledge can be defined in Ig. Being conscious of this relevant difference between Ig and the standard view in S 4 , we will identify them with respect to the behaviour of the modality for ignorance I in the following discussion.
We make a comparison, in Fig. 3, between SI and S 4 (and thus also Ig) in terms of logical truths explicitly involving I (all the listed formulas have been mentioned in the previous sections). The aim is to show the existing difference between approaching ignorance as lack of knowledge (standard view in S 4 ) and the type of content-theoretic ignorance analyzed here, according to the three-valued modal logic SI.
As already discussed in Section 2, SI and S 4 present remarkable differences, with respect to the behaviour of the modality I. Regarding, for instance, conjunctive statements, in our proposal, an agent who is ignorant of all the chapters of a book then is ignorant of the whole book (formula 2), which does not happen to be the case in S 4 . In the latter, perhaps, it does not make sense to express sentences like "ignoring a book". Indeed, one could say that "an agent does not know the content of a book", and not that "an agent does not know whether the content of a book". The converse implication (3) does not hold neither in S 4 nor in SI.
A remarkable difference distinguishes S 4 and SI relatively to the behaviour of I with respect to disjunctive statements, too. Severe ignorance is characterized by the principle that a disjunction is ignored if and only if one of the two disjuncts are ignored. This shall not appear strange in scientific contexts that inspire our notion of severe ignorance. Indeed, to make an example, Kepler, before investigating the astronomical data collected by Tycho Brahe, was ignorant of (as anyone else) the laws that today go under his name. After he discovers the first law, we might think that he still was ignorant of the others, and we might say that he also was ignorant of the disjunction (of the three laws), because such disjunction contains scientific terms (the second and third law) which Kepler could not imagine nor understand.
This does not happen to be the case in S 4 , because lack of knowledge is different from severe ignorance. In a toy example: suppose that I do not know whether my aunt yesterday went to the cinema but I know that s/he went out for dinner. Thus, I do not know whether s/he went our for dinner (only) or also to the cinema (maybe before or after cinema), but surely I do not ignore that s/he went out to dinner or to the cinema.
The distribution of I over implication (6) fails in both SI and S 4 . Remarkably, this gives the occasion, once more, to illustrate the sense of severe ignorance (in the scientific context). To exemplify the failure of (6), we might reasonably think that in 1914, Einstein was ignorant of the fact that the curvature of space-time is the cause of the anomaly affecting the perihelion shift of Mercury. At the time, the implication is scientifically ignored, however scientists were conscious of the anomaly in Mercury's perihelion. This is a good reason why I should not distribute over implication, in case it models severe ignorance.
Formulas 7 and 8 witness that the two logics have the same behaviour with respect to the relationship between first-order (Iϕ) and second-order ignorance (IIϕ). Not surprisingly, the latter implies the former but not the other way round.
Formula 9 is also in common between S 4 and SI. As we already commented in Section 4, it expresses the very intuitive principle that being ignorant of a conjunction implies being ignorant of at least one of the conjuncts, a principle that must be common (together with 1) to any possible notion of ignorance.
Formulas 10-12 witness the main difference due to the choice of different propositional basis. Indeed ϕ ∨ ¬ϕ (ϕ ∧ ¬ϕ, respectively) is true in every world, of every model of S 4 (false, respectively), hence can not be ignored. On the contrary, in a three-valued setting, those formulas can be indeterminate (when ϕ is indeterminate) and, consequently ignored. The validity of this formula tells us that the agent who is ignorant of ϕ is ignorant also that ϕ ∨ ¬ϕ coincides with the truth (something that is possible only in non-classical cases). This confirms that the notion of severe ignorance in SI stands quite far from lack of knowledge.
Formula 12 (which is not expressible in the language of S 4 ) states that ignoring a formula implies that the formula takes the value 1/2 and this characterizes the class of reflexive models (see Proposition 13). Formula 13 expresses the "factivity of ignorance" (the analog of the usual notion of factivity for the modality K for knowledge). The importance of this property for the notion of ignorance has been recently discussed in literature, where some authors look for logics of ignorance where it holds (see [28] and [22]). In the context of severe ignorance, as a content-theoretic notion, we are not surprised that the formula does not hold. However, we highlight that a modal approach, based on a three-valued logic, can be adopted also for logics of ignorance admitting the factivity, by choosing a different set of designated values ({1,1/2}).
Finally, both logics suffer the phenomenon that Fine [17] calls the "black hole of ignorance". In his paper, Fine shows that second-order ignorance and higher-orders of ignorance are tightly tied together: once second-order ignorance is present, an agent is doomed to the black hole of higher-order levels of ignorance. This is captured by formula 14.
We are conscious that much logical and epistemological work remains to be done and that also the choice of SI to model severe ignorance presents some difficulties. For instance, it could be argued that it is quite odd that being ignorant of a formula ϕ implies being ignorant of also ϕ ∧ ψ, when ϕ and ψ are totally unrelated formulas (this happens to be the case in SI). Nevertheless, the present exploration highlights that interesting aspects of ignoring are not successfully captured by the standard logical approach to ignorance, based on lack of knowledge. Interestingly, disconnecting ignorance from knowledge allows for the logical modelling of severe ignorance, a notion which is common in the everyday practice of science. We have decided to introduce a modal logic grounded on a peculiar non-classical propositional basis. A choice essentially motivated by the willingness of modeling a severe notion of ignorance. Clearly, many other options are available, in the realm of non-classical logics: a possibility that we leave for further research.
Proof By induction on the length of the derivation of ψ (from ).
Proof (⇒) By induction on the length of the derivation of ϕ (from ), by checking that axioms (A1)-(A29) are evaluated to 1 in every Bochvar algebra A and that the rule (MP) preserves this property.
(⇐) We reason by contraposition. Suppose B e ϕ and provide a counterexample to such inference by constructing the Lindenbaum-Tarski algebra. Let be the smallest set of formulas including and closed under B e (from now on we will simply write instead of B e ). For any pair of formulas, define ϕ ∼ ψif and only if ϕ ≡ ψ ∈ . Theorem 7 follows from Theorem 34 by observing that BA 3 is the quasi-variety generated by WK e ([18, Theorem 3.3]).
It is natural to wonder whether the quasi-variety of Bochvar algebras is simply an algebraic semantics for B e . Actually the relationship between B e and the class BA 3 is tighter. Recall that a logic is algebraizable with K as equivalent algebraic semantics (where K is a class of algebras of the same language as the logic ) if there exists a map τ from formulas to sets of equations, and a map ρ from equations to sets of formulas such that the following conditions hold, for any pair of formulas ϕ, ψ and set of equations E.
Examples of algebraizable logics include, among many others, classical, intuitionistic logic, all substructural logics and global modal logics. Not all logics though are algebraizable: examples of non-algebraizable logics can be found in the realm of Kleene logics, such as the Logic of Paradox (see [37]), Paraconsistent weak Kleene (see [6]) and Bochvar internal logic (see [7][8][9]). The above definition of algebraizable logic can be drastically simplified: is algebraizable with equivalent algebraic semantics K if and only if it satisfies either ALG1 and ALG4 (or, else ALG2 and ALG3). 6

Theorem 35
The logic B e is algebraizable with BA 3 as equivalent algebraic semantics.
The usefulness of the above result will be explored in a fore-coming paper, focused on a deeper understanding of the properties of Bochvar algebras [10].
Horizon 2020 research and innovation programme under the MSCA grant agreement No. 801370 and GNSAGA (Gruppo Nazionale per le Strutture Algebriche e Geometriche). Vincenzo Fano and Pierluigi Graziani's works were 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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