1 Introduction

Both ignorance and knowledge seem to require truth, at least on the face of it. Alan Hazlett writes,

We find it plausible that truth is a necessary condition on knowledge. Is truth likewise a necessary condition on ignorance? There seems to be a sense in which it is: you cannot be ignorant without there being some fact of which you are ignorant. The right kind of connection with some fact constitutes knowledge; the lack of such connection constitutes ignorance (2012, p. 465).

Despite its initial plausibility, this symmetrical treatment of ignorance and knowledge is incompatible with orthodox philosophical theories. In particular, the Standard View of ignorance must deny that ignorance entails truth. The Standard View holds that ignorance is merely the absence of knowledge (Le Morvan 2011, p. 336; Fine 2018, p. 4032). Since false propositions are not known, it follows that one can be ignorant that P when P is false. There is thus an asymmetry between the Standard View of ignorance and the orthodox conception of knowledge. Knowledge is generally thought to entail truth, but ignorance does not entail truth, according the Standard View.

This asymmetry is untenable. After some preliminaries (Sect. 2), I argue that the Standard View of ignorance is at odds with the claim that knowledge entails truth. In particular, if knowledge entails truth then we cannot explain away some apparent absurdities that arise from the Standard View of ignorance (Sects. 3, 4). I then discuss a modified version of the Standard View, which simply adds a truth requirement to the original Standard View (Sect. 5). Although this modified view allows ignorance to entail truth, I show that the two main arguments for the original Standard View fail to support this modified view.

2 Preliminaries

The claim that knowledge entails truth can be stated as

(KT) Necessarily, for every proposition P, S knows that P only if P.

(KT) is an important part of traditional theories of knowledge, as well as nontraditional views (e.g. Williamson 2000).Footnote 1 It therefore might be surprising, indeed problematic, if a given theory of ignorance is at odds with (KT). I argue that the Standard View of ignorance is at odds with (KT). As we’ll see, this tension arises because the Standard View must deny that ignorance entails truth.

First, however, we must get clear on what it would mean for ignorance to entail truth. The claim that ignorance entails truth is supposed to be about factual ignorance, as opposed to other forms of ignorance (e.g. procedural ignorance, objectual ignorance, etc.). But epistemologists have expressed factual ignorance in two different ways:

  1. (a)

    S is ignorant of the fact that P.

  2. (b)

    S is ignorant that P.

Which claim should we have in mind when considering whether factual ignorance entails truth?

We should not have (a) in mind, because it’s trivial that (a) entails the truth of P. For (a) to be true, ‘the fact that P’ must refer; and if this description does refer, then the fact that P must exist, which means that P must be true. So, (a) entails P merely by virtue of the description ‘the fact that P’. But it shouldn’t be so easy to establish that ignorance entails truth, given that this thesis is rejected by, what Le Morvan and Peels call, the Standard View of ignorance (2017). Moreover, if (a) were the sentence we used for determining whether ignorance entails truth, we would also have to grant that thinking entails truth—‘S is thinking of the fact that P’ entails that P for the same trivial reason as (a). But thinking does not entail truth. So, the construction in (a) is a misleading basis for considering whether ignorance entails truth.

It is an open question as to whether (b) entails the truth of P, given that (b) doesn’t contain any description such as ‘the fact that P’. Can we frame the question of whether ignorance entails truth in terms of (b)-type constructions? I believe so. But there’s an objection to consider.

Some epistemologists contend that there’s a grammatical problem with (b). Peter Unger claims that (b)-type constructions are not “English sentence[s]” (1975, p. 175), and Berit Brogaard says they’re “grammatically ill-formed” (2017, p. 58). Both philosophers are mistaken. Although (a)-type sentences are more common than (b)-type sentences, the latter are certainly grammatical. It’s plainly false that Elizabeth Anscombe (and her editors) failed to see that she wasn’t writing grammatical English in the following passage:

Butler exalts conscience, but appears ignorant that a man’s conscience may tell him to do the vilest things (1958, p. 2; italics added).

Indeed, ignorant-that constructions are fairly common within eighteenth- and nineteenth-century published writing. The following is from an insurance digest:

She was ignorant that her attorney had refused that amount in settlement and the attorney was ignorant that the agent had gone to her to induce her to take the draft. She was also ignorant that the agent was acting for the adjuster and not as her friend and for her best interests (Deitch 1917, p. 337; italics added).

These were surely grammatical English sentences in 1917, and our grammar rules have not changed in such a way as to make them ungrammatical today. Witness a recent instance of the construction in Politico Magazine:

[…] if one lacks such knowledge and intelligence, one remains ignorant that one is not good at that task (Dunning 2016; italics added).

Most likely, if the last clause were ungrammatical, it would’ve been corrected by either the author or editor. There is thus copious evidence against the contention in question. I will assume it is erroneous.Footnote 2

The claim that factual ignorance entails truth can thus be framed with (b)-type constructions. The precise statement of this principle is analogous to (KT):

(IT) Necessarily, for every proposition P, S is ignorant that P only if P.

(IT) seems true on the face of it. Let P be an obviously false proposition, such as that the Earth is flat. It seems implausible that someone could be ignorant that the Earth is flat. (IT) explains why.

As Daniel DeNicola notes, one could challenge (IT) with claims such as “Sherman is ignorant of the principles of phrenology.” To deal with this, DeNicola paraphrases such claims along the following lines: Sherman is ignorant of the truth that phrenology is impossible (2017, pp. 201–202). DeNicola’s paraphrase, however, is untenable. The locution ‘ignorant of x’ should mean the same thing regardless of whether the principles referred to by ‘x’ are true or false.Footnote 3 And the paraphrase fails when those principles are true. Suppose Sherman is ignorant of the principles of relativity theory (which we can assume are true). Obviously, this claim doesn’t mean that Sherman is ignorant of the truth that relativity theory is impossible.

The challenge should rather be met by first noticing that (KT) faces an analogous objection. Suppose that Sherman knows the principles of phrenology. Clearly, this claim doesn’t falsify (KT), because it doesn’t attribute factual knowledge to Sherman. If it did attribute factual knowledge, then it would be incoherent to say that Sherman knows the principles of phrenology and rejects them all. Instead, the claim attributes some other kind of knowledge, such as objectual knowledge (where the relevant object is the set of principles constituting phrenology) or knowledge-wh (where Sherman knows the answer to the question ‘What are the principles of phrenology?’).Footnote 4 Regardless of which kind of knowledge we appeal to, the point is just that a similar reply can be given to DeNicola’s challenge. After all, these two challenges deal with closely-related statements. According to this reply, the claim that Sherman is ignorant of the principles of phrenology doesn’t attribute factual ignorance to Sherman. It instead attributes some other kind of ignorance, such as objectual ignorance or ignorance-wh.Footnote 5 The objection therefore fails to address (IT).

Granting the initial appeal of (IT), let’s now notice that (IT) is incompatible with the Standard View of ignorance. The Standard View takes ignorance and knowledge to be contradictories, where not knowing is both necessary and sufficient for ignorance. This view is accepted by Kit Fine (2018, p. 4032) and Pierre Le Morvan (2011, p. 336); according to DeNicola (2017, p. 17), Timothy Williamson accepts it too. For ease of reference, the necessity and sufficiency conditions will be stated separately:

Standard View: Necessarily, for every proposition P,

(NEC) S is ignorant that P only if S does not know that P; and

(SUF) S is ignorant that P if S does not know that P.

It is uncontroversial that not knowing is necessary for ignorance. So (NEC) will be assumed throughout this article. But a potential problem arises if (NEC) is combined with (SUF).

Together, (NEC) and (SUF) preclude ignorance from entailing truth. Notice that the Standard View quantifies over every proposition P, regardless of whether P is true or false. So, let P be a false proposition. From (KT), it follows that S does not know that P. On the Standard View, then, S is ignorant that P while P is false. (IT) is thus incompatible with the Standard View. Le Morvan and Peels are explicit about this implication of the Standard View—“we can be, and in fact are, ignorant that P if P is false” (2017, p. 25).

I have here assumed (KT) in showing that the Standard View is incompatible with (IT). But this can be established without assuming (KT). We can instead assume something weaker:

(POS) It is possible that P is false while S does not know that P.

The Standard View together with (POS) entails that it’s possible that P is false while S is ignorant that P. So, the Standard View is incompatible with (IT), provided (POS) is true, and regardless of (KT).

The only way to reconcile (IT) with the Standard View is to reject (POS). Although it seems absurd to reject (POS), there is one potential strategy. We could claim that the sentence ‘S doesn’t know that P’ lacks a truth-value whenever P is false (just as ‘The present King of France is wise’ may lack a truth-value). Assuming the Standard View, this strategy requires that ‘S is ignorant that P’ also lacks a truth-value whenever P is false.Footnote 6 So, this approach is inconsistent with how Le Morvan and Peels understand the Standard View, since they explicitly claim that “we can be […] ignorant that P if P is false” (2017, p. 25). Nevertheless, is the approach tenable?

No. It’s quite implausible that ‘S doesn’t know that P’ lacks a truth-value whenever P is false. If I claim your lottery ticket is a loser, but later learn you’ve won, I could retract my earlier statement as follows: ‘It turns out I didn’t know you had a losing ticket’. And you could rebut my earlier statement by saying ‘See, you didn’t know I had a losing ticket’. Both utterances seem true, even though the propositional content of the knowledge-denial is assumed false.Footnote 7

For another example, consider a complex formula containing a knowledge-denial:

(1):

S knows whether P but doesn’t know that P.

It’s plausible that instances of (1) are true. Consider

(1*):

She knows whether the Earth is flat, but doesn’t know that the Earth is flat.

This sentence is true (assuming ‘she’ refers to the right person). But the present approach requires that it’s impossible for any instance of (1) to be true. Notice that the truth of (1) would require ¬P: if S doesn’t know that P (per the second conjunct), then the only way for S to know whether P (per the first conjunct) is for her to know ¬P, which entails ¬P. However, if ¬P is true, then P is false, which means that (1) must lack a truth-value due to its second conjunct lacking a truth-value. So, on the present view, it’s impossible for any instance of (1) to be true. But this is implausible. It’s easy to see how (1) could be true, provided S knows ¬P (e.g. provided S knows the Earth is not flat).

In short, knowledge-denials can be true when their propositional contents are false. We cannot reject (POS), and cannot reconcile (IT) with the Standard View of ignorance. There is thus a genuine asymmetry between the orthodox view of knowledge and the Standard View of ignorance. Truth is entailed by the former but not the latter.

3 Apparent absurdities

To show that this asymmetry is problematic, I now argue that the Standard View of ignorance entails some apparent absurdities (and it does so precisely because it’s incompatible with (IT)). I then clarify what proponents of the Standard View must do to explain away these apparent absurdities.

Recall the sentence mentioned above:

(1):

S knows whether P but doesn’t know that P.

We saw that (1) would be true in cases where S knows ¬P. However, if the Standard View is true, then (1) is equivalent to

(2):

# S knows whether P but is ignorant that P.

And (2) seems problematic, as indicated by ‘#’. Consider a specific instance of (2):

(2*):

# She knows whether the Earth is flat but is ignorant that the Earth is flat.

If the Standard View were correct, then (2*) could be true. The trouble is that it seems incoherent on the face of it.

Various other problematic sentences can be generated from the Standard View. When a subject knows a proposition P, she thereby fails to know any propositions that are incompatible with P. So, the following is possibly true:

(3):

She knows that the Earth is spherical, and doesn’t know that it’s flat.

But, if the Standard View is true, then (3) is equivalent to another seemingly incoherent claim:

(4):

# She knows that the Earth is spherical, and is ignorant that it’s flat.

More generally, whenever a subject knows a proposition P, it follows from the Standard View that the knower is ignorant of all propositions incompatible with P. So, it’s necessarily true that an omniscient being would have an abundance of factual ignorance—for each known proposition there are indefinitely many incompatible propositions of which she is ignorant.

Sentences (2) and (4) involve contrasts between knowledge and ignorance, but the Standard View can also generate problematic sentences that don’t involve any such contrast. To see how, let us first notice that the following is necessarily true:

(5):

No one knows that they don’t exist.

But if the Standard View is correct, this necessary truth is equivalent to something that seems necessarily false:

(6):

# Everyone is ignorant that they don’t exist.

Sentence (5) is obviously true while (6) seems incoherent, and yet the two claims are supposedly equivalent.

The sentences designated with ‘#’ are examples of, what I’ll call, apparent absurdities. This notion can be defined as follows: a sentence S is an apparent absurdity if and only if, in every context, an utterance of S would carry an air of inconsistency (i.e. there would be something inconsistent about uttering S).Footnote 8 Moorean sentences, such as ‘P and I do not believe P’, are apparent absurdities in this sense (Koethe 1978, pp. 303–304). And Moorean sentences reveal two things about the notion of an apparent absurdity. First, a sentence can be an apparent absurdity even if it’s true. Moorean sentences are sometimes true even though their utterances ring inconsistent in every context. It’s therefore not immediately problematic that the Standard View entails the truth of each #-sentence; it’s only problematic if there’s no feasible explanation for why these supposed truths are apparently absurd. And second, Moorean sentences reveal that not all apparent absurdities are apparent contradictions (i.e. sentences that seem to entail logical contradictions). Those familiar with Moore’s paradox can truthfully claim that Moorean sentences do not seem to entail contradictions and so are not apparent contradictions. But they would still admit that Moorean sentences are apparent absurdities—there’s always something inconsistent about uttering such sentences. Some apparent absurdities are therefore not apparent contradictions. To my mind, all of the #-sentences are apparent contradictions as well as apparent absurdities, but my argument only assumes they are apparent absurdities, in the sense just defined.Footnote 9

In particular, I assume that the above #-sentences are apparent absurdities while their corresponding equivalences are not.Footnote 10 For simplicity, let’s focus on the following pair:

(1):

S knows whether P but doesn’t know that P.

(2):

# S knows whether P but is ignorant that P.

Readers may detect some awkwardness in (1), but this doesn’t mean (1) is an apparent absurdity. After all, there are contexts where instances of (1) can be consistently uttered. Imagine a climate-change skeptic asking “Does Pelosi know that climate change is a hoax?” A climate-change believer could retort:

(1′):

She knows whether it’s a hoax, but doesn’t know that it’s a hoax.

There’s nothing inconsistent about an utterance of (1′) in this context. The utterance can be heard as a backhanded way of implying that climate-change is not a hoax. Notice also that we can abridge (1′) by simply removing the second occurrence of ‘know’, as in

(1″):

She knows whether it’s a hoax, but not that it’s a hoax.

In the context just described, an utterance of (1″) would be completely normal. (1″) resembles other felicitous sentences that contrast two kinds of knowledge, such as ‘She knows who the President is but not where the President is.’ However, if the Standard View is correct, then (1′) and (1″) are equivalent to

(2′):

# She knows whether it’s a hoax, but is ignorant that it’s a hoax.

And an utterance of (2′) would ring inconsistent in the above-mentioned context. Indeed, there appears to be no context where (2′) could be uttered without an air of inconsistency. While (1′) and (1″) can be used to communicate that climate-change is not a hoax, there doesn’t seem to be a coherent message behind an utterance of (2′). And although we can make (1′) seem more natural by removing the second occurrence of ‘know’, (2′) cannot be similarly abridged.

No doubt, proponents of the Standard View might not share the above judgments about (1) and (2). This paper only assumes that these judgments would be widely shared among neutrals—i.e. those who don’t take a stand on whether the Standard View is true. Their judgments are least likely to be impacted by their own theoretical commitments. If it turns out that neutrals would not share these judgments about (1) and (2), we can easily shift discussion to another pair of sentences, such as (5) and (6). It would be highly surprising if neutrals thought that all #-sentences can be consistently uttered—for example, that the following could be uttered without any air of inconsistency:

(6):

# Everyone is ignorant that they don’t exist

Although I focus on (1) and (2), everything I say can be applied analogously to (3) and (4), (5) and (6), and similar pairs.

Thus far, (2) has been likened to a Moorean sentence—both are apparent absurdities. Unlike Moorean sentences, however, we cannot automatically assume there’s no logical contradiction that would explain the apparent absurdity in (2). Indeed, we can maintain that (2) is a logical contradiction, if we accept (IT) and reject the Standard View. If (IT) is true then (2) entails both P and ¬P. First observe that (2) entails (1). This holds merely by virtue of (NEC), the uncontroversial part of the Standard View. Now recall that (1) entails ¬P (as established earlier). So, by transitivity, (2) entails ¬P. However, if (IT) is true, then (2) also entails P. Therefore, if (IT) is true then (2) entails a logical contradiction.Footnote 11 This is a straightforward explanation of the apparent absurdity in (2). Instances of (2) carry an air of inconsistency in all contexts because they entail logical contradictions. Let’s call this the IT Explanation, since (IT) is an integral part of it.

It should be noted that the IT Explanation explains why (2) is an apparent absurdity without falsely predicting that (1) should also be an apparent absurdity. According to the IT Explanation, the apparent absurdity of (2) arises because its ignorance-ascription entails P. But the corresponding knowledge-denial in (1) does not entail P. So, the IT Explanation does not predict that (1) should be apparently absurd.

Since the Standard View is incompatible with the IT Explanation, proponents of the Standard View must seek another way of explaining the apparent absurdity in (2). There are two constraints on any such explanation. First, the explanation cannot allow that (2) is a logical contradiction. After all, (2) is equivalent to (1) on the Standard View, and (1) is not a logical contradiction. And second, any explanation of the apparent absurdity in (2) cannot falsely predict that (1) should also be apparently absurd. In the next section, I show that the best way of meeting these two constraints is undermined by the claim that knowledge entails truth.

4 Explanations

I begin by briefly rejecting two unpromising strategies for explaining the apparent absurdity in (2). I then discuss the most promising strategy and show that it’s undermined by (KT).

4.1 Unpromising strategies

The first potential explanation quibbles with the sort of ignorant-that construction found in (2). This explanation claims that ‘ignorant that P’ is markedly different from other constructions, such as ‘believes that P’ and ‘knows that P’. The word ‘that’ can be removed from the latter constructions but not the former. ‘S knows P’ is perfectly fine, but ‘S is ignorant P’ is not. This suggests ignorant-that constructions do not involve appropriate uses of ‘that’. For this reason, it’s expected that we should find (2) to be somewhat odd, but not (1).

Let us first notice that this explanation predicts that the second conjunct of (2)—i.e. ‘S is ignorant that P’—is by itself an apparent absurdity. This is because the explanation attributes (2)’s oddity solely to the ignorant-that construction, and not at all to how this construction relates to the first conjunct of (2). But this prediction is false. ‘S is ignorant that P’ is not an apparent absurdity—as noted earlier, there are many contexts where it can be used felicitously. The second thing to note is that, although ‘ignorant P’ may seem strange in the abstract, many of its instances are unproblematic. Consider ‘Sue was ignorant we had a meeting’ and ‘Tyson is ignorant we’re parked behind him’. Consider also some sentences from online sources:

Lokko claimed so-called proxy marriages were common in his country, and he was ignorant they are illegal in the UK.Footnote 12

The man that rented us the U-Haul was very ignorant we had to bring the truck back buy 4 pm.Footnote 13

As these samples suggest, ignorance-that constructions operate much like believes-that and knows-that. The word ‘that’ can be omitted without rendering the sentence ungrammatical, and without changing the construction’s meaning.

A second explanation would claim that we’re prone to misinterpret the ascription of factual ignorance in (2) as a different sort of ignorance. Consider the difference between factual and propositional ignorance.Footnote 14 A subject has factual ignorance when she’s ignorant that P—i.e. when she’s ignorant that P’s truth-conditions obtain. But a subject can also be ignorant of P’s truth-conditions themselves, if, say, she doesn’t even have the concepts needed for grasping P. The latter is propositional ignorance—it requires that the subject has never entertained the proposition in question (Le Morvan 2012, p. 388). Now return to (2). Although (2) is meant to express factual ignorance, one might claim that we’re prone to misinterpret (2) as expressing propositional ignorance. This explains the apparent absurdity in (2). If S knows whether P, per the first conjunct, then S must have entertained P. But if the second conjunct expresses propositional ignorance, then S has not entertained P.

In reply, notice that there is a parallel distinction between factual knowledge and propositional knowledge. And it’s unclear why we aren’t equally prone to misinterpret the second conjunct of (1) as expressing the denial of propositional knowledge rather than the denial of factual knowledge. In that case, there should also be an apparent absurdity in (1), since the denial of propositional knowledge requires that S hasn’t entertained the proposition, according to the Standard View (Peels and Le Morvan 2017, p. 24). We need an explanation for why we’d be prone to misinterpret (2) but not equally prone to misinterpret (1). It’s unclear what this would be.

4.2 Promising strategy

A better strategy would hold that (2) involves a pragmatic illusion. Although proponents of the Standard View cannot accept (IT), they can accept something similar:

(IT*) The sentence ‘S is ignorant that P’ implies P without entailing it.

Henceforth, the word ‘implies’ will mark out a generic inference relation that includes entailment as well as various pragmatic inferences that are not truth-preserving, such as conversational implicature, conventional implicature, pragmatic presupposition, etc. (IT*) is briefly suggested by Le Morvan and Peels. After recognizing that the Standard View is incompatible with (IT), they acknowledge “that S is ignorant that p conversationally implies that p is true” (2017, p. 25).

(IT*) helps to explain why (2) is apparently absurd. The apparent absurdity in (2) arises because we confuse the correct principle—(IT*)—for a mistaken one—(IT). We mistakenly think ‘S is ignorant that P’ entails P, when it really only implies P via some non-truth-preserving inference.

But there’s an initial problem, which forces us to supplement the explanation so far. ‘S doesn’t know that P’ also implies P via a non-truth-preserving inference. Le Morvan and Peels grant this implication of ‘not know’ (2017, p. 25).Footnote 15 And if both ‘S is ignorant that P’ and ‘S doesn’t know that P’ imply P via a non-truth-preserving inference, then our explanation predicts that both (1) and (2) should be apparently absurd, which they aren’t.

To explain the difference between (1) and (2), we must supplement (IT*) with an additional claim. We must claim that P is implied by the ignorance-ascription in (2) but not by the knowledge-denial in (1). One way to do this is to argue that the inference to P is cancelled in (1) but not in (2). For example, since conversational implicatures are cancellable, it could be claimed that P is a conversational implicature of ‘S doesn’t know P’, and that this implicature is cancelled in the context of (1). Conventional implicatures, however, are not cancellable. So, it could be claimed that P is a conventional implicature of the ignorance-ascription in (2). Regardless of the details of this particular proposal, the main point is just that we can in principle claim that P is implied by (2) but is not implied by (1).

With this additional claim, let’s call the resulting approach the IT* Explanation. I will assume that the IT* Explanation involves three parts—it accepts (IT*), the Standard View of ignorance, and the claim that P is implied by (2) but not by (1). This explanation meets the constraints explained in Sect. 3. But we must pause on two important features of the IT* Explanation.

The first is that the IT* Explanation requires that P is a detachable implication of (2). Sentence A carries P as a detachable implication if and only if A implies P and there could be another sentence truth-conditionally equivalent to A that doesn’t imply P. Consequently, the fact that P is implied by A must be explained by the particular linguistic expressions in A, and cannot be explained by the truth-conditional contents of those expressions (Grice 1989, p. 43; Huang 2007, p. 57; Platts 1997, p. 78). Now recall that (1) and (2) are truth-conditionally equivalent, according the Standard View. And on the IT* Explanation, (2) implies P while (1) does not. So, the IT* Explanation requires that (2) carries P as a detachable implication. Naturally, this detachable implication is explained by the particular expression ‘ignorant that P’, which occurs in (2) but not in (1).

The second feature is that the implication from ‘ignorant that P’ to P is constant under negation. That is, P is a typical implication of the affirmative sentence in

(7):

S is ignorant that P.

And P is also a typical implication of the negation in

(8):

S is not ignorant that P.

In this sense, the implication cited by the IT* Explanation is constant under negation.

Essentially, the IT* Explanation requires that P is a detachable implication of (7), rather than an entailment of (7). In what follows, I argue that the IT* Explanation is plausible only if (8) also doesn’t entail P. And if (8) doesn’t entail P, then knowledge doesn’t entail truth, on the Standard View. After all, (8) is equivalent to ‘S knows P’, on the Standard View. So, if (8) doesn’t entail P, then ‘S knows P’ doesn’t entail P, and (KT) is false. On the other hand, if (KT) is true, then its truth is evidence against the IT* Explanation.

The above reasoning hinges on my claim that the IT* Explanation is plausible only if (8) doesn’t entail P. What supports this? Recall that P is supposed to be a detachable implication of (7). What this means is that P arises solely because (7) contains the form of words ‘ignorant that P’, and not because of the truth-conditions associated with the expression. But (8) contains the same form of words. So, the IT* Explanation predicts that P will also arise from that form of words as they occur in (8), regardless of (8)’s truth-conditions.Footnote 16 In other words, the IT* Explanation predicts that P should also be a detachable implication of (8). But detachable implications are not entailments. So, the IT* Explanation predicts that P is not an entailment of (8). On the other hand, if (8) does entail P, and equivalently, if knowledge entails truth, then the IT* Explanation makes a false prediction. Thus, the truth of (KT) would be evidence against the IT* Explanation. I do not claim that the IT* Explanation is logically incompatible with (KT), but only that the latter provides evidence against the former in the following sense: if (KT) is true, then the IT* Explanation makes a false prediction.

Above, I argued that if P is a detachable implication of the affirmative in (7) then P should also be a detachable implication of the negation in (8), and so shouldn’t be entailed by (8). There is a general basis for this premise. Generally speaking, if P is constant under negation, then P is detachable from the affirmative only if P is also detachable from the negation.Footnote 17 To see why we should accept this general claim, let’s examine some classic examples of implications that are constant under negation and detachable from the affirmative.Footnote 18

First consider the implicative verb ‘manage’, as used in the affirmative sentence ‘John managed to solve the problem’ (Abbott 2016, p. 11). This affirmative sentence implies that John faced difficulty in trying to solve the problem. And the same implication is carried by the negation ‘John didn’t manage to solve the problem’. Most importantly, this implication is detachable from both sentences. The affirmative is equivalent to ‘John solved the problem’; the negation is equivalent to ‘John didn’t solve the problem’. And neither equivalent sentence implies that John faced difficulty.

Now consider the connective ‘but’, as in ‘Shaq is huge but agile’ (Abbott 2016, p. 10). This affirmative sentence implies that there’s a contrast between hugeness and agility. The same implication is carried by the negation ‘Shaq is not huge but agile’. And this implication is detachable from both sentences. Equivalent sentences can be obtained by replacing ‘but’ with ‘and’ in each case. And neither equivalent sentence implies a contrast between hugeness and agility.

Now consider pejorative expressions, as illustrated by the affirmative sentence ‘Jan is Boche’ (Williamson 2009). This sentence implies contempt for Germans. And contempt is also implied by the negation ‘Jan is not Boche’. And again, this implication is detachable from both sentences. By simply replacing ‘Boche’ with ‘German’ for each sentence, we obtain an equivalent sentence that lacks the implication.

Consider also the kind of conversational implicature that arises from Grice’s manner maxim, which Grice allowed to be detachable (1989, p. 39). These implications are not always constant under negation, but some are. Consider an example, from Jessica Rett (forthcoming), where a parent spells out ‘ice cream’ so as to mask her message from the children. The affirmative ‘We have I-C-E-C-R-E-A-M’ implies that the speaker wants to hide her message from a third party, and so does the negation ‘We don’t have I-C-E-C-R-E-A-M’. This implication is obviously detachable from both sentences—simply saying the word, rather than spelling it, will result in equivalent sentences that lack the implication.

Finally, consider parenthetical constructions, as in the sentence ‘Armstrong, the cyclist, battled cancer’ (Potts 2007, p. 668). This affirmative sentence implies that Armstrong is a cyclist, which is also an implication of the negation ‘Armstrong, the cyclist, did not battle cancer’. And this implication is detachable from both sentences. By simply omitting ‘the cyclist’ from each sentence, we obtain an equivalent expression that lacks the implication.

In short, these examples serve to confirm the general principle stated earlier: if P is constant under negation, then P is detachable from the affirmative only if it’s also detachable from the negative. I’ve focused on the clearest examples of detachable implications arising from affirmative sentences, and have suggested that the same should be said of the corresponding negation. Of course, one can raise doubts about whether these implications are really detachable from the negative sentence in each case. But such doubts can be equally raised about the corresponding affirmative (Bach 1999). So, no such doubts should infect my claim that detachable implications of the affirmatives are also detachable implications of their negations.

One might object by claiming that (7) is not really an affirmative sentence, as I’ve assumed, because it contains the prefixal negation ‘ig’. However, (7) is affirmative in the sense that it doesn’t contain any negations that can be removed without rendering the sentence ungrammatical. The expression ‘ig’ cannot be removed from (7) because ‘norant’ is not an English expression. So, if (7) were not already an affirmative sentence, it would be altogether impossible to state a corresponding affirmative, which is absurd. One might claim that ‘S knows that P’ is the affirmative corresponding to (7). But this is false, since (7) does not contain the word ‘knows’ or any of its grammatical forms.Footnote 19

Another objection might rely on Robert Stalnaker’s explanation for why the implication from ‘S knows that P’ to P is constant under negation. Stalnaker assumes that the affirmative ‘S knows that P’ entails P, and then uses this entailment to explain why the sentence’s negation implies P. Assuming knowledge requires both truth and belief, it would be “inappropriate” for a speaker to assert that S doesn’t know that P in a context where the speaker thinks P is false. In that case, the speaker’s assertion would be “gratuitously weak,” leaving it open that the speaker may only be denying that S believes P rather than denying the truth of P (Stalnaker 1999, p. 55). This provides a conversational explanation for why ‘S doesn’t know that P’ implies P, and it assumes that knowledge entails truth. Now, proponents of the Standard View can claim that ‘S doesn’t know that P’ is equivalent to (7), and that ‘S knows that P’ is equivalent to (8). Given these equivalences, they may wish to employ Stalnaker’s approach to explain why both (7) and (8) imply P. Roughly, (8) entails P, for the same reason that knowledge entails truth. And (7) implies P for the same reason that ‘S doesn’t know that P’ implies P—i.e. it would be gratuitously weak for a speaker to assert (7) in a case where she thinks P is false, because this would leave it open that the speaker may only be denying that S believes P rather than denying the truth of P.

However, even if Stalnaker’s approach works in the case of knowledge, it makes a false prediction when applied to our present concerns. It predicts that (2) should be apparently absurd if and only if (1) is. If ignorance-ascriptions and knowledge-denials imply truth via the same mechanism—as the above explanation holds—then either (1) and (2) should both be apparently absurd, or neither of them should. Yet only (2) is the apparent absurdity. To explain the difference between (1) and (2), we need something in addition to Stalnaker’s account.

So far, I’ve argued that the IT* Explanation requires that P is a detachable implication of (2) and (7); otherwise it cannot explain why (2) is apparently absurd while (1) is not. I’ve also argued that the entailment from knowledge to truth is evidence against the detachability of P, and likewise against the IT* Explanation. In short, proponents of the Standard View must either deny that knowledge entails truth, or leave the apparent absurdity in (2) without proper explanation. Both options are undesirable.

The best option, I think, is to accept the original IT Explanation, and thereby reject the Standard View of ignorance. On this approach, we accept (IT)—the claim that factual ignorance entails truth. This allows us to explain the apparent absurdity in (2) by claiming that (2) is a logical contradiction. This explanation makes no appeal to pragmatic principles. And it allows us to accept both (KT) and (IT), thereby eliminating the problematic asymmetry between knowledge and ignorance.

5 Modified standard view

A natural reaction to the above critique is simply to modify the Standard View in a way that’s consistent with (IT). The simplest approach is to claim that factual ignorance is not merely the failure to know, but is rather the “failure to know what is true” (Zimmerman 2008, ix). This approach merely supplements the original Standard View with the requirement that P is true.

Modified Standard View: Necessarily, for every proposition P, S is ignorant that P if and only if it’s true that P and S does not know that P.Footnote 20

Since the Modified Standard View is so similar to the original Standard View, one might think it’s harmless to retreat from my critique in Sect. 4 by accepting the Modified Standard View. But this retreat is not harmless. I do not here argue that the Modified Standard View is false.Footnote 21 I instead argue that the Modified Standard View gleans no support from the two main arguments that have been used to support the Standard View—the Common Usage Argument and the Unification Argument.Footnote 22 If anything, these arguments only support the original Standard View, not the Modified Standard View. It’s therefore not harmless to retreat from my critique in Sect. 4 by accepting the Modified Standard View.

5.1 Common usage

The first argument—called the Common Usage Argument—begins with a cluster of linguistic facts about the word ‘ignorance’. For example, dictionaries typically define ‘ignorance’ as the lack of knowledge. And the etymology of the English word roots in a Latin expression referring to the opposite of knowledge. Moreover, there are numerous languages that take cognates of ‘ignorance’ to be antonyms of cognates of ‘knowledge’ (e.g. in Danish ‘uvidenhed’ is an antonym of ‘viden’). Together, these linguistic facts support the claim that ‘ignorance’ and ‘knowledge’ are antonyms in English. Moreover, it’s assumed that if two terms are antonyms then they probably pick out contradictory features—i.e. features such that everything must possess one of them but nothing can possess both. Therefore, ‘ignorance’ and ‘knowledge’ probably pick out contradictories, just as the Standard View proposes (Le Morvan and Peels 2017, pp. 15–16).

However, this argument fails to support the Modified Standard View; indeed, the argument indicates that the Modified Standard View is false. Notice that the conclusion of the argument is that (probably) ignorance and knowledge are contradictories. But, according the Modified Standard View, ignorance and knowledge are not contradictories—the Modified Standard View allows that one may neither know that P nor be ignorant that P, if P is false. So, even if the Common Usage Argument supports the original Standard View, it does not support the Modified Standard View.

Of course, there’s no reason to think the Common Usage Argument is successful in even supporting the original Standard View. This is because there’s absolutely no reason to accept the assumed premise that if two terms are antonyms then they probably pick out contradictories. Many antonym pairs don’t pick out contradictories. ‘Happy’ and ‘unhappy’ are antonyms but it’s possible to be neither happy nor unhappy—e.g. one might be so-so. ‘Tall’ and ‘short’ are antonyms but it’s possible to be of average height. ‘Courageous’ and ‘cowardly’ are also antonyms but many actions are neither courageous nor cowardly—e.g. brushing one’s teeth.Footnote 23 Similar points hold for many other antonym pairs, such as ‘big’/‘small’, ‘smart’/‘stupid’, ‘beautiful’/‘ugly’, and so on. The upshot is that there are many antonym pairs that don’t pick out contradictories. And it would be surprising if there were more antonym pairs that do pick out contradictories. We thus have no reason to accept the key assumption behind the Common Usage Argument.

To be sure, one could revise that assumption as follows: if two terms are antonyms then they probably pick out contraries if not contradictories. However, this revised argument does not support either version of the Standard View. The conclusion of this revised argument is that ignorance and knowledge are probably contraries (if not contradictories). But if they are contraries, this simply means that one cannot be ignorant that P while one knows that P, which is entirely uncontroversial. Indeed, this conclusion is amenable to even non-standard theories of ignorance. For example, consider the Belief View of ignorance (a.k.a. the New View):

Belief View: Necessarily, for every proposition P, S is ignorant that P if and only if it’s true that P and S does not believe that P (Peels 2012, p. 743).

The Belief View entails that one cannot be ignorant that P while one knows that P, because knowledge requires belief, and ignorance requires the absence of belief. Thus, the revised argument is amenable to even a non-standard theory of ignorance such as the Belief View. It therefore fails to support any version of the Standard View, let alone the Modified Standard View.

Can we bypass all mention of antonyms, and base the argument solely on the linguistic facts mentioned above? No. None of these facts supports the Modified Standard View, because none of them supports the claim that factual ignorance entails truth. For example, the dictionary definition of ‘ignorance’ as the lack of knowledge says nothing about truth; it therefore fails to support the claim that factual ignorance requires truth. Similarly, we can grant that cognates of ‘ignorance’ in other languages are structures that involve not-knowing. But since not-knowing doesn’t entail truth, the claim in question does not by itself support the view that ignorance entails truth.Footnote 24

Suppose we take these linguistic facts as merely supporting the claim that not-knowing is necessary for factual ignorance, but then use my argument in Sect. 4 as a reason for adding truth as an additional necessary condition. Wouldn’t this support the Modified Standard View? No. As already noted, the Belief View can claim that not-knowing is necessary for factual ignorance. It also claims (explicitly) that truth is a necessary condition for factual ignorance. So, the present approach does not support the Modified Standard View over a non-standard theory such as the Belief View. There appears to be no version of the Common Usage Argument that supports the Modified Standard View.

5.2 Unification

The second argument—called the Unification Argument—starts with a general hypothesis, and then claims that this hypothesis is plausible because it can unify theories of ignorance with theories of knowledge. The general hypothesis is as follows:

General Standard View (GSV): Each kind of ignorance is merely the absence of some kind of knowledge.

(GSV) deals with all types of ignorance, including factual ignorance, but also other types such as objectual and procedural ignorance. Objectual ignorance is ignorance of an entity, and is expressible with the locution ‘ignorant of x’, where ‘x’ refers to an entity such as a person or thing (e.g. Sue is ignorant of the car parked behind her). Procedural ignorance is ignorance of how to do something, and is expressible with the locution ‘ignorant of how to φ’, where ‘φ’ refers to an action (e.g. Sam is ignorant of how to fly an aircraft). In short, (GSV) is a hypothesis about various types of ignorance, in addition to factual ignorance.

The Unification Argument claims that we have reason to accept (GSV), because (GSV) “unifies” theories of ignorance with theories of knowledge in that “insights into the nature of knowledge automatically yield corresponding putative insights into the nature of ignorance” (Le Morvan and Peels 2017, pp. 16–17). How does (GSV) unify these theories? Corresponding to objectual ignorance, there is also a notion of objectual knowledge—e.g. Jill has knowledge of the car parked behind her. And corresponding to procedural ignorance, there is also a notion of procedural knowledge—e.g. James knows how to fly an aircraft. If (GSV) is correct, we can simply claim that objectual ignorance is merely the absence of objectual knowledge, and that procedural ignorance is merely the absence of procedural knowledge. So, once we have accounts of objectual and procedural knowledge, those accounts automatically yield corresponding accounts of objectual and procedural ignorance, provided (GSV) is true. (GSV)’s ability to unify such theories is taken as evidence for (GSV).

Finally, notice the Standard View is just another instance of (GSV)—i.e. the Standard View takes factual ignorance to be merely the absence of factual knowledge. So, according to the Unification Argument, we have reason to accept (GSV) because it unifies theories of knowledge and ignorance, and this lends indirect support to the Standard View of factual ignorance.

5.2.1 Problems

The first thing to note is that the Unification Argument does not in its present form support the Modified Standard View, because the Modified View is incompatible with (GSV). On the Modified View, factual ignorance is not merely the absence of factual knowledge, because factual ignorance also requires truth. So, for the Unification Argument to support the Modified Standard View, we must modify (GSV). Can we modify (GSV) in a way that will accommodate the claim that factual ignorance entails truth?

There’s an initial barrier to this. Objectual and procedural ignorance do not require truth, at least not in the same way as factual ignorance. It’s not immediately clear what would fill the blanks in the following schemata:

S is ignorant of x only if ___.

S is ignorant of how to φ only if ___.

Since ‘x’ and ‘φ’ cannot function as declarative sentences, they cannot fill the blanks. This situation contrasts with the corresponding schemata for factual ignorance, where each occurrence of ‘P’ can function as a declarative sentence: S is ignorant that P only if P.

Nevertheless, as Nottelmann points out, it is plausible that each kind of ignorance has a success condition that’s identical to the success condition for the corresponding kind of knowledge (2017, pp. 37–42). Consider that objectual knowledge-of-x seems to require that x exists. And procedural knowledge-of-how-to-φ seems to require that there’s a correct answer to the question of how to φ.Footnote 25 And, of course, factual knowledge-that-P seems to require the truth of P. Let’s call these success conditions. The basic idea is that each kind of knowledge seems to require that the world contains something closely pertaining to the object of knowledge.

Ignorance seems to have the same success conditions as knowledge, and the Modified Standard View can appeal to this. Corresponding to the Modified Standard View, we can thus frame a general hypothesis analogous to (GSV):

General Modified Standard View (GMSV): For some kind of knowledge K, each kind of ignorance is the absence of K when K’s success condition obtains.

(GMSV) can unify our theories of knowledge and ignorance in a way that rivals (GSV). On this view, objectual ignorance-of-x is the absence of objectual knowledge-of-x when x exists. So, for example, one cannot be ignorant of the present King of France. Similarly, procedural ignorance-of-how-to-φ is the absence of procedural knowledge-of-how-to-φ when there’s a correct answer as to how to φ. Thus, no one can be ignorant of how to draw a square circle. In short, (GMSV) can also gain support by unifying theories of knowledge and ignorance, and the Modified Standard View is just a specific instance of (GMSV). So far, then, we have a version of the Unification Argument that seems to provide some support for the Modified Standard View.

But there’s a problem. Now that we’ve allowed success conditions, even non-standard views of ignorance can be made to correspond to hypotheses like (GMSV), which can also unify theories of knowledge and ignorance. This means the Unification Argument doesn’t support the Modified Standard View over these non-standard theories.

As an example, notice that the Belief View of ignorance can be made to correspond to a general hypothesis like (GMSV). Since knowledge entails belief, we can recast the Belief View in an equivalent way that makes it explicit that ignorance requires an absence of knowledge. Consider the following, which is equivalent to the Belief View: S has factual ignorance that P if and only if P is true, S does not believe that P is true, and therefore does not have factual knowledge that P. This equivalent statement of the Belief View makes it apparent that factual ignorance requires an absence of factual knowledge; it also makes it clear that this absence of factual knowledge arises from a lack of belief. Furthermore, notice that the belief in question has the same content as the success condition for factual knowledge—the success condition that P is true. The point of recasting the Belief View is just to show that the Belief View is an instance of a general hypothesis:

General Belief View (GBV): For some kind of knowledge K, each kind of ignorance is an absence of K that arises from a failure to believe K’s success condition when it obtains.Footnote 26

My equivalent statement of the Belief View, provided above, is an instance of (GBV). On the Belief View, factual ignorance is an absence of factual knowledge arising from a failure to believe P when P is true.

How does (GBV) account for objectual and procedural ignorance? Now that we’ve acknowledged success conditions, objectual ignorance can be explained as follows: S has objectual ignorance of x if and only if x exists, S does not believe that x exists, and therefore does not have objectual knowledge of x. This theory of objectual ignorance makes the feasible assumption that objectual knowledge-of-x requires the belief that x exists. Here, again, we have an instance of (GBV). Objectual ignorance is the absence of objectual knowledge arising from a failure to believe that x exists when x does exist.Footnote 27

(GBV) can also explain procedural ignorance: S has procedural ignorance of how to φ if and only if there’s a correct answer to the question of how to φ, S does not believe any correct answers to this question, and therefore does not have procedural knowledge of how to φ. This view makes the plausible assumption that knowing-how-to-φ requires one to believe a correct answer to the question of how to φ.Footnote 28 Here again we have an instance of (GBV). Procedural ignorance is an absence of procedural knowledge arising from a failure to believe any correct answers to the question of how to φ.

So far, I have used (GBV) as a rival hypothesis that can unify theories of ignorance and knowledge just as well as (GMSV). I should clarify that I do not take this as reason to accept (GBV) or the Belief View. Rather, (GBV) and the Belief View are only used to undermine the attempt at using the Unification Argument to support the Modified Standard View. If (GBV) can unify theories of ignorance and knowledge just as well as (GMSV), then the Unification Argument provides no reason to accept (GMSV) over (GBV), and no reason to accept the Modified Standard View over the Belief View.

5.2.2 Objections

One might object that (GBV) doesn’t unify theories of knowledge and ignorance in the appropriate way, because it doesn’t yield theories of ignorance merely by negation. We cannot simply negate each kind of knowledge and thereby obtain a corresponding type of ignorance. However, if this objection works against (GBV), it works equally against (GMSV). On (GMSV), we cannot simply negate each kind of knowledge and thereby obtain a corresponding type of ignorance; we must also assert that the relevant success condition obtains.

Of course, there is a reasonable sense in which both (GMSV) and (GBV) unify theories of knowledge and ignorance—if either hypothesis is true, then our theories of the various kinds of knowledge automatically entail theories of each kind of ignorance. But, again, this holds for both hypotheses, which means the Unification Argument doesn’t support (GMSV) over (GBV).

One might object that there are counterexamples to specific instances of (GBV). Without detailing such counterexamples, we can already see that this objection is irrelevant. Nowhere have I claimed that (GBV) is true, or probably true, or even remotely plausible; and nowhere have I claimed that (GBV) can be defended from counterexamples.Footnote 29 Rather, the point of this section is only that (GBV) unifies theories of ignorance and knowledge just as well as (GMSV), which means the Unification Argument fails to support (GMSV) over (GBV). Thus, for the sake of argument, let’s suppose that instances of (GBV) have counterexamples. In no way does this show that (GBV) does an inferior job of unifying theories—i.e. it’s still true that if (GBV) is correct then our theories of the various kinds of knowledge entail theories of each kind of ignorance.

Of course, one might reply that facts about unification only supply a defeasible reason to accept a hypothesis; but in the case of (GBV) that reason is defeated by the fact that there are counterexamples to particular instances of (GBV). However, this reply ignores the fact that there are many putative counterexamples to instances of (GMSV), specifically to the Modified Standard View itself. Rik Peels has pointed out that certain Gettier-style cases seem to be counterexamples to the Modified Standard View (2011, p. 352). And elsewhere I have provided a different sort of counterexample, where a person suspends judgment regarding the true proposition P but still doesn’t seem ignorant that P (Kyle 2015, p. 1498). More generally, it is well-attested that there are cases about which we’re inclined to deny that a subject knows that P and also inclined to deny that she’s ignorant that P (where P is a true proposition) (e.g. Goldman and Olsson 2009, p. 21). So, if we take counterexamples at face value, there’s no reason to think (GMSV) fairs any better than (GBV).

Of course, one might think that (GMSV)’s putative counterexamples should not be taken at face value, because our intuitions about such cases can be explained away. But no one has shown that these intuitions can be explained away.Footnote 30 Indeed, I have argued elsewhere that they cannot be explained away (Kyle 2015, pp. 1497–1510). And, for all we know, we might be able to explain away the intuitions about the putative counterexamples against (GBV)—whatever those examples are. So, there’s no automatic reason to assume (GMSV) fairs any better than (GBV), even if counterexamples are not taken at face value.

In short, even if unification only provides defeasible reasons, it appears such reasons are defeated in the case of (GBV) only if they’re also defeated in the case of (GMSV). Either way, the Unification Argument fails to support (GMSV) over a non-standard theory of ignorance.

6 Conclusion

This paper has attacked two versions of the Standard View of ignorance. I have argued that the original Standard View is at odds with the claim that knowledge entails truth. If knowledge entails truth, then we cannot explain away some apparent absurdities that arise from the original Standard View. This may lead proponents to retreat to a modified version, where the Standard View is supplemented with a truth requirement. I’ve argued that this is an unsafe retreat, because the Modified Standard View gleans no support from the two main arguments in favor of the original Standard View. I have not addressed all possible arguments that could be advanced for a version of the Standard View. But I have addressed all arguments that have so far been advanced in the epistemology literature. None of them succeed in supporting the Modified Standard View. We’re thus left to wonder what reason there is to accept such a view.Footnote 31

I have not proposed an alternative theory of ignorance. The present paper is merely a small part of a larger effort to make epistemologists question a view that they’ve so far taken as axiomatic (Kyle 2015; Peels 2012). Still, it’s worth noting that there are at least four alternatives to the Standard View, and all of them remain untouched by the critiques advanced in this paper. In addition to the Belief View, Berit Brogaard has argued that ignorance is the complement of being knowledgeable (2017, p. 59). And Daniel DeNicola has proposed that ignorance is not a mere negation, but “may be manifested as a distinctive mental state, or more accurately, a cluster of related states” (2017, p. 18). Finally, Duncan Pritchard has recently argued that ignorance “essentially involves […] an intellectual failing of inquiry” (forthcoming). The considerations I’ve alleged against the Standard View neither support nor undermine these alternative theories. Those on the market for an alternative theory are advised to give due consideration to these views, but are also warned that theorizing about the nature of ignorance is still in a nascent stage.Footnote 32