Incomplete Understanding of Concepts
Abstract and Keywords
This article discusses the thesis that a subject can have a concept, think thoughts containing it, that she incompletely understands. The central question concerns how to construe the distinction between having a concept and understanding it. Two important versions of the thesis are distinguished: a metasemantic version and an epistemic version. According to the first, the subject may have concept C without being a fully competent user, in virtue of deference to other speakers or to the world. According to the second, the subject may have a concept without being able to provide a proper explication of it. It is argued that whereas the epistemic version is plausible, the metasemantic version faces some challenges. First, it needs to be explained precisely how deference enables a speaker to have C. Second, metasemantic incomplete understanding is in tension with the idea that concepts serve to capture the subject’s cognitive perspective.
An influential idea in contemporary philosophy of mind and language is that it is possible to think with a concept that one only understands incompletely. This idea plays a central role in externalist theories of mental content, in particular in the tradition of Tyler Burge, but it has also been suggested that it is required in order to account for concept learning and for the epistemology of reflection. This survey presents and examines the central theories of incomplete understanding of concepts.
What all theories of incomplete understanding of concepts have in common is the idea that understanding a concept cannot simply be identified with having the concept. That is, it can be true of a subject that she has concept C, in the sense that C figures in her thoughts, even though her understanding of C is less than complete, thesis ICU:
(ICU) S may think thoughts containing concept C, even if S has an incomplete understanding of C.
The central question is how to construe the distinction between having a concept and understanding it. In the debate, a comparison with incomplete understanding of word meaning is often suggested, since it seems plausible that I can use a word in the public language without fully understanding it. The question is how illuminating this comparison is: To what extent can the incomplete grasp of concepts be modeled on the incomplete grasp of word meaning? And what happens to the idea that concepts play an essential psychological role (in reasoning and classification) if individuals have an incomplete grasp of the very concepts they think with? The survey is divided into four main sections. Section 1 sets the stage by looking at the related notion of incomplete grasp of word meaning, sections 2 and 3 examine some common accounts of what it is to have an incomplete grasp of concepts, and section 4 discusses whether incomplete understanding can be reconciled with the idea that concepts should capture the subject’s cognitive perspective.
1. Incomplete Understanding of Word Meaning
Central arguments for incomplete understanding of concepts take as their starting point the idea that there is incomplete understanding of word meaning. This is quite explicit in Tyler Burge’s famous 1979 thought experiment. It involves a subject who misunderstands the term “arthritis,” as it is used in the common language, thinking it applies to a disease of the joints and ligaments. Incomplete understanding of word meaning, Burge suggests, is an extremely widespread phenomenon and does not just afflict technical terms, such as those of medicine or the sciences: “What I have called ‘partial understanding’ is common or even normal in the case of a large number of expressions in our vocabularies” (1979, 83). A quick look in the dictionary, Burge suggests, illustrates that the list is long: “brisket,” “contract,” “recession,” “sonata,” “deer,” “pre-amplifier,” “carburetor,” “gothic,” and so on.
There is a sense in which it is trivially true that there is incomplete grasp of word meaning. The common language, be it English or Swedish, contains a large number of words that individual speakers may have a more or less firm grasp of. However, Burge is not defending this trivial thesis, but a much stronger one: individuals may have an incomplete grasp of the meaning of the words in their own idiolect.1 That is, his claim is that the word as used in the idiolect of an individual, under certain conditions, will take on the meaning of the common language—even when the individual only has a partial understanding of this meaning. The relevant conditions are two:
(i) The individual defers to the practice of his community
(ii) The individual has a minimal competence with the word
The first condition is essential, since it is the commitment to the community that is supposed to imply that the meaning of the individual’s word is determined by factors beyond her. That an individual defers, according to Burge, is shown in her disposition to stand corrected. In the thought experiment the patient stands corrected when the doctor tells him that one cannot have arthritis in the thigh and this is the normal reaction: when caught in errors of this sort the subject does not normally respond by saying that she has been misunderstood (1979, 94). This, according to Burge, shows that the subject intends the word to have its standard meaning.
Individualists do not question the claim that speakers intend to use their words as others do and stand corrected when they deviate. What they question is the idea that this has any semantic implications. For instance, Davidson (1986, 1987) has argued that the fact that speakers are willing to stand corrected in this way merely shows that they are willing to adjust their language to the community practice—perhaps because of a desire to communicate with ease or just a desire to conform—and does not have anything to do with the meaning of their words. On Davidson’s view, therefore, the deviant speaker does not have an incomplete grasp of the meaning of the word used but simply uses it with a slightly different meaning. A proper interpretation of her language should reflect this, capturing the meaning of the word in her idiolect.2
Burge argues against this reinterpretation strategy on several grounds. First, he suggests, it does not fit with our intuitions about these cases: our intuition is that the patient has said something false when uttering “I have arthritis in my thigh,” not something true, as the strategy would imply.3 Second, in ordinary practice we do not typically reinterpret people, but explain the deviance as deriving from a difference in belief, and any attempt to provide such reinterpretations, perhaps by introducing a new term for the deviant use (“tharthritis”) would fail, since there would be no recognized standards for the new term (1979, 94).
Burge does not, however, argue for the stronger claim that the meaning of an individual’s words must be that of the community language. Here condition (ii) is essential. There is an important distinction, he says, between cases of incomplete understanding and cases where someone utters something without any comprehension (Burge 1979, 89). We standardly reinterpret when it comes to foreigners who lack a command of the language, we adjust for regional dialects, and there is a class of radical errors (tongue slips, Spoonerisms, malapropisms, serious misunderstandings) where we reinterpret even when the individual is part of our community and committed to it. For instance, Burge argues, if a competent and reasonable speaker thinks that “orangutan” applies to a fruit drink, we will not take his utterance of “I have been drinking orangutans for breakfast the past few weeks” to have its standard meaning (90‒91). An important task for Burge, therefore, is to motivate the idea that there is a distinction between the radical cases, which do require reinterpretation, and the non-radical ones, which instead involve incomplete understanding of (standard) meaning. The challenge is to do this while at the same time uphold a distinction between the case where the speaker is fully competent but makes an empirical error, and the case where the error is such that it involves incomplete understanding rather than empirical error. Addressing this challenge requires spelling out the underlying semantic assumptions.
It is important to keep distinct three different questions. First, the semantic question: What is meaning? Second, the metasemantic question: What is it to use a word with a given meaning? Third, the epistemic question: What is it to know the meaning of a word? The semantic question concerns the individuation of meaning or semantic value. For instance, someone may hold that the meaning of a term is given by its inferential role, or by a set of definite descriptions, or that meaning is nothing over and above reference. The metasemantic question concerns the facts in virtue of which it may be true that someone uses a word with a given meaning (construed along the lines suggested by the semantic theory). For instance, it may be held that e means M as used by S in virtue of facts about S’s dispositions to use e, or in virtue of facts about the causal-informational links between S and her environment. Finally, the epistemic question concerns what it is to know that e means M. One answer might be that knowledge of meaning simply involves the ability to use e correctly, in accordance with its application conditions. Another idea is that knowledge of meaning consists in having correct metalinguistic beliefs about the meaning of e, tacitly or consciously.
With these distinctions in hand, it is possible to see how the three issues can come apart. Assume, for instance, that someone endorses an inferential role semantics, holding that the meanings of theoretical terms, such as “electron,” are given by a certain inferential role. This is naturally coupled with a semantics according to which “electron” means electron, as used by speaker S, if and only if S is disposed to use “electron” in accordance with its constitutive inferential role (i.e., the role that serves to individuate this particular meaning).4 If so, clearly, there can be no incomplete understanding of meaning: a subject who does not use the term (or is disposed to use it) in accordance with the meaning (inferential role) of “electron” in the public language, uses it with a different meaning.5 To allow for incomplete understanding, it has to be argued that the subject may use “electron” to mean electron, despite the subject’s usage in itself being insufficient to determine that “electron” means electron. That is, it has to be held that meaning is determined by facts beyond individual use. The result is a form of metasemantic incomplete understanding, applied to the case of linguistic meaning: in the case where S’s own dispositions are insufficient to determine that she means M by e, other meaning determining facts may pick up the slack such that e means M in the language of S.
Which facts? According to the deferential account, proposed by Burge, facts about other people’s usage. Thus, S’s reliance on other speakers whose use does meet the full metasemantic conditions ensures that her word has the standard meaning despite her deviance. The upshot is that there are two distinct sets of facts in virtue of which S may mean electron by “electron”: a set of “complete” inferential dispositions (in the case of the expert) and a set of less complete such dispositions coupled with the further meaning determining factor (deference to expert use). The function from the meaning determining facts to meaning will thus be many-one:
“electron” means electron for S
full inferential disposition partial disposition + deference
This also illustrates that what it is to fully understand the meaning of a term, and therefore to incompletely understand it, depends on the semantics. In the case of Burge, the semantics in the background does not seem to be inferential role semantics but rather some version of traditional definitionalism according to which meaning is to be understood in terms of necessary and sufficient conditions given by the linguistic conventions.6 He suggests that the patient uttering “I have arthritis in my thigh” does not make an ordinary empirical error but a linguistic or conceptual error, since in the actual community “arthritis” does not apply to ailments outside the joints “by a standard, non-technical dictionary definition” (Burge 1979, 78). Similarly, Burge (1978) argues that someone may doubt an analytic truth, such as “a fortnight is a period of fourteen days,” and still use the term “fortnight” with its standard meaning because of her commitment to the community standards. In such a case, the individual has an incomplete understanding of “fortnight.” On Burge’s view, therefore, the subject has an incomplete understanding of a word when she fails to use it in accordance with its definition (but still has a minimal competence with the word).
A corollary to this is that someone who rejects inferential role semantics (and traditional descriptivism) in favor of a direct reference semantics, will deny that failing to draw certain inferences (or assenting to certain sentences) constitutes having an incomplete grasp of the meaning of the expression. Since the inferences (or descriptions) are not constitutive of the meaning, failing to draw them does not imply incomplete understanding of the meaning of the expression. Indeed, on a direct reference theory it is difficult to see how to make room for incomplete understanding of this sort in the first place: there is simply nothing in the meaning of the term to fail to grasp.7 Direct reference theories require some version of a causal-externalist metasemantics, since the determination of meaning is meant to be independent of associated descriptions. For the expression to have a meaning in the first place, therefore, the right causal connections need be in place. But once those are in place there is no such thing as grasping the meaning more or less fully.8 An Earthling who dissents from “Water is H2O” dissents from a necessary truth, but her error does not imply that she has an incomplete grasp of the meaning of “water,” since it is not part of the meaning of “water” that it is H2O.
For there to be incomplete understanding of meaning in the metasemantic sense, thus, two conditions need be met. First, it has to be possible for S to use expression e with meaning M, even when S’s use in itself is insufficient to determine that e means M. Second, the semantics of e must be sufficiently rich to allow for a distinction between factual errors and meaning errors. The first condition is metasemantic, the second semantic. The most common strategy to meet the first condition, again, is by appealing to the idea that the speaker defers to others, along the lines suggested by Burge. An interesting question is whether there is some other way of meeting the condition. Could the experts themselves have an incomplete grasp of meaning? Could it be the case that no one in the linguistic community fully grasps the meanings of the terms used? I shall return to this question in section 2.
Another question is whether there can be other types of incomplete understanding that do not depend on the metasemantic condition. One type of incomplete understanding concerns the notion of knowledge involved. There is an important distinction between conscious, personal level knowledge, and tacit, subpersonal level knowledge. Thus, a speaker may have a full tacit grasp of a concept (in the sense of using it fully in accordance with its meaning), and yet fail to have conscious person level knowledge of meaning. For instance, James Higginbotham (1998) has suggested that we need to distinguish between knowledge of meaning in the sense of tacit conceptions of meaning and in the sense of having an adequate conscious view of meaning. Assuming that the subject has knowledge of meaning in the first sense (i.e., has an accurate, tacit conception of the meaning of e), she may still fail to have person level, conscious knowledge of meaning in which case she can be said to have an incomplete understanding of meaning.
The existence of this type of incomplete understanding should be uncontroversial. Competent speakers rarely, if ever, bother to form conscious views of the meaning of their terms—that is something left to linguists and philosophers—and if they were to try they may easily go wrong.9 At the same time it should be clear that this way of drawing the distinction between full and incomplete understanding of meaning is orthogonal to the first one: even a speaker who has full understanding in the metasemantic sense, using the expression in accordance with its meaning, may have incomplete understanding in the sense of not being able to provide an explication of the meaning of the term (spelling out its inferential role or providing a definition). Nevertheless, the distinction between these two types of knowledge, between implicit conceptions and explicit awareness, is of relevance when we turn to the case of concepts.
2. Metasemantic Incomplete Understanding of Concepts
2.1 Deference to Experts
As noted above, Burge bases his arguments for incomplete understanding of concepts on the case of incomplete understanding of word meaning. He is quite explicit on this, noting that “many concepts (or mental content components) are like words in that they may be employed before they are mastered” (1979, 117). However, given the important differences between words and concepts it is to be expected that the notion of incomplete understanding of words cannot be straightforwardly applied to the case of concepts. Words are conventional signs, standing in an arbitrary connection with what they express, and we can therefore distinguish between employing a term with understanding and employing it without understanding. Concepts, by contrast, are not conventional signs, they do not stand in an arbitrary connection to what they express, and it is not obvious what it means to “employ” a concept without understanding it. Indeed, the very notion of “understanding” appears misplaced here, since there is no obvious vehicle in place to be understood. On the contrary, it would seem, using a concept just is grasping it, thinking a thought containing it. How, then, could there be a gap between employing or possessing a concept and understanding it?10
Burge’s motivation, in part, is simply the assumption that concepts stand in a close connection with meaning. Indeed, his view seems to be that they are identical. Concepts, he suggests, constitute the meanings of the speaker’s words: “When we say ‘That’s a chair’, we express a thought that that’s a chair; we express the concept chair with the word ‘chair’. The concept is not to be distinguished from the meaning of the word ‘chair’” (1993, 312). If so, incomplete understanding of meaning, in the metasemantic sense, will imply incomplete understanding of concepts in the same sense. That is, incomplete grasp of concepts results from the existence of cases where the speaker has a concept C, thinks thoughts containing it, without meeting the full metasemantic conditions for having C. Thus, Burge subscribes to a metasemantic version of thesis (ICU):
(ICUMS) S may think thoughts containing concept C, even if S’s use fails to meet the full metasemantic conditions for having C.
The patient who assents to “I have arthritis in my thigh” fails to meet the full metasemantic conditions for the concept arthritis, since it is a conceptual truth that arthritis afflicts the joints only.11 If one were to look to the patient in isolation, there would therefore be no reason to attribute the concept of arthritis rather than some alternative concept, such as tharhritis (the concept of a disease afflicting the joints and the ligaments)—indeed, there would be greater reason to attribute the latter concept to him. Nevertheless, because of his commitment to the community practice, and because he has basic competence with the concept, he is to be attributed the concept used by the experts and ascribed a conceptual error.
A similar idea lies behind Christopher Peacocke’s (1992) distinction between possession conditions and attribution conditions. Peacocke proposes an inferential role semantics that is closely tied to what it is to possess a concept (i.e., think thoughts containing it). To possess the concept and, for instance, S needs to find certain transitions primitively compelling—those constitutive of the concept (such as the transition from p, q to p and q, etc.). Since possession conditions serves the task of individuating concepts, meeting the possession conditions is meeting the full metasemantic conditions and involves having full understanding of the concept.12 However, Peacocke suggests, one can have the concept without meeting these conditions, since one can have the concept by meeting a set of weaker conditions—the attribution conditions.13 For instance, whereas the concept red can be individuated by a set of possession conditions (such as S having the disposition to apply the concept in response to a certain perceptual experience), a subject can be said to think thoughts involving the concept red in virtue of the following, weaker attribution conditions:
a. The subject is willing sincerely to assert some sentence of the form “___red___” containing the word “red” (or some translation of it).
b. He has some minimal knowledge of the kind of reference it has (e.g., that it is a color word).
c. He defers in his use of the word to members of his linguistic community.
Conditions (b) and (c) correspond to Burge’s conditions (ii) and (i) above. And just like Burge, Peacocke stresses the essential role of deference: “It is what distinguishes the case we are interested in, partial understanding (and partial misunderstanding) of a word in a communal language, from the quite different case of an individual’s taking over a word from his community and using it in his own individual, different sense” (1992, 29).
How is the notion of deference to be understood in the case of concepts? In the debate, it is often assumed that it can simply be modeled on the case of linguistic deference. Thus, while Peacocke’s conditions (a)‒(c) are meant to spell out the attribution conditions of the concept red they are formulated in terms of the word rather than the concept. Peacocke notes this and says that while possession conditions are given for concepts “attribution conditions and deference-dependence primarily concern words and sentences” (1992, 30).14 There is a reason for this. To defer to the community the subject needs to have a deferential intention. In the case of word meaning, she needs to have the intention that her word “red” has the meaning and extension that the community takes it to have. This requires ascribing metalinguistic intentions to speakers and that is not trivial, in particular since it is meant to explain how small children learn their first language—children, presumably, do not have metalinguistic intentions. In the case of concepts, however, matters are worse, since it seems to require that the individual has metalevel thoughts about concepts.15 Indeed, it is not even clear how such an intention is to be spelled out in the case of concepts, since it is not clear what the equivalent to the word is in the case of concepts. The only plausible way to do it, it would seem, is via the linguistic expressions: “My word ‘arthritis’ expresses the same concept as is expressed by experts in my community using this word.”
Assuming that it is unproblematic to attribute intentions of this sort to ordinary speakers, the challenge is to explain how this intention can succeed: How can it be the case that the subject uses the same concept as an expert merely in virtue of intending to use the same concept? After all, having an intention does not in general guarantee its success. Adding that the subject has a minimal competence with the concept, notice, does not help, since she also has a minimal grasp of any number of alternative concepts (such as tharthritis, a disease of the joints and the ligaments). Precisely how is it that facts about other people can serve to determine the concepts of an individual who may not even have interacted with the relevant group of people, and may only have the vaguest idea about who they are and how the concept is to be used?
Mark Greenberg (2001, 141‒143) argues that standard theories of content do not address this important question. The theory, he argues, must be able to explain how deference enables a person who lacks full grasp of a concept nevertheless comes to have the concept. For instance, a dispositional view must explain how someone who lacks the required dispositions nevertheless has the concept in virtue of deference, but any such explanation will in effect involve rejecting the tenet that having a concept is having the appropriate disposition. If the theory simply ends up by saying that there are two different ways in which a subject may have a concept, it in effect gives up on its claim what it is in virtue of which an individual thinks thoughts with a given concept. The dispositionalist, Greenberg argues, thus gives up on the prospects of a unified view.
It is not clear how problematic the lack of a unified theory is. As suggested above, it is possible to give a many-one construal of the function from the concept determining facts to concepts, and if so it is possible to provide an account of concept determination which is not unified in Greenberg’s sense.16 The real challenge facing this proposal, arguably, is not that the metasemantics is many-one but that it remains unclear precisely how the subject’s intention to defer to other people can have the result that facts about how they use concepts serve to determine the concept she thinks with.
There are attempts to spell out the details of the mechanism of semantic deference. One proposal is that this type of linguistically mediated deference implies that the subject does not have the concept of the expert but, rather, a type of metalinguistic, deferential concept such as the condition called “arthritis” (Donnellan 1993, 167). This avoids the attribution of idiosyncratic concepts to the deferring individual (e.g., the concept tharthritis), without having to face the challenge of explaining how deference serves to make it the case that the subject has the precise same concept as that of the community. Of course, it also means that it is not true that the patient believes that he has arthritis in his thigh. What he believes, rather, is that he has a condition called “arthritis” in his thigh. This account of deference, therefore, is not compatible with Burge’s account.
An alternative construal of deference is proposed by Recanati (1997, 2000). According to Recanati there are genuinely deferential concepts—concepts that have an extension only if there are experts who need not rely on others. These are syntactically marked and are to be understood as a form of indexicals: they have a character that serves to bring in other speakers whose usage determines the full content expressed. Burge’s thought experiment, according to Recanati, involves deferential concepts of this sort. Although the patient does not possess the concept arthritis, he does entertain a mental representation whose content is that he has arthritis in his thigh. This is possible because he possesses another, deferential concept whose character is such that it allows him to have thoughts about arthritis via reference to a competent speaker.
Recanati therefore provides a detailed account of the semantic function of deference that secures the conclusion that when a subject relies on others she can think thoughts whose content is determined by other people’s usage. At the same time, the appeal to special deferential concepts does not support thesis ICU, since it does not support the distinction between having a concept and fully understanding it. For this reason it is also doubtful that Recanati’s deferential concepts can be fitted into Burge’s framework. It is an essential aspect of Burge’s thought experiment, and of his social externalism, that an individual who has incomplete mastery of a concept has the very same concept as the experts do. Moreover, on Burge’s view, concept grasp is a matter of degree, developing gradually, whereas Recanati’s account implies a sharp distinction between having a deferential concept and having another, non-deferential one.17
To reach Burge’s conclusion, therefore, the role of deference must not be to introduce special deferential concepts. Rather, its role must be metasemantic, explaining how facts about other people’s usage can serve to determine the concepts of an individual whose usage does not meet the full metasemantic conditions. This raises the question whether it is possible to account for incomplete understanding of concepts without relying on the idea that individuals defer to others. Leaving aside the problems concerning deference, there are other reasons one might want to make room for incomplete understanding of concepts without deference. In particular, it might be held that it should be possible also for the experts to have an incomplete understanding of the concepts they think with—after all, incomplete understanding as a result of deference is necessarily limited to non-expert understanding.18
2.2 The Role of the External Environment
The idea that experts can have an incomplete grasp of concepts plays an important role in Burge’s later writings. To illustrate this possibility Burge (1986) designs a different thought experiment, involving the term “sofa.” Burge imagines an individual, A, who is a fully competent speaker and has a full grasp of the conventional meaning of “sofa.” Nevertheless, he develops a nonstandard theory about sofas (according to which sofas are religious artifacts) and as a result comes to doubt that sofas are furnishings to be sat on. In our community such an individual would be wrong, but in a counterfactual community (where sofas are religious artifacts), A would be right and his term “sofa” would express a different concept. Since A is an expert in the actual community, he does not have an incomplete grasp of the conventional meaning, but merely a nonstandard theory. The difference in concept derives not from the community practice but from differences in the environment in the two worlds. This, Burge suggests, makes the thought experiment extremely comprehensive: “Nearly anything can be the topic of nonstandard theorizing. Similar thought experiments apply to knives, clothing, rope, pottery, wheels, boats, tables, watches, houses” (1986, 709).
The upshot, according to Burge, is that we need to distinguish between conventional meaning and concepts: If the doubt that sofas are furnishings meant for sitting proves to be well founded, then there will be a change in the conventional, meaning giving definitions, but the concept expressed (the concept sofa) would be the same before and after this change.19 While community use determines conventional meaning it does not determine “cognitive value,” the real concept expressed. For this reason, even if the expert has a complete grasp of conventional meaning she may not have a complete grasp of the concept expressed. Something similar takes place, according to Burge, in the case of theoretical change in science. Although Dalton defined “atom” in terms of indivisibility, and this definition was later discarded, he nevertheless had our concept atom (Burge 1986, 716). The revision of meaning giving definitions provides an improved understanding of the concept expressed and at a final, ideal point the experts may achieve full understanding of the concept: “Understanding of this sort bears comparison with the sort of ultimate insight, championed by the rationalist tradition, that was regarded as concomitant with deep foundational knowledge” (718).
This type of incomplete understanding, just as in the case of deferential incomplete understanding, derives from the idea that individual use does not fully determine meaning or concepts. However, it does not depend on the assumption that the metasemantic determination relation is many-one (although this will typically be allowed) but, rather, on the assumption that features in the individual’s environment serve to determine meaning:
“atom” expresses the concept atom for S
individual dispositions + environment of S
The gap between individual understanding and full understanding, thus, is a result not of there being more than one way of having a concept, but of the physical environment playing a role in determining the concept expressed.
The appeal to environmental factors, and the suggestion that even the experts may be wrong about the essential nature of a kind picked out by a general term, is reminiscent of what Kripke (1980) and Putnam (1975) says about natural kind terms. At the same time, there cannot be incomplete understanding unless the semantics goes beyond direct reference theories: there has to be some content that the subject has an incomplete grasp of.20 This, also, seems to be Burge’s view. His proposal is that there is an ultimate, metaphysical, essence-determining definition to be had, explicating the concept expressed, and that until such a definition is reached subjects will have an incomplete grasp of the concept expressed.21 These definitions provide ultimate, “intellectual norms” for inquiry and serve to individuate the concepts expressed.22
This is a controversial claim. Standard natural kind externalism does presuppose that natural kinds have essences, but not that these essences provide ultimate definitions of the concepts expressed. Moreover, the talk of hidden essences is arguably limited to the case of natural kinds. What are we to make of the idea that sofas, knives, clothing, rope, pottery, wheels, boats, tables, watches, houses have essences of this sort—essences that may not be known even by the most competent speakers? Someone skeptical of this idea might accept Burge’s explanandum (that conventional definitions can be rationally doubted) but provide an alternative explanans: All “definitions” can be rationally doubted, since there is no such thing as sentences that are true in virtue of meaning alone, and no such thing as sentences that one must assent to in order to fully understand the meaning of the component expressions. If so, there will not be a distinction between empirical error (non-standard theory) and incomplete understanding and thesis ICU will not be upheld.23
3. Epistemic Incomplete Understanding
As noted in the case of linguistic meaning, someone may have a complete grasp of the meaning of a word (using it correctly) and yet fail to provide a proper explication of it. Similarly, someone may be fully competent at using a concept and yet be quite unable to explicate it—being able to use a concept is one thing, reflecting on the concept quite another.24 This introduces a distinct notion of incomplete understanding of concepts, one that is not metasemantic but epistemic. To have an incomplete grasp of a concept in this sense is just to be unable to explicate it properly:
(ICUEXP) S may think thoughts containing concept C, despite the fact that S is unable to provide a proper, metalevel explication of C.
This notion of incomplete understanding also plays a role in Burge’s writings and is related to his distinction between concepts and conceptions of concepts. According to Burge conceptions of concepts are tied to explications, to the epistemic definitions the subject would give of the corresponding word, spelling out the conditions that a speaker treats as most basic for a concept. These are distinct from metaphysically correct definitions, spelling out actual necessary and sufficient conditions (Burge 1993, 314). Epistemic definitions may therefore be incorrect and get replaced with more accurate ones without a change in meaning or concept expressed: “One may think with a concept even though one has incompletely mastered it, in the sense that one associates a mistaken conception (or conceptual explication) with it” (317).25
It should be clear that one could accept thesis ICUEXP without accepting ICUMS. Even if one holds that concepts are determined by facts about the individual, it does not follow that the individual will be infallible when it comes to explicating the concepts employed. Assume, for instance, that one takes concepts to be individuated by inferential role. In such a case, acquiring metalevel knowledge of one’s concepts would involve reflecting on one’s use of the concept: the role it plays in one’s reasoning, under what circumstances one would and would not apply it and so on. It is quite clear that such a process is cognitively demanding and that one might easily go wrong, for instance, by ascribing some inferences as constitutive of the concept while they are not and vice versa.26
At the same time, there is a connection in the other direction between the two types of incomplete understanding: ICUEXP follows from ICUMS, since an individual who fails to meet the full metasemantic conditions will not be in a position (prior to investigations of her social and physical environment) to provide a proper explication of her concepts. Burge’s view, therefore, is most likely that the two are connected and that there is widespread epistemic incomplete understanding because there is widespread metasemantic incomplete understanding.
In the debate there is also a further notion of “conception of concepts,” one that does not correspond to Burge’s epistemic definitions, but rather to implicit conceptions construed at the subpersonal level. Conceptions in this sense are meant to capture how the subject conceives of her concepts, as revealed in her usage, and may or may not correspond to the explicit definitions she is disposed to provide. Thus, Higginbotham (1998) suggests that we distinguish between having a conscious view of a concept (corresponding to Burge’s conceptions) and having a tacit conception of a concept. Tacit conceptions serve to explain usage but need not be immediately accessible to the subject, whereas views are conscious and explicit. Like Burge, Higginbotham suggests that our conscious views of concepts may be mistaken, but he also suggests that when there is metasemantic incomplete understanding there is a subpersonal level on which we have a mistaken view of the concept. Higginbotham cites the example of a child who refuses to apply “fork” to other forks than those of her own household. In such a case, Higginbotham suggests, we should say that the child has a mistaken conception of the concept FORK, and this conception would have (in part) the content:
(CON) FORK is true only of tined utensils in my house
Conceptions, in this sense, are therefore meant to explain the individual’s application of concept C and may be correct or incorrect.
A similar view can be found in Peacocke’s account of implicit conceptions (2003, 2008). According to Peacocke, implicit conceptions are unconscious psychological states, typically subpersonal, corresponding to the individual’s understanding of a concept (as well as to her linguistic understanding and grasp of logical principles). One of Peacocke’s aims is to explain how one can apply a concept, and follow logical principles, prior to having any explicit understanding of them. For instance, understanding the connective “or” involves having an implicit conception with the content that any sentence of the form “A or B” is true if and only if either A is true or B is true. By reflecting on her pattern of semantic evaluation (as well as using simulation to gain knowledge of how she is disposed to evaluate) the subject may come to have explicit knowledge of the concepts she applies and the principles she follows. The process is fallible, but when done right the subject gains reflective understanding and the resulting belief is a priori. Peacocke suggests that a very good illustration of this process is how Leibniz and Newton struggled with the concept of the limit of a series. Although they were unable to provide an explicit definition of the concept, they clearly had the concept and their application of it was guided by an implicit conception of it (2003, 121).27
As noted earlier, Peacocke also allows for metasemantic incomplete understanding, and like Higginbotham he construes this as involving mistaken implicit conceptions of the concept employed. He stresses, however, that the case of Leibniz and Newton is not an example of incomplete understanding in this sense, since that can occur only where there is deference—Leibniz and Newton themselves were experts and not deferring to anyone.28 Rather, Peacocke argues, they grasped a definite concept (had a correct implicit conception of it) while failing to grasp it “sharply” (they failed to have a correct reflective understanding of the concept) (2003, 122).29
According to Peacocke and Higginbotham, therefore, we need a threefold distinction between conscious views of concepts (Burge’s explications), the concept employed, and implicit conceptions (corresponding to the subject’s actual grasp of the concept). Burge, by contrast, denies the need for implicit conceptions in this sense. While he grants that mentalistic elements are required to explain the subject’s use of a concept, he questions the assumption that these need be conceptions of concepts. In many cases, he suggests, the subject may just use perceptually stored material, not unified under some conceptualized principle, combined with memory information, similarity principles, and so on (Burge 2003, 384). Similarly, Smith (2015) questions Peacocke’s idea that there need be determinate implicit conceptions of concepts underlying a subject’s usage of a term. Discussing the case of mathematical concepts, in particular that expressed by the term “derivative,” Smith argues that there is no reason to assume that Newton and Leibniz had one determinate implicit conception rather than another, since their reasoning and classification behavior is not sufficient to decide between equally possible alternative implicit definitions. Taking explicit articulations of the concept into account, Smith suggests, would not settle the matter either. For example, Leibniz articulates his own concept in terms of infinitesimals but “there are several known derivative concepts in real analysis that appeal to ‘infinitesimals’ rather than limits” (2015, 16).
Unlike Burge, however, Smith also takes this to throw doubt on the idea that there is a definite concept shared by Leibniz and Newton. If it is unclear which implicit conception Leibniz and Newton had, he argues, it is equally unclear which concept they were thinking with. A central point made by Smith is that once we pay attention to the details of mathematics there are any number of possible “derivative”-concepts involved—mathematical reality, he says, “pulls the word ‘derivative’ in different directions” (2015, 3). There are several derivative concepts and this means that the expression “the derivative concept” has no referent. The proposal that, somehow, Leibniz and Newton could defer to a future, optimal derivative concept is of no help, Smith argues further, since it is not clear what would count as the optimal derivative concept: “each derivative concept is optimal for some purposes, but none is overall globally optimal” (2015, 22).
This puts some pressure on the anti-individualist’s line of argument, even if she is willing (like Burge) to reject the appeal to implicit conceptions. Smith’s argument is not limited to the case of mathematical concepts. Similar worries apply to a term such as “arthritis.” It plays a complex role in medical theory and once one pays attention to the details of the case it becomes increasingly hard to support the idea that there is such a thing as “the concept of arthritis.” Arguably, there is a large range of cases of this sort. For instance, biologists disagree on how to individuate species and use different species concepts, and this makes all talk of the species concept problematic.30 This poses a further difficulty for the picture of deference and the claim that there is a definite concept which non-experts have an incomplete grasp of it. It also poses difficulties for the idea that appealing to the external facts can serve to determine the concept: just as mathematical reality pulls a word in different directions, so reality may pull a word like “arthritis” in different directions, and it is not obvious that physical reality itself can determine which is the “right” direction, independently of human interests and pragmatic concerns. Rather, the conclusion would seem to be that there is a great deal of indeterminacy, as Smith also suggests.
4. Incomplete Understanding and Cognitive Role
The most common challenge to thesis ICU concerns the role of concepts in capturing the individual’s cognitive perspective, her reasoning and actions. To say that concepts are incompletely understood in the strong metasemantic sense, again, is to say that there is a certain gap between the individual’s grasp of the concept (how she uses it) and the concept used. On Burge’s construal, the gap consists in the individual misapplying her concept, on the implicit conceptions view the gap also involves mistaken implicit conceptions about the concept. Either way, the result is that the individual reasons in a way that is not in accord with the concept employed. The challenge, then, is to explain how the attribution of incompletely understood concepts at the same time allows us to attribute thought contents that serve to make sense of the subject’s reasoning and actions.
Burge (1979, 90–91) is quite aware that anti-individualism appears to be in tension with the idea that content attributions should respect the assumption that individuals are by and large rational. How, he asks, can a rational person believe that she has a disease of the joints only in her thigh? However, he suggests that the anti-individualist has a reply readily available: the apparent irrationalities can be explained away by appealing to the notion of linguistic misinformation. Considering the objection that charitable interpretation requires that we do not attribute beliefs of this sort to a rational person he responds that this is not a good argument, since “there is nothing irrational or stupid about the linguistic or conceptual errors we attribute to our subjects. The errors are perfectly understandable as results of linguistic misinformation” (1979, 100).
The question, again, is whether the parallel between concepts and meanings holds up. That the patient misunderstands the meaning of the word “arthritis” may explain why he uses the word in such an odd way (uttering “I have arthritis in my thigh,” for instance), but it does not explain how he can have the belief that he has arthritis in his thigh. The appeal to linguistic misinformation helps rationalize the subject’s perspective only to the extent that it suggests a gap between meaning and content, such that the subject cannot be said to literally believe what he says. But this, of course, is precisely what Burge disallows, since the very point of his anti-individualism is that there is no such gap—not only does the misapplied word have its standard meaning it also expresses the standard concept. Thus, the appeal to linguistic misinformation does not help rationalize the subject’s cognitive perspective.
A related strategy is to take the appeal to linguistic misinformation one step further and apply it to the level of concepts: The patient believes that he has a disease of the joints only in his thigh, but only because he has the mistaken thought that the concept applies to a rheumatoid disease of the joints and the ligaments. A version of this strategy has become quite popular among externalists in the recent debate where it has been argued that problems concerning externalism and cognitive perspective can be solved by giving up the assumption that mental content is transparent—the idea that a subject can always tell, a priori, whether or not two thought contents are the same. By giving up this principle, it is held, we can explain why apparently rational individuals hold apparently irrational beliefs without having to give up on the externalist commitments.31 For instance, it can be explained why someone takes a different attitude toward the thoughts expressed by two synonymous sentences such as “Cilantro should be used sparingly” and “Coriander should be used sparingly”: The subject fails to realize that the contents are the same. Similarly, in a Twin earth scenario the subject may mistakenly think that two thoughts are the same (Water is wet and Twater is wet) and this explains why she draws certain invalid inferences.
This strategy, it should be clear, presupposes a richer notion of conceptions than Burge allows for, along the lines suggested by Higginbotham and Peacocke. What is proposed, in effect, is that the subject’s mistaken metalevel beliefs about the concept involved explain her apparent irrationalities. If one is skeptical of the attribution of such beliefs on the part of the subject (as Burge is), then this strategy is not available. Moreover, it is a good question whether the attribution of mistaken beliefs about concepts really helps meet the challenge. The proposal at hand is not that there is a second type of content (“narrow” content) that serves to explain the subject’s reasoning and actions, but that the mistaken metalevel beliefs explain why her reasoning is less than rational. At most, therefore, such beliefs provide exculpations: the subject is not to be blamed for her irrational reasoning, since she is mistaken about the content of her beliefs. The challenge, however, was not to exculpate but to explain how the subject’s reasoning can be made rational (avoiding ascribing contradictory beliefs to the subject, etc.) and the attribution of metalevel beliefs cannot do that.32
Sanford Goldberg (2002) suggests a rather different strategy, based on an alternative construal of conceptions. Goldberg focuses on Burge’s sofa-thought experiment and considers the objection that the subject (Adam) cannot coherently doubt that sofas are large, overstuffed pieces of furniture meant for sitting, and yet be using “sofa” literally, since “sofa” just means piece of furniture … meant for sitting. Burge, Goldberg sums up the objection, is “in effect asking us to imagine that Adam’s perspective is incoherent” (2002, 606). To avoid this conclusion, according to Goldberg, the anti-individualist should reject the assumption that conceiving of something as a sofa, is conceiving of it as a piece of furniture … meant for sitting. Instead we should adopt a minimalist construal of conceptions, according to which Adam simply conceives of sofas as sofas, and reasons his way to the conclusion that sofas are not large, overstuffed pieces of furniture meant for sitting. The assumption that there would be something illogical about this is simply based on the mistaken idea that Adam is rejecting some kind of conceptual truth. This is just mistaken, Goldberg argues, since the concept of sofa is distinct from the concept of large, overstuffed piece of furniture made for sitting. Instead of being illogical, Goldberg concludes, a person such as Adam “may simply have false empirical beliefs about sofas (conceived, in the standard way, as sofas)” (614).
Now, it should be clear that if indeed the individual simply has a false empirical belief about sofas, then there is nothing incoherent about his perspective and there is no reason to say that he has an incomplete grasp of the concept sofa. Similarly, on the assumption that the arthritis patient simply has a false empirical belief that he has arthritis in his thigh, there is nothing incoherent about his perspective and no reason to ascribe incomplete understanding to him. However, again, the anti-individualist is not in a position to describe Adam and the patient this way. If Adam is merely having a false empirical belief, then there is no reason to accept Burge’s claim that “sofa” expresses a different concept in the counterfactual world. The difference between the two worlds would just be that people in the counterfactual world have a different theory about sofas and use sofas for different purposes. Similarly, if the patient is merely making an empirical mistake, there is no reason to accept the claim that his twin must have a different concept.33
This brings out the precise nature of the challenge concerning incomplete understanding and cognitive role. The challenge can be put in terms of a dilemma: Either the subject’s deviance must be such that it warrants the conclusion that she has an incomplete grasp of the concept, rather than making an empirical error, and this requires saying that the subject fails to apply the concept in accordance with its content which raises the worry that her perspective will not be coherent. Alternatively, the subject does not fail to apply the concept in accordance with its content, but is merely developing a nonstandard theory (or making an empirical error), in which case her perspective will not be incoherent but then there is no reason to say that she has an incomplete grasp of the concept employed.
One response to this challenge is to question the assumption that concepts should capture the subject’s cognitive perspective: believing that one has arthritis in one’s thigh is holding an incoherent belief, but the theory of content need not be constrained by the assumption that subjects are (by and large) rational.34 The question is whether this response is acceptable. Naturally, people are sometimes irrational—every theory of content has to recognize the existence of a variety of psychological failures of rationality as a result of illness, confusion, intoxication, repression, priming, etc., etc. However, the thesis that there is incomplete understanding in the metasemantic implies ascribing failures of rationality also in cases where there is no psychological malfunctioning, simply as a result of the commitments of the theory of content. This illustrates that whether thesis ICUMS is ultimately cogent depends on what the theory of content should do—on whether the function of content is to rationalize the subject’s perspective, her reasoning and actions.
The focus of this survey has been a version of the thesis of incomplete understanding according to which a subject may have a concept while failing to meet the full metasemantic conditions, thesis ICUMS. This version of the thesis has been distinguished from a purely epistemic one, ICUEXP, according to which one may have a concept while failing to have full explicative, metalevel, knowledge of the concept. The two types of incomplete understanding, I have stressed, are distinct: It is possible to hold that there is epistemic incomplete understanding without metasemantic incomplete understanding. However, the converse does not hold: Metasemantic incomplete understanding leads to epistemic incomplete understanding, since a subject who fails to meet the full metasemantic conditions will not be able to provide a proper explication of the concept employed (prior to investigations of her social and/or physical environment).
There are other versions of thesis ICU. For instance, David Bourget (2015) argues that there is a difference between believing a proposition (having a concept) and grasping it, as when someone believes that smoking is dangerous but does not really grasp this proposition until a colleague dies in lung cancer. This shows, according to Bourget, that there is a mental act of understanding that goes beyond mere belief and plays a role in rational behavior, and he suggests that this act involves a phenomenal experience with the proposition in question as content (2015, 19). This construal of thesis ICU, therefore, depends on the idea that there is a type of cognitive phenomenology. This is an interesting, and much debated idea that cannot be properly addressed in this survey. The focus here, instead, has been on versions of ICU that do not depend on phenomenological assumptions but construe incomplete understanding purely as a cognitive failure: either on the level of first-order beliefs (as when the subject fails to meet the full metasemantic conditions) or on the level of metabeliefs (as when the subject fails to have explicative knowledge of her concept) or both.
The main challenge to ICUMS is that it fails to capture the subject’s cognitive perspective: a subject who has incomplete understanding in this sense is a subject who fails to use the concept in accordance with its content. As a result, she will be described as being irrational, despite no psychological malfunctioning. However, even if this challenge can be met, the problems concerning deference remain. Anti-individualism need explain precisely how it is that the subject’s willingness to defer to others implies that she has the very concept that they think with—even if she has a rather vague grasp of who these other subjects are and what concept it is that they employ. According to some, deference is not to the experts but to “facts,” features of the world, in which case this challenge is even greater. The attempt to show that there can be incomplete understanding without deference depends on the controversial idea that concepts have metaphysical definitions, independently of the epistemic definitions we (including the experts) are able to give.
In conclusion, thesis ICU is a central and challenging thesis in contemporary philosophy of mind and language. The thesis is more or less problematic depending on how it is spelled out. While there are versions of it that are fairly innocuous there are other versions that face serious difficulties.
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(1) See also Higginbotham, who argues that the autonomy of meaning (its independence of communicative intentions) implies that one can have “partial knowledge of one’s own idiolect” (1989, 155).
(3) Burge’s talk of “our intuition” can be contested. Not all philosophers share Burge’s intuition (for a critical discussion, see Crane 1991), and if Burge wishes to make a claim about ordinary speakers he would need empirical evidence concerning layspeaker intuitions.
(7) For a discussion, see Rattan (2010). Direct reference theories, he argues, do not have the resources to distinguish between factual error and conceptual error and hence no resources to account for incomplete understanding (19). See also Greenberg (2001, 135–137).
(8) Causal-externalist theories, such as Kripke’s account of how a name is first fixed on to an object, are typically combined with the idea that there is a chain of communication leading back to the initial baptism (Kripke 1980). This means that there is an element of reliance on others. However, it is not the kind of reliance that allows for incomplete understanding of meaning, since the same point applies: either the subject stands in the right causal-historical connection to the initial baptism or she does not. If she does, then the word has the meaning that was initially determined, but no matter how little the individual knows about the referent, there can be no such thing as incomplete grasp of meaning as long as the reference exhausts the meaning.
(9) Woodfield (2000, 434) suggests that this allows for a different type of deference, since an individual may defer to a definition provided by an authoritative linguist or academy. It should be stressed that this is quite distinct from the metasemantic deference appealed to by Burge, where the idea is that expert first-order applications of a term serve to determine the meaning and extension of the term.
(10) For some skepticism, see Bach (1988, 88 n. 3): “It is clear how one can use a word that one incompletely understands but I have no idea of what it is to think with a concept that one incompletely understands, for I have no idea of what it is to understand a concept over and above possessing it.”
(11) As noted by Brown (2004, 295) incomplete understanding need not involve actual error—it is sufficient that the subject is unsure whether the concept applies to certain kinds of things (where there is a determinate fact about whether or not the concept does apply to that kind of thing). For example, the subject may be unsure whether “arthritis” applies to problems of the thigh.
(12) Davis (2005) criticizes Peacocke’s epistemic account of concept individuation and argues that while full mastery of a concept is individuated epistemically, having a concept is not. It is possible to have the concept of a quark without believing anything about quarks, but having full mastery of the concept requires knowing everything one should believe (even know) about quarks (2005, 319). However, if the concept is individuated non-epistemically, and the metasemantic conditions are non-epistemic, it is not clear in what sense the subject who knows little about quarks has an incomplete grasp of the concept quark rather than simply incomplete knowledge about quarks.
(13) For a similar distinction between two ways of having a concept, one more demanding than the other, see Bealer (2000, 2008), Crimmins (1989), and Siebel (2005). On Bealer’s view, full understanding is of importance, since he takes it to involve truth-tracking intuitions, allowing for an account of a priori knowledge.
(15) It should be stressed, again, that it does not suffice to appeal to the fact that the speaker is disposed to stand corrected, since this willingness can be construed differently: as a willingness to adjust one’s usage to that of the community (along the lines suggested by Davidson). Deference therefore cannot simply be reduced to a disposition to stand corrected.
(16) Indeed, Burge (1982) suggests that there are three ways in which a subject may have a natural kind concept, such as water: in virtue of being an expert, in virtue of having been in causal contact with the kind, or in virtue of deferring to experts. Adopting a many-one account, notice, does not allow for a reduction of concepts to something else (such as dispositions), but it is compatible with the idea that concepts supervene on non-semantic facts, such as dispositions and the environment.
(18) An exception is the view called “temporal externalism” (Jackman 1999) according to which speakers defer to future communities so that meaning and mental content is determined by later linguistic developments. This would allow for incomplete understanding on the part of contemporary experts, but not on the part of the future experts who have reached the final, correct theory. Of course, the problems concerning the metasemantic role of deference are even more striking with respect to temporal externalism: How could it be that the linguistic behavior of speakers who do not even exist now serves to determine the concepts used by us today? For a criticism of temporal externalism, see Brown (2000) and for a reply Jackman (2005).
(19) Burge (1993) suggests that there is a notion of linguistic meaning corresponding to that of the concept—what he calls “translational meaning.” The upshot, therefore, is not that concepts are not meanings but that concepts are not conventional meanings. Instead, Burge suggests, we must make a twofold distinction between, on the one hand, conventional meaning and translational meaning, and, on the other, speaker conceptions and concepts. Conventional meaning and conceptions correspond to the subject’s grasp at a time (to epistemic definitions), whereas translational meaning and concepts correspond to the metaphysical or essence-determining definitions of meaning/concept possessed (1993, 314–315). See also Burge (1989).
(20) Since Putnam, unlike Kripke, holds that stereotypes are part of the meaning of natural kind terms he could allow for a type of incomplete understanding (although he suggests that the subject has to know the relevant stereotypes in order to use the word with its standard meaning).
(21) See also Burge (1989, 1993) and Rey (1983, 1998). Rey argues that Kripke’s and Putnam’s account of natural kind terms shows that the definition of a concept need not be known even by the most competent speakers: “the correct definition of a concept is provided by the optimal account of it which need not be known by the concept’s competent users” (1983, 255). The upshot, he suggests, is that an individual may be a competent user of a concept she doesn’t really understand.
(22) A related idea can be found in Greenberg (2001). Greenberg argues that concepts are determined by the standards to which a subject is responsible, and that facts play a role in determining these standards. This, Greenberg argues, allows for incomplete understanding also on the part of experts. See also Rattan (2010) for the idea that intellectual norms serve to individuate concepts in a way that is incompatible with pure use theories of content determination. Rattan proposes that this allows for a robust type of incomplete understanding applying also to experts (36‒37).
(24) That there is this type of incomplete understanding, notice, means that one can allow that the experts have an incomplete understanding without appealing to the notion of metaphysically correct definitions of concepts: An individual may be a fully competent user of a concept without having the required metalevel knowledge about the concept, without being able to fully explicate it. See Rattan and Wikforss (2015) for a discussion.
(25) See also Burge (1986): “One can know what one’s thoughts are even while one understands one’s thoughts only partially, in the sense that one gives incomplete or mistaken explications of one’s thoughts or concepts” (125).
(26) There are views of what it is to have a concept that are not compatible with this type of incomplete understanding, such as the claim (associated with the so-called Classical View of Concepts) that having a concept consists in consciously knowing its necessary and sufficient conditions. However, such views are widely recognized to be problematic and are not a necessary part of an individualist metasemantics. For a discussion, see Rey (1983).
(28) This case therefore illustrates (if Peacocke is right about it) how one can have a complete grasp of a concept in the metasemantic sense while having an incomplete grasp of it in the epistemic sense. Peacocke’s account of conceptual reflection is therefore an example of the type of account that is available to an individualist who rejects the metasemantic notion of incomplete understanding.
(29) Burge (1990) suggests that Frege held a similar view. According to Frege, Burge argues, the failure of mathematicians to lay down good definitions does not show that the relevant concepts are vague but only that even those with best understanding of the relevant concepts have not clearly grasped them.