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Four grades of ignorance-involvement and how they nourish the cognitive economy

  • Knowing the Unknown
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Abstract

In the human cognitive economy there are four grades of epistemic involvement. Knowledge partitions into distinct sorts, each in turn subject to gradations. This gives a fourwise partition on ignorance, which exhibits somewhat different coinstantiation possibilities. The elements of these partitions interact with one another in complex and sometimes cognitively fruitful ways. The first grade of knowledge I call “anselmian” to echo the famous declaration credo ut intelligam, that is, “I believe in order that I may come to know”. As construed here, one knows in this anselmian way that E = mc2 just in case one knows that sentence expresses a true statement, but without having to understand the proposition it expresses. Most epistemologists ignore the significance of this grade of epistemic involvement. In a second grade of epistemic involvement, knowing that E = mc2 is knowing what that sentence means and understanding the proposition it express. This is knowledge in the propositional or semantic sense, and is the dominant target of epistemological investigation. Tacit and implicit (TI) knowledge occupies another tier. A typical example would be something that someone has “known all along” but, until now, hasn’t had occasion to put her mind to it or formulate in words. TI-knowledge remains a minority interest in today’s epistemology. Operating at a fourth grade of epistemic involvement is what I call “impact”-knowledge, which is the knowledge of a matter at its deepest and most widespread. An example, to be discussed below, is the knowledge that was generated by the Wiles proof of Fermat’s last theorem. Its true importance lies not only, or even mainly, in its verification of a commonly accepted fact about numbers, but rather in its enrichment of the mathematics of elliptical curves and the promise it holds for greater advancement into the mathematical unknown. Knowledge of this fourth grade has yet to find a seat in the parliaments of epistemology. Knowledge of the anselmian sort is independent of the other three. Tacit and implicit knowledge is incompatible with anselmian and semantic knowledge but coinstantiable with impact-knowledge. Semantic knowledge is incompatible with tacit and implicit knowledge but coinstantiable with the others. Impact-knowledge is pairwise coinstantiable with the others. Below I will bring the ignorance partitions into such alignment as they have with these ones. In doing so, I’ll propose a naturalized causal response epistemology designed to give these interactive distinctions the theoretical air they need to breathe.

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Notes

  1. A welcome exception is Arfini (2019). For the unexceptional rest, see for example, the following: In a volume as authoritative as Goldman’s and Whitcomb’s Social Epistemology: Essential Readings, ignorance makes no appearance in the index. The same is true of Bernecker’s and Pritchard’s The Routledge Companion to Epistemology, save for one citation of Descartes’ argument from ignorance in the First Meditation, which hardly counts. Check Blackburn’s Oxford Dictionary of Philosophy and you’ll find nothing except “ignoratio elenchi”. Save for that one reference, there is no mention in these three places of autoepistemic inference or arguments from ignorance. Honderich’s The Oxford Companion to Philosophy comes to much the same end. It acknowledges the more general form of Descartes’ use of it in its scant entry on arguments from ignorance, and a briefer mention of ignoratio elenchi. See Goldman and Whitcomb (2001), Bernecker and Pritchard (2013), Blackburn (2005), and Honderich (2005).

  2. See here Kitcher: “Typically, a flourishing science is incomplete. At any time, it raises more questions than it can currently answer. But incompleteness is no vice. On the contrary, incompleteness is the mother of fecundity …. A good theory should be productive; it should raise new questions and presume that these questions can be answered without giving up its problem-solving strategies.” Kitcher (1983; pp. 46–48).

  3. Just to be clear, when I speak of the space-time world, I mean the world we ourselves inhabit as natural beings. For present purposes, considerations from relativity theory don’t apply. I am not invoking four-dimensional Minkowki space.

  4. Wiles (1995; pp. 443–551).

  5. Rescher (2007, 2009).

  6. The word “ignore” derives from the fourteenth century French “ignorer” (“to be unaware of”), derived in turn from the 14th c. Latin (“not to know, be unacquainted”), but also (“take no notice of, disregard”).

  7. See, for example, Arfini and Magnani (2016; pp. 612–627), and Arfini et al. (2018; pp. 37–50).

  8. For a discussion of how abductive reasoning operates in cognitive economies, readers could consult my (2012; pp. 148–161).

  9. For a good appreciation of the ups and downs of naturalizing the investigation of reasoning, see Urbanski and Klawiter (2018; pp. 583–597; Sect. 3), “What about the cognitive turn in logic?”.

  10. Goldman (1967; pp. 357–372). Woods (2013). A summary account appears as “Logic naturalized”, in 2016. See also Magnani (2015; pp. 13–36, 2018; p. 44).

  11. See, for example, Unger (1975).

  12. See, for example, Stich (1990).

  13. As formulated here, the conditions are most naturally and accurately read as characterizing propositional or semantic knowledge. It is also adaptable to the other grades of knowing, but the present formulation will do for illustrative purposes.

  14. Harman (1986; p. 2). In Gabbay and Woods (2005b; pp. 398–417).

  15. Later I will expand this notion (with suitable adjustments) to multiagents.

  16. The philosophical literature on consciousness also produces a deluge of rivalrous opinion. Two recent volumes of note are Paglieri (2012), and Jacquette (2018). It is a matter of interest that in neither of these volumes does ignorance appear in the index. Also of interest are Bachmann (2000), Breitmayer (2014), and Aru and Bachman (2017; pp. 128–135).

  17. Further details, can be found in chapter four of Errors of Reasoning. See also Doya et al. (2011), and Hohway (2013), Zimmerman (1989; pp. 166–175), Shiffrin (1997; pp. 49–64, 1976; pp. 177–236), and Mole (to appear).

  18. See further, Wilson (2002), Wegner (2001) and Carruthers (2011).

  19. This common sense view is nicely supported in Mole’s (2008; pp. 86–104).

  20. Dretske (1981). For reservations about the treatment of belief, see Gabbay and Woods (2003; chapter 7, Sect. 7).

  21. A phase transition in physics is a reversable change in a substance from a given state (e.g. liquid) to a qualitatively different state (e.g. gas) at a specific combination of temperature and pressure. See for example, Callen (1985; chapter 9). There is a link between Aristotle’s concept of potentiality (dunamis) and the phase-transitions of modern physics. For Aristotle, dunamis is a thing’s capacity to take on a new form without losing its identity. See Metaphysics, Book 8, 1 1046a 12, 1048a 25, 27. Frege thought the same about numbers. He thought that they could take on the cross-type form of sets without losing their identities. An item’s haecceity is that in virtue of which it is its self-same thing and not another thing. A thing’s quiddity is that in virtue of which it is the very kind of thing it is. Then the phase-transition thesis has it that there are ranges of cases in which a thing’s haecceity is unmolested by change and restoration of quiddity. An excellent modern treatment of kinds is Pelletier (2010).

  22. See here Orzumi et al. (2018; pp. 1–25).

  23. Jaakko Hintikka, “Who has captured the notion of information?”, in his Socratic Epistemology, pages 189–210; p. 189.

  24. Quoting Adriaans and van Benthem (2008; p. 12). A problem pressed by the first two kinds of information is that epistemic information can’t not be true, whereas probabilistic information needn’t be true. There is, however a route from the complexity conception to the probability conception by way of the set of all prefix-free programs under provisions of Kraft’s Inequality.

  25. www.cia.gov/library/publications/the-world-fact/docs/history.

  26. Here is Godfrey-Smith’s summary of Dretske’s view of the matter: “Information, for Dretske, is found where there is contingency and correlation. Any variable in the world which has a range of possible states is a source of information. When a state of information is correlated with the state of another variable, as a consequence of physical laws, the second variable carries information about the source. For Dretske, information is a resource that organisms use to make their way through the world; cognitive systems are information consuming, or information exploiting, systems.” Godfrey-Smith (1996; pp. 236–237).

  27. Errors of Reasoning, chapter 6, Sect. 9.

  28. Hintikka wrote this in ≈ 2007. A scant 11 years later, it strikes us how understated his admonitions were.

  29. Einstein (1905; pp. 639–643).

  30. I omit the upper case on the first letter of this name to discourage the misunderstanding that the present section is an exercise in Anselmian scholarship.

  31. Kunen (2013; p. 1).

  32. For example, we may take it that dialethically inconsistent mathematics is not of the conventional kind; but what about intuitionistic mathematics?

  33. It is important to keep in mind that the four-grades thesis is itself gradualistically instantiated. It is possible for an anselmian knower of p to have known its semantic content earlier. Consider two cases. In case one, p’s semantic content reposes in stirrable memory, if only thanks to some external stirrer. In case two, there is nothing in the anselmian’s memory that is semantically retrievable. In the first case we have a faded memory of something once grasped. In the second we have amnesia, that is, savoir perdu.

  34. An essential feature of the English common law is the inexpressibility of court-created precedents. Precedents play an essential role in determining future cases; but when judges and scholars try to write them down and make them explicit, invariably they get them wrong. Further discussion can be found in chapter five on “Unwritten law” in the second edition of my (Woods 2018a).

  35. When we grasp that 2 + 2 = 4 and put it into words, what we’ve said comports with Convention T: 2 + 2 = 4 if and only if “2 + 2 = 4” is true. When this knowledge is stored below, it loses its semantic and linguistic character and therewith its capacity to oblige Convention T. But if it has occasion to resurface, it will recover the truth-value T.

  36. Similarly, if a piece of information is semantically loaded and now transitions downwards, it loses its content. However, upon transitioning back up, the old content is recovered.

  37. Ryle (1929–1930; pp. 91–126, 1945–1946; pp. 1–16, 1949). Wittgenstein (1953). Heidegger (1962). Polanyi (1958, 2009). See also Lowney II (2017). A good survey of these and Polanyi’s works can be found in Gascoigne and Thorton (2013).

  38. See, in the first instance, Armour-Garb and Woodbridge (2015; pp. 84–90), and, in the second, Moss (2018; chapter 1).

  39. Zimmerman (2018; p. 1).

  40. The idea of ignorance-preservation first arose in my “Remarks on the logical structure of abduction”, presented at the Symposium on Abduction and Creative Inference Cog Sci 2004, Chicago. It was developed further in Gabbay and Woods (2005a); chapter 3 on “The structure of abduction”. An updated treatment can be found in my (2012; pp. 148–161). For an update of that update, see my (2017; pp. 137–149). The Gabbay-Woods model is given an attractive pragmatic dynamicization in Chiffi and Pietarinen (2018).

  41. Peirce (1931–1958; volume 5; p. 189). Line numbers are mine.

  42. Peirce (1992; p. 128; CP 5.171, 7.220).

  43. CP 5.59, 6.46–6.473, 7.202–219.

  44. For a recent discussion of how Peirce’s model as extended by the Gabbay-Woods schema allows for successful abductions that aren’t inferences to the best explanation, see The Reach of Abduction, chapter 5, Magnani (2009; chapter 2), Park (2017; Sect. 4; chapter 2), and Minnameier (forthcoming).

  45. See “Reorienting the logic of abduction” for details. Good information here is A’s excellent track-record under premissory licence. See also Urbański and Klawiter, “Abduction”.

  46. Thagard (1992; p. 170).

  47. It may interest some readers that none of the works cited in this section is mentioned in the Stanford Encyclopedia of Philosophy’s entries “Abduction” and “Peirce on Abduction” (Douven 2011, 2017). The gist of the latter is that Peirce’s views on abduction have lost whatever appeal they may have had earlier on.

  48. There is another large literature which investigates the standards that qualify sayso as reliable and the manner in which recipients of sayso attempt to apply them in their own situations. There are contexts in which this is the right way in which to proceed; for example, when judges determine the admissibility of expert testimony at trial. In the general case, it is the wrong way to go. It is but a version of justificationism as applied to the reliability of testimony-induced belief. See here Lackey (2011, chapter 29) in Bernecker and Pritchard, and the works cited there. For a different slant, see Errors of Reasoning, chapter 9, and Is Legal Reasoning Irrational?, 2nd edition, chapters 8 and 15, and Woods (2018b; pp. 1205–1257).

  49. See here Lewis (1975; pp. 3–35, 1983).

  50. The formal dynamics of telling haven’t yet had much of an innings in the philosophical mainstream. Of most direct significance for epistemology is the theory of telling investigated by public announcement logics (PAL), originating with Paya’s (1989; pp. 201–216). PAL extends multiagent epistemic logic to model the communicational consequences of announcements to multiagents. Think for example of the Prime Minister’s televised speech announcing on behalf of the governments its intention to place tariffs on U. S. steel effective a week thence. Further references can be found in the note on p. 323 of Errors of Reasoning, chapter 9 on “Being told.”.

  51. Hardwig (1985; pp. 335–349).

  52. In his (1997), Peter Galison asks how it is possible for different “scientific cultures” with different disciplinary backgrounds to coordinate and produce a knowledge of particle accelerators, for example. More generally, how is multidisciplinary cognitive multiagency possible? The four insets above are part of the answer to Galison’s question.

  53. For excellent historical and analytic coverage of the actual workings of what I’ve been calling mathematical multiagencies, see Ferreirós (2016).

  54. I’ve drawn here on Sect. 9.7 of Errors of Reasoning, pp. 318–320. In writing those pages I was told the Fermat part by Cornell et al. (1997), and Singh (1998). Now, for those of you who hadn’t been told it before, I am happy to be the teller. It enrolls you and me in multiagent sayso-manifold ensuing from mathematical antiquity.

  55. Kanamori (2013; pp. 21–35). Angus MacIntyre has claimed that the central Modularity Thesis in Wiles’ proof is provable in Peano arithmetic, prompting Kanamori to suggest that, if so, “there could be a new proof of the Fermat Last Theorem in Peano Arithmetic, possibly by passing through further analysis to Modularity Thesis”. (p. 32). See also MacIntyre (2011; especially the appendix to chapter 1), “The impact of Gödel’s incompleteness theorems on mathematics.” But as Ferreirós notes, “… such convictions are not mathematical facts; we are facing beliefs”, Mathematical Knowledge, p. 183.

  56. Let q(abc) be the quality of the triple (abc), defined as logc/log (rad (abc)). The radical rad (n) of a positive integer n is the product of n’s distinct prime factors. The abc conjecture puts it that, for every real number ε, there exist only finitely many triples (abc) of coprime positive integers with a + b = c such that q(abc) > 1 + ε.

  57. Castelvecchi (2015).

  58. Klarreich (2018).

  59. Graham Priest gave much the same advice to metaphysicians who have difficulty bending their minds to objects of which it is the case that nothing whatever is they. Dissenters, he proposed, should “get their understanding rewired.” Priest (2016).

  60. For an informal overview, see Weinberger https://www.wired.com/story/neverunderstand/.

  61. Again because space is stretched, I’ll briefly come back to what I take to be a fatally mistaken epistemological course. It is the course of postulating the ideal rational agent and defining it as a being which revises its belief in strict accordance with the rules of the probability calculus, which its endorsers invest with a normative authority that binds the belief-revisions of human life. One such rule is that a rational believer will close his beliefs under consequence. There being at least ω of them, there is no finite degree to which any human reasoner approximates to that standard. Another example is from Harman (1986). Suppose that you are head of a CID investigation of a serious crime. As of now, the evidence supports the judgement that Spike McGurk is a shade away from being charged with the crime. It is now early tomorrow morning. Twenty items of new evidence have been logged overnight. For that to have happened in accordance with the Bayesian belief-update rules, ≈ one million calculations would have to have been made. Had 30 bits of new evidence arrived overnight, ≈ a billion calculations would have been required for update. The very idea that these rules doom you and your team to irrationality is as amusing as it is bemusing. In matters of cognition, as in all things, it is always a good idea to watch out with whom we plan to do business.

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Acknowledgements

For sagacious advice and constructive encouragement I warmly thank the two editors of this special number, Lorenzo Magnani and Selene Arfini. I also thank my UBC colleague Alirio Rosales for many generous hours of fruitful instruction and productive criticism. Thanks, too, to Mike Bennett of the UBC mathematics department for welcome instruction on how the Mochizuki proof is structured. For their constructive suggestions, I also thank this journal’s two referees.

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  1. Department of Philosophy, University of British Columbia, 1866 Main Mall BUCH E370, Vancouver, BC, V6T 1Z1, Canada

    John Woods

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Woods, J. Four grades of ignorance-involvement and how they nourish the cognitive economy. Synthese 198, 3339–3368 (2021). https://doi.org/10.1007/s11229-019-02283-w

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