A nomological conception of knowledge

It could be unsettling for a teacher to be unsure of the nature of knowledge, yet questions relating to the concept appear frequently on Twitter. Here, using David Didau’s proposition: ‘Knowledge is what we think with and about as my starting point, I hope to provide some clarity on the issues and put forward a conception of knowledge that I believe could make a contribution to the philosophy of psychology, and thus to cognitive science.

Addendum (20/07/2017): To be clear, this is not a critique of Didau’s post, nor an attempt to undermine it. On the whole I agree with the above proposition. The aim of this post is to attempt to expand upon it by answering the questions: where do we go from here? how should a teacher make use of this proposition?

I shall take David Didau’s proposition as the trunk of this essay, and in the spirit of a philosopher-gardener, I shall attempt to prune three branches that I think unhealthy, and tend to a fourth which I believe will yield more fruit.

My question is this: what can ‘knowledge is what we think with and about’ actually mean, and how can we teachers make use of it?

1.   A propositionalist conception: Knowledge as propositions

One difficulty with Didau’s statement is that one could argue that I might think with and about my brain but my brain is not synonymous with knowledge, i.e. my brain is not knowledge, it is my brain. Whilst this may appear to be mere sophistry, it does raise an important point about distinctions between objects in the world, and facts about the world. One possible move at this point would be to refine Didau’s original proposition thus:

  • Knowledge is the collection of propositions which we think with and about.

Such a view is manifest in education in the form of the now-popular knowledge organisers. These are straight-forward collections of the propositions which the teacher would like the students to remember, so that they can think with and about them. The problem we now have, however, (as Chrysippus first pointed out in the third century BC) is that there is a distinction between the words that make up a proposition, and its meaning. A student could diligently learn to recite a proposition without having the first clue what it meant.

One option is to create yet more propositions that enumerate the connections between propositions. This is certainly reasonable in many situations, but not in all. If we were to apply this to all situations it appears to reduce all knowledge to the merely explicit, and clearly not all knowledge is explicit. It may be implicit, awaiting someone to make it explicit, or it may simply be inexpressible. Take the following example, I once taught a student who could count perfectly well, though it was fair to say she had no real idea what the numbers meant. Whilst she was learning her times-tables I had to explicitly tell her the answer to each question (NB: this was in an unusual educational setting). I could say to her: 2×2=4, 2×3=6, 2×4=8… but she had no idea how to go on; she did not know the rules. Wittgenstein described such rules appearing to lay in front of us like train tracks. —So then can we not just teach her the rules? With this student, no! I was unable to find any expression of the rules that she could understand. (An expression of a rule is no more a rule, than the words ‘my brain’ are my brain.) As Wittgenstein points out, this train-track analogy is misleading precisely because the rules do not exist without the content and context in which they are embedded – rules are embedded in practices. Not noticing this will lead one down a logical positivist path…

2.   A logical positivist conception: knowledge as the capability to perform

Last week on Twitter, I had an exchange with @crispinweston who was putting forward a logical positivist stance. His argument was that, in practical terms, we can only meaningfully define knowledge in terms of the way in which it is demonstrated, i.e. the capability to perform. @ChrisMWParsons then suggested that were Stephen Hawking no longer able to communicate, we could still meaningfully say that he knew things. @crispinweston rejected this saying that without demonstration, no one could meaningfully say that ‘Stephen knows X’. I responded to this by saying that by the same token, if there were no way of demonstrating it, neither could anyone meaningfully say that ‘Stephen loves…’

According to the logical positivist approach, all meaningful statements must be verifiable, and all ordinary sentences are reducible to an ideal logical expression. Thus, a logical positivist would respond to my argument by saying that what I really meant was that ‘were he able to demonstrate knowledge/love, he would’. I outright reject this approach, and the assumption that all utterances can be reduced to unifying logical rules for the following reasons:

  1. Different domains have different rules – compare the rules governing the writing of a police report with those of writing poetry.
  2. No two phrases are entirely synonymous in terms of their meaning; to treat them as such is to lose much of what is important. If we ‘translate’ the utterances of a toddler into this ideal language, we will lose all the cuteness. This cuteness is part of the meaning of the sentence. In more extreme cases, such a belief in ideal logical rules is simplistic to the point of parody: If my wife lay dying, unable to speak and I said, ‘I know loves me’, I would not be too impressed if someone were to say, ‘no, what you mean is that if she could demonstrate her love for you she would.
  3. The manner in which a logical positivist appears to require verification or falsification is very limited, and results in us believing that we cannot know something without an explicit test. There does, however, exist imponderable knowledge, which we may be unable to make explicit. We can tell whether someone is annoyed with us without necessarily being able to pin down why. Of course, I accept that in most cases this imponderable knowledge may be made ponderable, – it may be pointed out to me that the other person’s lips were curled when they spoke to me – but in such a situation I recognise the truth of this description after the fact, but may not be able to say before that ‘if their lips curl when they talk to me, then I will know that they are annoyed with me’.

There are various purposes of measuring things in schools, but often, it comes down to holding people accountable: the teachers, the students. Another way of looking at it is to try to get people to be better, to improve: to achieve an A* or an ‘outstanding’. But this presupposes that one knows what ‘improvement’ looks like. (If we know what improvement looks like, then surely we need to simply tell people what to do and then get them to do it.) However, the level of complication expands with the number of possible answers that could be judged passable. If the question was merely ‘2+2=?’ Then we can clearly state that 4 is a better answer than anything else. With, for example, an essay, there are more possible answers than there are people who answer it, multiplied by the number of occasions on which each person answers it. This level of complication then tempts people into creating generalised algorithms for marking and levels, but as one realises the poverty of one’s first attempt, such a positivist approach will soon get out of hand as one develops more and more strands to one’s algorithms; strands that flit in and out of the core levels in ever more complicated arrangements. These strands are often what are referred to as ‘skills’ but these are at best heuristics, rules of thumb. There is nothing wrong with heuristics, and I am very much in favour of them. They are tools that can be taught and that can be very helpful, but to view them as being mechanical elements of a working mind is probably fallacious. All this was summed up very neatly by Dylan Wiliam: ‘the more neatly we specify what we want, the more likely we are to get it, but the less likely it is to mean anything.’ Humanity is present in the subtle inflexions of life, not dull-footed scientism.

3.   An idealist conception: knowledge as that about which one thinks

A further route open is to treat David Didau’s proposition as an expression of idealism (similar in ways to George Berkeley’s Master Argument). If this were the case, which I don’t think it is, then Didau would simply be arguing that ‘we cannot think about that which we do not know’. Again, I have a great deal of sympathy with this interpretation; the problem with this is that it gets us nowhere: every object of thought becomes knowledge and we are offered no means of connecting it to the world outside our minds. If we cannot connect it to the world outside our own minds, this is not that useful for teachers: a. the minds of the children appear to be entirely hidden from the teacher, and so we are again unable to know what they know, and b. we have no way of distinguishing between a truth and a falsehood and thus we will end up in a post-modern mess.

4.   A nomological conception: knowledge as expectations embedded in ‘cases’

I propose that we can make most sense of Didau’s proposition, if we see it through, what I shall call a nomological conception of knowledge in education.

NB: I am not legislating for language here! I am simply offering a way of looking at ‘knowledge’ within the context of education that avoids the difficulties above. I fully accept and understand that the word ‘know’ can be used in a huge variety of different senses!

If we accept that there is a. a difference between the words that constitute a sentence (declarative knowledge) and its meaning, and b. that a sentence might have different meanings in different contexts, then the difficult task is to achieve some clarity regarding the meaning of a sentence as opposed to merely the words. Wittgenstein described the meaning of a sentence (in most cases) as its use. This seems appropriate for any item that may find its way into a knowledge-organiser: We do not simply want the students to be able to repeat that ‘the battle of Hastings occurred in 1066’, we also want them to understand how this sentence might be used. It might be used (to employ a silly example) to explain why Harold did not use guns. These possible uses are defined and limited by rules.

In order to clarify the kinds of rules with which we are concerned, I shall employ Hayek’s[1] distinction between two different conceptions of Law, nomos and thesis: Thesis are rules that have been intentionally invented by people, whereas nomos are rules that are simply recognised as the way things are. Consider the rules of English grammar, they have not been decided by anyone, they have simply survived. Of course, they evolve and change, and of course, retrospectively we might be able to assign a reason or purpose to their being as they are, but no single individual can cause this to happen. Hayek called this way of looking at things evolutionary rationalism. It is rules in the sense of nomos and not thesis with which I am concerned here. (Hence the name nomological.)

My expansion on this is the following: the rule that a particular person is following in their use of a sentence can be described as that person’s expectations concerning that sentence -what does the person expect to occur as a result of using the sentence? I am using expectation in a particular way here:

  • By expectation, I mean ‘that which is conceived as following from…’ There would appear to be some form of continuum with regards expectation that has great relevance for cognitive psychology and teaching:
    1. There is that which one would, without thinking, expect to follow.
    2. There is that which one would expect to follow, once certain facts have been brought to mind – these other facts acting as heuristics or rules of thumb.
    3. There is that which one would not find inconceivable, but nonetheless unexpected.
    4. There is that which one would find inconceivable, impossible to see how it might follow

… and many other levels in between.[2]

  • I would also distinguish between an expectation in the sense I am using it, and a belief. An expectation is seen, or embedded in the aspect of a thing that one observes. One has no choice about what one expects in this sense: I see a vase wobbling precariously on the edge of a table – I have the expectation that it will fall. (As opposed to ‘having high expectations of the class’ which I would often class as ‘belief’ in this context. See note [4])
  • Obviously, one’s expectations can develop and change:
    • I may have an expectation that I will not burn in the sun, but experience will prove me wrong. I do not mean that we need to directly experience something to develop and refine our expectations: someone who has been blind from birth might still be surprised if someone told her that the sky was green.
    • A child may learn to recall, ‘the battle of Hastings occurred in 1066’ but initially have no expectations regarding it, or she may have the wrong expectations regarding it – these can then be taught and refined.

What we desire is that our expectations (the rules that we follow) and those of the students accord with what is actually the case – with nomos and not with legislation or thesis.

—How then, do we establish what is in accordance with nomos (what is an accurate expectation) and what is not? Again, I think that a most fruitful analogy is to be made with how judgements are made in law, specifically common law. In common law, the court looks to past decisions, establishes analogy with the current facts (or shows that no analogy is relevant) and synthesises and applies the past decisions. Clearly, the more cases that a lawyer knows, the better they will be able to do their job. In education, we do much the same: our past cases are the declarative knowledge that we wish the students to learn, and the judgements are the expectations embedded within those cases and examples. These cases of declarative knowledge do not say the rules (since an expression of a rule is not a rule) but the rules are shown in them; they are exemplified in them. To educate a child, therefore, is to refine their expectations through the apprenticeship with cases.

Note how, I have not stated this in terms of a purpose of education. This would be nonsensical – one could never reach a stage where one had accurate expectations about everything! Thus, this process of refinement, and education generally, has an inherently negative character. Nomos does not so much tell us what to do but rather what is wrong. This seems to me to reflect well our psychological approach to education:

In the eyes of those lovers of perfection, a work is never finished — a word that for them has no sense — but abandoned;[3]

Hence, there can be no ‘rubric’ to follow to achieve one’s aims, since one’s aims cannot be described, and hence it would be perfectly reasonable for exams to contain unpredictable questions (see this by Michael Fordham): an exam board would merely outline the cases to be learnt, and the student, would then try to apply them to any question that might be asked. The key would be in enabling the student to properly pay attention (in the sense described by Simone Weil) to the cases being given. Imagine a student trying to complete a tessellation of some complicated shape but not realising that they could rotate the piece. To pay attention to a case is to rotate it in one’s mind, to imagine it from various angles, and in different contexts and to simply get to know it – not change it, or force it where it won’t go. (One can see a parallel here with Diderot’s Memory, Reason and Imagination delineation of knowledge, brought to my attention by Tomas White. Instead we have cases, expectations, and attention)

5.   Implications for teaching

Firstly, the implications of this nomological conception for cognitive load theory, I would imagine, are obvious: those expectations that strike us most strongly as in (1) are automatic, whereas (3) requires some explanation. Secondly, one can also relate this to Hebbian learning and the way associations are brought about. However, given the frequently negative nature of expectations – i.e. one might not have an expectation about what will happen, but only what will not happen – then it is not simply a matter of mere association. Thirdly, there is also the obvious implication for marking and improvement – away from rubrics, and more towards the comparative methods championed by Daisy Christodolou. Fourthly, the tackling of misconceptions can also be seen, I would argue more fruitfully, in the light of expectations. And finally, clearly the quality of the cases matters a great deal for the nature of the expectations embedded in them. Thus, I would argue in favour of looking at the content we teach through the prism of expectations.

There are also ethical aspects running through this conception of knowledge as expectation. Of all visions of education, I find those that impute it with some form of explicit purpose, the most insidious. Someone made the comment on Twitter that: ‘there is no such thing as a neutral education’. Whilst this may be true in the sense that different forms of education will have different consequences, it is the height of hubris if someone believes that they can predict and therefore control the future and know what those consequences will be. Justice is not a state to be achieved, but an obligation to fulfil one’s own expectations[4] of oneself. Woven throughout this nomological conception is an imperative to pay attention, to refine one’s expectations and be more aware of the expectations of others, our students in particular. (Not least because expectations have been shown to be crucial to one’s happiness![5]) To make this aspect more explicit, I shall finish with Simone Weil writing about the most sacred of expectations:

At the bottom of the heart of every human being, from earliest infancy until the tomb, there is something that goes on indomitably expecting, in the teeth of all experience of crimes committed, suffered, and witnessed, that good and not evil will be done to him. It is this above all that is sacred in every human being.

The good is the only source of the sacred. There is nothing sacred except the good and what pertains to it.

This profound and childlike and unchanging expectation of good in the heart is not what is involved when we agitate for our rights. The motive which prompts a little boy to watch jealously to see if his brother has a slightly larger piece of cake arises from a much more superficial level of the soul. The word justice means two very different things according to whether it refers to the one or the other level. It is only the former one that matters.

Every time that there arises from the depths of a human heart the childish cry which Christ himself could not restrain, ‘Why am I being hurt?’, then there is certainly injustice. For if, as often happens, it is only the result of a misunderstanding, then the injustice consists in the inadequacy of the explanation.[6]

[1] F. A. Hayek, Law, Legislation and Liberty: A New Statement of the Liberal Principles of Justice and Political Economy (Oxford: Routledge, 2012).

[2] One can see here an obvious connection with the psychological concept of priming. See http://nautil.us/issue/48/chaos/shakespeares-genius-is-nonsense-rp?utm_source=pocket&utm_medium=email&utm_campaign=pockethits for an interesting article discussing psychology and Shakespeare.

[3] Aaron Copland, ‘Creativity in America’, Copland on Music (1944), p. 53

[4] I shall perhaps write a little addendum about the possible confusion between ‘beliefs’ and ‘expectations’, specifically expectations in the classroom, and the ‘Pygmalion effect’ – I’ve already written too much!

[5] Robb B Rutledge and others, ‘A Computational and Neural Model of Momentary Subjective Well-Being.’, in Proceedings of the National Academy of Sciences of the United States of America, 2014, cxi, 12252–57.

[6] Simone Weil, Simone Weil: An Anthology (London: Penguin, 2005). pp.71-72


11 thoughts on “A nomological conception of knowledge”

  1. Hmm. My problem with this is that you went wrong right at the beginning. I categorically do not think that “Knowledge is the collection of propositions which we think with and about.” This is a very limiting view of what knowledge is as the propositions we know make up just the tiniest part of our knowledge. Most of what we know is impossible to formulate, articulate or even identify. Very often we don’t know what we know. Nevertheless, we know it because it forms the mental architecture with which we think; it is the structure within which the silt of our propositional knowledge washes up.

    If you really want to critique my definition my saying you think with and about your brain, then, wearily, we can refine the definition thus: knowledge is the stuff within our minds which we think both with and about.

    If we begin from that premise, I’m not sure the edifice of your argument quite works. What do you think?


  2. Ah that’s a shame. I certainly wasn’t clear enough in the introduction. The tenor of my argument wasn’t a critique of your proposition, or indeed the position you outlined in your post – I agree with it! It was an attempt to answer the questions: what should a teacher do with it? Where should we go from here? (An expansion rather than an attempt to undermine it.)
    The first three sections are what I perceive to be dead ends (and not what I believe you think) and the fourth was something more fruitful.
    (I’ll explain this in an addendum in the introduction.)

    [My problem with the ‘stuff within our minds…’ bit is that I’d be afraid it would lead someone to seek a more physicalist interpretation, i.e. it sounds as if the proper description of knowledge can only be given by a neuroscientist. Again, that’s not what I think you’re saying as such, but the incorporation of the preposition ‘within’ certainly ‘places’ knowledge in such a materialist fashion.]


    1. I’m all for a “physicalist interpretation” and suspect that, at some unspecified point in the future, it will be neuroscience that provides us with the most complete answer of what knowledge is. But, whatever they come up with will, probably, be the wrong level of description. We’ll continue to find such abstractions as working and long-term memory useful as they speak more to what knowledge *does*.


      1. Yes, I don’t agree with this common assumption that we will one-day discover what knowledge ‘really is’ in terms of neuroscience. The difficulty is that there are fundamental differences between 1st person uses of words and 2nd or 3rd person uses. ‘I know’ is not analogous to ‘you know’, or ‘he knows’, and they cannot be unified with reference to a ‘state of the brain’:
        Consider the following:
        • I want to get knowledge of his knowledge by observing his brain, using my knowledge of observing brains.
        Now translate knowledge as ‘a particular state of the brain’
        • I want to get ‘a state of the brain’, of his ‘state of the brain’ by observing his brain, using my ‘state of the brain’ of observing brains.
        It should be clear from this, that any state of the brain that *I* have concerning *his* state of the brain cannot be the same as his state of the brain. Thus, if ‘I’ want get some knowledge of the knowledge of 3rd by observation of brain structure, there will forever appear to be a veil between the one and the other. Knowledge cannot be both the instrument and the object of investigation. A social description of knowledge does not have this problem. All we need to know is right in front of us as a ‘naturalism of the surfaces’ as Raimond Gaita calls it.


  3. Yes, I’d be (wearily) wary of placing knowledge within the mind. I think there’s much to be gained from a theory of knowledge that claims its objectivity from its social nature. For example, we know the world is round because a scholarly community has come to some form of consensus on this fact. That same community used to know something else (that it was flat). In each case, the knowledge was derived from its social intersubjectivity. Hope this makes sense.


  4. I should clarify, here I am concerned with knowledge in education. Clearly the knowledge I possess about how I’m currently feeling relies much less on social intersubjectivity (although even this is debatable). But the kind of knowledge that is powerful and therefore merits being taught in schools is of the social kind.


    1. I both agree and disagree with you here. One one level * of course* knowledge is only in the mind and on another *of course* it also has an independent physical and social reality. No one is suggesting the curriculum be made up up of what you or I are currently feeling. *Of course* we will select information which we deem to be useful and culturally significant. And that is why I distinguish between knowledge (that which we possess individually) and Knowledge (the accumulation of human culture).


  5. I’m writing this comment as much to work on my read of the post as I’m writing it to add.

    When I first started teaching, I had a moment of philosophical confusion. In my undergraduate philosophy courses, knowledge was always treated as (justified) belief in a true proposition (or a collection of propositions). What I was reading about learning — and what I was experiencing as a math teacher — was something else. Often kids intellectually accede to some proposition, but they don’t deploy it, e.g. they believe that negative powers don’t make the base negative, but then their work will show 2^-3 = -8. My job as a teacher was to help break students of this tendency — not to help them believe in some proposition with justification.

    I never studied epistemology per se in university, so I didn’t know of other theories. But I knew that we’d need a better way of describing knowledge than belief in propositions in order to think accurately about teaching.

    I take it that this is your first point: propositional knowledge is only a small part of what teaching involves.

    Surely, though, this isn’t a problem just with school learning. Moral action shouldn’t be modeled as primarily a matter of living in accordance with various propositions (though I know it has often been). Really, any lived performance is going to involve non-propositional learning. Perhaps, in the past, knowledge was primarily conceived of as a kind of belief in propositions because a primary function of teaching was to maintain a body of knowledge. Now, the primarily value of learning is seen in a certain kind of behavior — to perform in a certain way. It becomes more important to think of knowledge as something closer to the training needed for excellent action.

    The description of knowledge you end up with is grounded in the psychological state of expectation. It’s the sort of form behind the matter of fact, the sense behind the references. (Another good thing: expectations can infinitely generate true beliefs. If knowledge is just a matter of belief in propositions, then we have to assume that students only can use propositions with logical quantifiers, e.g. “Every chemical reaction with X, Y, Z has properties A, B, C.” But this is trouble, because students often reason from cases without explicit belief in the generalized propositions. And, in general, the general propositions are often incomprehensible to students, even if they assent to the underlying cases and act in accordance with that knowledge. So it’s great to avoid ascribing a belief in a generalized proposition to students.)

    (It’s also good that expectations allow for a sliding scale of confidence, as knowledge is often partial in a way that the propositional version can’t really allow.)

    I’m still getting used to your idea. My first concern is whether this really captures performance. Going back to 2^-3, my students might under certain contexts express an expectation that 2^-3 should be a fraction. Then, later, under the pressure of a quiz or in the midst of a tougher problem, they might fall back on their cognitive biases and declare that 2^-3 is -8. How should we model this inconsistent performance? Is it an inconsistent expectation? Should we say that their expectation is weakly held, so they expect the result to be negative at times, fractional at times? (This doesn’t seem to me what is going on with the kids.)

    The problem, if there is one, is that expectations are another psychological factor. While I haven’t read intensively about philosophy since finishing college, I’ve kept up with a few writers and bloggers. One of my favorite bloggers is a fan of (what he calls) a disopositioalist take on belief, where to believe something is to be disposed to behave in certain ways. I wonder if that might get at a lot of what you’re saying here, but with a behaviouristic grounding.


    Sorry for the long comment! I’m looking forward to thinking more about this, and looking forward to reading more.


    1. I love this comment, thanks! It’s just the kind of practical, close reading of/attention to the realities of teaching it would be nice to clarify- e.g. the 2^-3 example. (It’s for this reason that I enjoy your maths posts despite not being a maths teacher!)
      I’ve read the dispositionalist post but will need to re-read it, and around it, and let it settle in my mind a bit before I can make any sensible comments.
      My initial thoughts are that you’re quite right that ‘expectation’ doesn’t quite cut it, though it has some advantages. I’d certainly want to avoid a sense of ‘adding’ a further (or accompanying) psychological process- I think it’s more accurate if ‘meaning’ is construed as being ‘what is seen’, I.e. as immediate as any sense-impression. (Raimond Gaita talks of a ‘naturalism of the surfaces’ in ‘the philosopher’s dog’, which is well worth a read! And shows how we can avoid the behaviourism/dualism difficulties so present in psychological accounts)
      I think the idea of ‘inconceivability’ I describe in s1 of my post about ‘debates on edutwitter’ might have more legs- I.e. it is not so much that your students haven’t developed the correct expectations, but that they don’t yet see ‘2^-3 is -8’ as ‘inconceivable’- which is, perhaps a better description of what meaning (rule following) is, I.e. not so much describing the set of things that might follows, as what is not in that set, what *cannot* follow, -the limitations. (For a maths ‘master’ it would be inconceivable, but for a ‘novice’ perhaps not.)
      With regards the moral aspect, it does seem to me that anything ethical can only be ‘shown’ rather than ‘said’ (propositionally) hence the difference between the spirit and the letter of a law. I think that there is some mileage in seeing the ethical as ‘adverbial’. i.e. someone can perform all the correct actions but somehow not manage to do so virtuously, yet we this is only apparent in the details that (may be recognised after the event but) cannot be stipulated prior. (Again, I think that Gaita’s view is valuable here)
      As I say, I’ll have to ruminate over the disposition idea, but I will follow that blog closely.
      Thanks again for your thoughts, you’ve given me plenty to think on!


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