German Museum of Technology Berlin - 07TM-3438

By Jorge Royan (Own work) [CC-BY-SA-3.0 (], via Wikimedia Commons

UPDATE 2016: I wrote a draft paper about this some time ago (in 2014 I guess), I don’t think I’m going to do anything with it (I don’t think it’s quite right either) so download the draft: Assessing the Measure of Knowledge. ———– One area in which epistemology has very practical purchase in everyday life is assessment. One way in which this manifests is with respect to the delimiting of ‘knowledge’ claims, and [“The Measure of Knowledge” (Treanor, 2012)]1 Treanor puts forward two suggests (cardinality and counterfactual distance) which he dismisses, before proposing a third (similarity). Cardinality Treanor outlines that cardinality (i.e., counting truths) cannot be the way we compare two agent’s knowledge states (let’s call them Simon and James). There are various reasons for this, briefly 1. Given many are comfortable with there being an infinite number of truths, there are an infinite number of truths that I do not know; yet clearly I know more than I did when I was 10, thus appeals to cardinality fail 2. any atomic claim is associated with (perhaps entails) many other truths and decomposing this relation may be impossible (the holistic argument); it may not make sense to claim that we know ‘some number’ of truths 3. decomposing natural language claims into their atomic truth veritic (i.e., verifiable) parts may be impossible, moreover, clearly there are issues of type in comparison of knowledge states; my knowledge of 1000 trivial truths is not the same as your knowledge of a smaller number of signifiant ones.  Counterfactuals Another possibility might be to take one’s knowledge to relate to the epistemic space in which one works – that is, as one comes to know more, one’s space of possibilities (one’s epistemic space) becomes smaller (e.g. as one learns the rules of physics, or [normative issues aside] that two wrongs don’t make a right, or whatever) One way of dealing with such a claim is to use the language of counterfactuals or possible worlds. In this case, the claim is that the possible-world that is furthest away from the actual world while still being compatible with Simon’s epistemic-space is closer to the Actual world than James’ (that is, his beliefs are compatible with views that are further away from actuality than Simon’s).

Such an approach is attractive for various reasons, including that it allows that some beliefs are more important than others. However it is problematic because:

  1. It cannot deal with knowledge about necessary truths, because such truths are true in all possible worlds and as such having or failing to have such beliefs would not make a difference to one’s furthest away possible world.
  2. It has the consequence that once one knows that ‘p’, coming to know what ‘p’ entails has no bearing on the measure of one’s knowledge (because once one knows ‘p’ one’s furthest possible world has already been altered)

Moreover, we don’t really know what such ‘distance’ would look like, and it is entirely possible that such distance would require a quantification of the kind discussed above.


Another approach is to look at similarity. Here we’re looking at the similarity between the content of a representation, and what it is representing. Thus, a painting of a thing (James) is in many ways different to the thing (James himself), yet despite this there are similarities in form. This approach is stronger than that of possible worlds, because similarity is sub-maximal – representations do not need to be taken to represent complete possible worlds.

Now, at this point Treanor’s claim is not that the problem has magically gone away – ‘similarity’ measures are opaque, and at least to my mind some of them rely on counterfactual distance and/or cardinality! However, the claim is that ‘similarity’ is a term that is already widely accepted in everyday language and philosophy, and that this shifts the ‘bubble’ under the rug in an acceptable way.  His claim is that “…it is a reductive move: the measure involved in knowing more is the measure involved in similarity.”

Implications for inquiry

Treanor finishes the paper by noting that this metaphysical discussion has implications for epistemology – what it is to know more, or be less ignorant (and these are surely standard aims). Given an increasing focus on virtue epistemology this is an interesting point.  One of Nick’s points here is that, while it is now commonplace to hold that the aim of inquiry is not ‘truth’ (because if it were truth alone, we would not discriminate re: the types of truths held, counting threads would be equal in value to quantum mechanics), yet actually just because ‘inquiry aims at truth’ it does not follow that inquiry need treat all truths equally – as above. As he notes, if gold mining aimed at gold, it would aim at all gold (flakes, dust, nuggets, veins) equally is patently false. However, its falsity is not an indicator that gold mining does not aim at gold; it is an indicator that it does!


So what are the implications of this for education?  I’m still thinking about this (suggestions please) but I can think of a few:

  1. Assessment cannot only be about the counting of atomic knowledge
  2. Clearly we make decisions re: importance of knowledge taught, to some extent this could be couched in the language of counterfactuals (we want students to have beliefs such that the distance between a and their furthest pw is minimised)
  3. Similarity can obviously be assimilated into educational language (although as I note above I think this is a) something of a copout and b) might rest on cardinality)
  4. We clearly are not only looking for ‘the measure of truth’ – while the aim of inquiry might be ‘truth’, education is aimed at inquirers, and equipping them to undertake inquiry, as such. This is not so surprising – the aim of gold mining is the discovery of gold, and we expect a certain set of skills in so doing including (as discussed above) requisite knowledge re: normative and ‘measure’ principles including that some gold (or knowledge) is worth more than others
  5. It is not clear how the approach deals with the knowledge how/that distinction (note that I think I’m minded to view knowledge-that as a type of knowledge-how…this is controversial and I haven’t spent huge time thinking about it)
  6. There are also some interesting implications around the Extended Knowledge project with respect to one’s knowledge states under various extension conditions, for example does Simon (with a calculator) know more than James (without) and under what conditions? Here the similarity measure has implications but it shifts the question around which extensions embody knowledge states, and what those knowledge states are (e.g. we do not have to hold that S-calc and J-mental have the same knowledge when they make the same computations, but might they have the same quantity of knowledge?)

Update (23/01):

I was thinking about this a bit more yesterday, and I wonder if I have an approach here:

  1. There is a temptation (as above) to think about the measure of knowability (as in, one’s ability to know) v. the measure of knowledge.
  2. But, we can imagine that both S and J are equally virtuous with respect to their epistemic behaviour, yet J is more epistemically unlucky than S.
  3. Therefore, we need a way to distinguish between S and J based on knowing not knowability
  4. Approaches to such ‘measure’ are problematic both for truths (as above) and falsehoods (where the above arguments also hold)
  5. However, there may be something in an approach which says that “similarity to that knowledge state of a virtuous agent” is the target comparator
  6. In this case, the knowledge state (rather than virtue) is what matters, but it is related to social/virtuous standards.
  7. Sometimes this will mean that less virtuous agents ‘know’ more (insofar as they coincidentally have knowledge that is closer to that of some virtuous agent, or social epistemic standard), however generally
  8. the two would be related, but distinct
  9. Such an approach would also have interesting consequences for extensions, if we think that either virtue epistemology or social epistemology are good for extensions (i.e. we’d need to work out which external elements we hold to be ‘knowledge states’ of the [extended] agent), then such comparison to an ideal (or ‘many mind’ or whatever) comparator is also available to us…
  10. This shifts the ‘similarity’ measure from a represented-respresenting relation, to a comparison of representations

The implications of this for education are probably broadly outlined above – we should care about inquiry (and virtue or ability, etc.) but it gives us a means  to describe the similarity relation, and perhaps motivates an increased focus on the types of challenge presented in assessment contexts.

Further thoughts 24/01 (getting draftier)

Three possible complaints

  1. The bubble is shifted such that we now need to measure the ways in which an agent is virtuous, such that we can decide what a maximally virtuous agent is
    1. But this is not a good complaint, because we’re not interested in how virtuous any given agent is, we’re interested in how their knowledge state compares to that of any maximally virtuous agent
    2. Having said that, there is a new problem, namely that we’d need to work out a) what a maximally virtuous agent would look like and b) what they would therefore believe (including, for example, whether or not there might be one or many possible knowledge states arriving from such an agent
  2. The other horn is that we would need to enumerate the virtuosity of any given agent such that we can compare their similarity
    1. Again, this isn’t true though, we’re looking at the knowledge states of agents not their virtues. This allows, for example, ‘lucky knowledge’ to be brought in.
  3. There is actually a third possible concern, that such an approach doesn’t give us the same measure of ‘similarity’ that Treanor intends in suggesting a comparison between representation and the thing being represented. Treanor’s approach allows some sort of comparison between a maximal set of knowledge. This approach has appeal, but it’s hard to see how it doesn’t fall into various familiar traps (off the top of my head it looks potentially Platonistic, I don’t know how we’d deal with normative/pragmatic features) and it’s probably unnecessary for the work we want to do.

Another possible appeal of the approach is that it allows comparison over possible worlds (or actual cultural norms, etc.). So for example, I’ve heard Miranda Fricker talk about the epistemic standard for blameworthiness and whether or not we should hold historic agents blameworthy for their action when such things were common. The argument there is that we should hold them to the best epistemic (or moral) standards of the time with respect to blame, while holding that such standards have changed (improved) with respect to right/wrong.

So a case is this, S1 and S2 exist in World1 and World2 (W1/W2) and have the same knowledge state, so they exactly match (we don’t need comparison, don’t need to worry about cardinality, etc.). However, in W1 there is an epistemic norm to phi, while in W2 it is not. So, any comparison of the maximally virtuous epistemic agent in W1/W2 might give us a different ‘similarity measure’ in active comparison of S1/S2 with V1 and V2, despite the fact that S1 and S2 have the same knowledge state.

So the argument would go that this is not what we engage in when we seek to measure knowledge; but such comparison is precisely what we do. The only case in which phi matters for this case is where it’s completely arbitrary whether to phi is a norm or not, and the whole point of epistemic norms (particularly if we take, for e.g. Lynch’s stance, which I do) is that they are not arbitrary; in such an (arbitrary) case we would presume that phi’ing has no impact on V1/V2. So we would assume that there is a good reason for W1 and W2 having different norms of epistemic inquiry. In such a case, if to phi would lead to different epistemic outcomes for our maximally virtuous agent in W1, then – insofar as comparison across W1/W2 makes sense – despite the identical knowledge states of S1 and S2 we absolutely should say that an outcome of S1-V1 and S2-V2 comparison indicates a different similarity relation in each, despite the fact that S1 and S2 have the same knowledge.