Considerate la vestra semenza:

Fatti non fosse a viver come bruti

Ma per seguir virtute e conoscenza³

The preceding reflections on truth should suggest that while a concern for truth is a concern that may well stir up much of what is taken for granted by most people, it is one that will focus our attention on the various different sorts of proposition that we claim to know. Truth is a simple, and sharp-edged weapon, but it is not itself a notion that we need to dwell on. Rather we have to turn to the different things that we claim to know.

Epistemological Questions

While some of the questions we looked at in the first chapter were phrased in terms of truth, what is common to virtually all of them is the question of the nature of our knowledge: what is it like and where is it to be found? The teachers Haes questioned differed on these issues: some thought that knowledge is fixed and certain, others said it was provisional; some thought that science and mathematics are our paradigm cases of genuine knowledge, but many other people would wish to include moral or religious claims as equally reliable. We can see large parts of the philosophical tradition as answers to such questions. Many philosophers have told us what knowledge is like - how it is based, or what its logical relations are to other concepts such as belief. Again a great deal of philosophy focuses on the reliability of knowledge - in some cases, philosophers have questioned virtually everything we believe: 'stage' philosophers are still often shown wondering whether they are indeed standing on a stage. But most philosophers have in fact accepted many of the beliefs they found around them. Their scepticism has been partial, a matter of querying or rejecting some of the beliefs in common currency, not all. Lucretius praised Epicurus for freeing mankind from the terrors of superstition; Locke sought to free science from 'pretenders to a knowledge they had not' ([1690] 1961, III, viii. 2); and modern philosophy is full of the same concern for cognitive health.

As I said earlier, I cannot here go into particular cases such as psychoanalysis or Marxism or theism. Nor do I think there is much for you to gain from an extended discussion of the logical links between our concept of knowledge and our other concepts. But we can usefully survey the fundamental logical structure of our knowledge itself. This will yield insight into its status and so help in deciding what would be the most appropriate attitudes for teachers to adopt towards the transmission of our knowledge. While we shall not be able to discuss any of the more detailed questions fully you will see that the general survey of the structure of knowledge does provide a suggestive framework in which to take those discussions further.

The Word 'Know'

Before beginning with that task, however, I must indicate some of the restrictions on the scope of the discussion. I am concerned with the sort of knowledge that involves truth; the sort that when you know something, you know the truth, or at least a bit of the truth. In English we use the word 'know' and its cognates much more widely. One crude classification of our usage distinguishes between

As is commonly the case, virtually the same information can often be conveyed in what are logically or linguistically very different ways. We could report that Tom knows Pythagoras' theorem, or that Tom knows how to calculate the area of the square on the hypotenuse of a right-angled triangle, given the squares on the other two sides, or that Tom knows that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. In other cases, such as 'Fred knows London like the back of his hand', it is not so easy to find these alternative paraphrases. But my main concern at the moment is for you to realize that I am only going to talk about the third of the kinds of knowledge, propositional knowledge.

Empirical and A Priori Knowledge

Restricting ourselves to propositional knowledge there is an important prima facie distinction to be made between items of knowledge that can be established simply on the basis of reflection or calculation and items of knowledge that require some sort of experience or observation. Pythagoras' theorem, or more strictly the fact that Pythagoras' theorem is deducible from certain sets of geometrical axioms, is something that can be established simply by reasoning. The area of the cricket pitch at Sabina Park is something you would have to go and look to find out. The fact that all bachelors are unmarried is again something that can be established (assuming certain restrictions on the sense of the word 'bachelor') simply by reflecting on the standard meaning of the words used, whereas the claim that all such bachelors are generally more contented with life than their married counterparts is something whose truth or falsity requires extensive and complex investigation of the world, after one has clarified the senses of the words it contains.

Traditionally philosophers have used the terms 'a priori' to label the first sort of knowledge, the sort that can be established by reason alone, and 'a posteriori' or 'empirical' to label the other sort. (Notice that this is a much wider use of the word 'empirical' than the one with which you may be familiar, and that it has no evaluative connotations.) I said above that the distinction is a prima facie one because there is a great deal of involved argument in modern philosophy about whether there really is such a distinction, although everyone can agree about where many items of knowledge would go if there were. In terms of school subjects, a priori knowledge is represented pre-eminently by mathematics (to which we may add logic), while everything else that is clearly propositional knowledge would count as a posteriori - chemistry, history, etc. As far as we are concerned, it is not necessary to enter into the debates about the distinction; again my only concern is that you recognize I am talking about a posteriori or empirical knowledge. I have nothing to say about mathematics or logic. (I suspect a lot of what I do say could be applied without much revision to those subjects, but the issues are very complex and controversial, and anyway I do not think they raise problems for teachers in the way that empirical knowledge does.)

Empirical Knowledge and Experience

So our concern from now on will be with empirical knowledge. That is a big enough field. It includes items like 'Hydrogen has three isotopes', 'Mozart wrote 41 symphonies', 'I have a feeling that I'm going to sneeze', 'Rewards promote learning more than threats', and so on and on.

As was obvious from the account I gave above, empirical knowledge has got something to do with our experience or observation. Our reasons for accepting particular items of empirical knowledge will often be a matter of sensory experience, though in practice we both take a lot on trust and would anyway be unable to have direct sensory inputs - neither the past nor the hidden structures of physical or some sociological theories are directly observable. But let us pause a moment to consider those cases where we do have fairly direct sensory input.

Perhaps the main point to be made is that we can distinguish between thinking that p and having an experience that p. For us, having an experience usually combines these two aspects: sensory experiences and judgments about the kind of experience it is. This point cropped up earlier when we glanced at one of Hirst's arguments about truth. You can think that the cat is on the mat and you can see that the cat is on the mat (and you might be able to hear or feel by touch that the cat is on the mat). And these are distinguishable things, even when they are bound together in reflective self-conscious experience. All of them involve the employment of concepts, as we say. Thinking about cats on mats involves the use of concepts of cats, mats and spatial relations, and experiencing cats on mats similarly employs one's conceptual resources. Experience only rarely comes unconceptualized, and when it does what have been called 'raw feels' do not tell you anything; we have to start thinking to get anywhere with them.

I hope those last remarks sounded plausible. We easily talk about concepts, but it is by no means clear what that talk really amounts to. In the case of thinking, and in the human case, we can get a long way by replacing talk of concepts with attention to how people use language; but the two things are not the same. And in the case of perception or sensory experience it is by no means clear how what we call concepts relate to language or how they develop. While much epistemological work has tried to solve this problem of the origin of concepts, we can afford to leave it aside. The point to remember is that we don't really meet with unconceptualized experience, so concepts of some sort are a given of our story. But as I said earlier, the main point I want to stress is that there is a difference between an experience and the thought (knowledge) that is associated with it. There is then a gap between seeing and believing, in logic if not in our too credulous practice.

Public Knowledge

There is, however, one aspect of the question about the origin and development of our concepts that is worth a brief mention. Even though each of us has to acquire our concepts somehow, they are not our private creations. This is patently obvious when we translate talk of concepts into talk about the languages we use. Children have to learn their mother tongue, but they don't invent it; it is there already, publicly available and publicly learnable. Languages, and by extension our concepts, are social facts, not purely personal. While it is important in general philosophy to ponder precisely what this implies, and what it doesn't imply, I am concerned now to make an analogous point about our knowledge. Most of our stock of knowledge is not derived from our own unaided investigations. We take it on trust; we take it over from other people, books, and other sources. While schools still insist on individuals' cramming their heads full of diverse scraps of information, in the rest of real life what often matters much more is what is known rather than what John or Mary knows. Popper (1972) has appropriated the popular but unclear contrast between subjective and objective matters to label this impersonal knowledge stored in libraries and data banks 'objective knowledge'. Perhaps it is safer to call it 'impersonal knowledge'; but the point remains that whatever you call it such knowledge is what matters most, both to institutionalized science and to bureaucracies. This fact reflects one important aspect of the social basis of our knowledge; another that we shall meet later is the tradition of criticism or testing. In both respects, the progressive development of knowledge is not usually a solitary pursuit, but a social one. It flourishes in certain kinds of social environment and not in others; indeed it has flourished in a very unusual context, though one that has now become so entrenched for most of us that we may have difficulty realizing its peculiarity.

These facts underlie one aspect of this discussion that may be worrying you. We have plunged right in the middle of the cognitive enterprise. We have not tried to find some specially reliable starting point or foundation. In philosophy the idea that we must have such foundations has been very influential; but it does not correspond to most people's engagement with knowledge. Rather we start from where our peers are. Most people, most social groups, stay there, but we are focussing on the procedures of groups who are committed to criticizing and revising their starting points.

The facts that the crucial kinds of knowledge are impersonal and that changes are socially produced do not make some kinds of concern for 'personal' knowledge irrelevant. While it is more sensible to go to the metereological office to find out whether the sun was shining on July 16th, 1982, in Kingston, than to rely on one's memory, that is not the end of it. Knowledge gets into the impersonal stores from people, or instruments. And the knowledge in the data banks is only as reliable as the 'personal' knowledge of those people or machines when they fed it in. If one learnt that the person making the records was incompetent, or the machine taking readings was malfunctioning, then out goes that impersonal knowledge.

So the importance of impersonal knowledge, or more generally, the pervasiveness of our reliance on testimony, does not mean that we should ignore some traditional concerns for reliability. We have procedures for checking whether machines are working properly, we have rough guides for deciding when to accept and when to doubt what other people tell us. Later on I shall say a little about how I would prefer to see these caveats about the move from evidence or testimony to our conclusions; for the time being the point is that they do not cease to be important because it does not matter which of us, if any, carries the knowledge in our heads. What does drop out is the philosophical doctrine that genuine knowledge carries some special 'inner' mark or feeling of certainty.

Knowledge and Authority

I have just said that we have guidelines to tell us when to accept and when not to accept what other people report. Different groups have different guides. To put it another way, different groups recognize different sorts of authority in their dealings with knowledge. Many people require claims to be consistent with the 'testimony' of their own senses. Many people allow claims made in certain special places to override all other considerations - so what it says in the Bible, or in Aristotle, is accepted, whatever other reasons there are against it. On the account I shall be giving of the structure of knowledge, none of these moves is acceptable; but of course I am making a normative claim, I am saying that you ought not to give authority to these kinds of factor; I am not describing what you do, since I have just acknowledged that many people do things that I regard as irrelevant if knowledge is their goal. But that doesn't mean we have an arbitrary choice here. Rather the account I give is meant to be an accurate description of the logical structure of at least a central part of the knowledge all of us have; the normative claims arise from my belief that this description applies to all the knowledge we could have and that therefore the kinds of authority I am rejecting have no reasoned basis in the nature of knowledge, however useful they might fortuitously have been.

Singular and General

With these preliminaries briefly mentioned, let us move on to giving this description of the logical structure of our knowledge. First we need to grasp a logical distinction. It is, if you like, a numerical distinction: between one and more than one. A proposition can tell us about one individual - a place, person, or thing - or it can tell us about more than one individual. Part of a proposition can pick out one individual - a definite description such as 'the present Prime Minister of Jamaica', or a proper name in context, such as 'Edward Seaga' - or it can introduce more than one into our thinking, like plural noun phrases such as 'capitalist running-dogs'. We can label these contrasts, 'singular' on the one hand, 'general' on the other. When we are dealing with whole propositions we can go on to distinguish several sorts of general statement, or generalization. So we can have a series starting with a logically singular proposition such as 'Today is sunny', and then moving through various generalizations from the weakest, 'existential' generalization, 'At least one day is sunny' (this is the precise content that modern logic gives to English sentences such as 'Some days are sunny') through various proportional generalizations such as 'A few days are sunny', 'Many days are sunny', 'Most days are sunny' or their more sophisticated statistical counterparts like '84% of days are sunny', to the strongest, 'universal' generalization, 'All days are sunny'. Here you can see that the singular statement picks on one particular day; the other statements do not in that way focus on one particular day, since even the existential generalization doesn't tell you which day or days it is talking about. I have indicated the different kinds of generalization we can form and the names I would call them, but for this discussion the two most important kinds are the existential and the universal generalizations.

One small point which is worth getting out of the way is the fact that we can generalize in two contexts, which can be called 'open' and 'closed'. The sentences I have used above about sunny days might well be used in a conversational context in which it was clear that only the last five years at one particular place were in question. They would then be talking about a limited set of just over 1,800 days, and one could go through every day to check on their truth. This sort of context yields closed generalizations; what is called the 'universe of discourse' is finite and enumerable. But in many other cases we generalize about an unlimited set which we could not go through one by one. If we say that aspirins relieve headaches we are talking about a potentially endless stream of aspirins and we are prepared to say of other things that if they were aspirins they would relieve headaches too. In such cases we are using open generalizations; and these cases are the norm.

Concepts, Generality and Revision

Having got the distinction clear, we can now see the first important point that our concepts are, at least in most cases, general. We have a concept of fraternity or of sugar or of a whale. These do not introduce individuals into our thinking but rather kinds of relation between people, or a kind of stuff, or a kind of animal. There can be many such animals or many lumps of the stuff, and the use of the concept is not tied to any one particular animal or lump. We can say that the use of such concepts is tantamount to accepting some implicit generalizations about different individual animals or lumps of stuff all having certain properties in common. (This might be an oversimplification, but we are always having to simplify to make progress.)

A very important aspect of the pursuit of knowledge is the attempt to make explicit these sorts of generalization embedded in the use of concepts. This attempt almost always leads us to change our use in some respects. It is well-known that whales were once classified as fish, but are now grouped under mammals. You might want to say that we changed our concept of fish or of whales or perhaps of both; the point is that such a change, however described, was encouraged by our attempts to arrive at perspicuous generalizations about the animals in question. In fact, revision of the network of concepts and making statements using those concepts go hand in hand; it is usually fruitless simply to set out a taxonomy without at the same time using it to make new and better generalizations. As Whewell said, 'the establishment of a right definition of a term may be a useful step in the explication of our conceptions; but this will be the case only when we have under consideration some proposition in which the term is employed' (quoted in Mill, 1886, Book IV, Chapter IV; see also Flew, 1975, ch.5, for some trenchant and sensible remarks about definitions). And so one may wonder, for instance, whether anything is gained by arguing about the class structure of a society if nothing more is to be said about that society and its workings. Similar doubts arise about the ritual 'definitions of terms' students are often encouraged to put at the beginning of their essays.

I have said that it is typical of the advance of knowledge in any area to revise the conceptual scheme that common sense provides. Of course, common sense doesn't stand still, and for some people in some areas it is more sophisticated, more up to date, than for other people or in other areas. But the general point is obvious enough, and no one could deny its applicability at least to the natural sciences and mathematics. In other areas it may not seem so clearly relevant, but I suspect that this is due to the greater complexities of such areas and our general lack of success in developing theoretical understanding of them. But whether or not general agreement is achieved, it is clear that people do seek to revise and stipulate technical meanings for terms like 'revolution' or 'learning' or 'social class'. All this transmuting of ordinary language into technical terms does, however, create a general problem for teachers. People do not normally like to revise their conceptual inheritance. What is implicit takes a lot of painful extracting, and so it tends to remain undisturbed as long as possible. It is also, of course, what is learnt first. I suspect that by no means enough is done to make students face the fact that they are being required to change their concepts, to learn a bit of new language, if you like (cf. Holton, 1984, esp. p. 103). Teachers are often encouraged to make new learning appear 'relevant' to the learner and this is also likely to underplay the discontinuities between the learner's common sense and the actual content to be learnt.

In any case, there is plenty of evidence that students do not learn what they are expected to learn in such cases of conceptual revision. It has been found, for instance, that high school students of physics think about everyday problems in the pre-Newtonian implicit physics of our ignorant common sense rather than in the somewhat more accurate terms of their school specialization (McCloskey, 1983). And, to mention a case I have observed, some students in maths are determined to believe that division by zero is a possible operation, even if it doesn't yield any answer, rather than grasp the stipulation of the subject that such an operation is simply undefined. In these and many other cases, it seems that students persist in thinking in terms that common sense allows rather than face the fact that they have to uproot and destroy these bits of their inheritance.

Aiming at Understanding

I have said that typically conceptual or taxonomic revision proceeds hand in hand with the reformulation, correction or discovery of generalizations. Typically, again, we seek such generalizations in order to understand more of what is going on. This is true, I think, even in areas that may seem preoccupied with taxonomy such as some parts of biology, but it is perhaps more obvious in subjects like chemistry. You find two lumps of stuff that look and smell alike and that you initially classify as the same substance, but then you discover that one lump behaves one way in a certain solution while the other does something different. You are likely to reclassify your lumps into at least two substances so that you can get a firmer grip on understanding these and other reactions. And when you have a generalization, you want to understand it too in its turn. So you are likely to put forward further generalizations to do so, since understanding, as I briefly mentioned earlier, is in large part a matter of seeing how things connect and can be unified. The fact that disparate items can be unified in this way is in turn one of the main reasons for thinking that the explanation is on the right lines.

If you want to understand why acids turn litmus paper red you have to do something to link acids and litmus paper and colour changes and all these to other things you know. The strategy that we have found to be of enormous value in doing this is to postulate initially hidden levels of structure - so now we would talk of molecules, or ions, or shells of electrons surrounding the nucleus, and these notions serve to link the odd case of litmus paper to a vast range of other phenomena. We can see the same sorts of connections in a quite different case: we want to understand why poorer families tend to have more children, and we may seek to show how children are a more sensible investment for poor people than for more affluent groups. Whether or not this is the correct account in any particular case, it exemplifies the way we try to put the phenomenon to be explained into a wider picture; in this case, of means-end rationality. Here we may not be going to a different level of structure as we did in moving from everyday substances to their molecular structure, but we are at least moving to a more general factor, something that can be seen at work in other instances besides family size.

Explaining Away

One central purpose then in our cognitive endeavours is to get hold of explanations and the understanding they bring. But as I have said, many of our successes here have involved moving away from the way things appear to us to be towards underlying structures or mechanisms that certainly do not appear on the surface. Accompanying these moves we also very often find that the appearances are 'explained away'; the deeper explanatory picture we arrive at allows us to do without notions that might have been suggested by the surface appearances. Just as we can successfully explain away the strong impression we all have that the earth is standing still and go on to endorse a theory which says that it is moving in a very complicated and, by our ordinary standards, remarkably rapid manner, so we can equally well see how to dispense with many of the appearances around us, from intrinsic colours-as-seen to the equal opportunities many people imagine they have. Our best explanations do not need to postulate these items; the story they tell is, then, in conflict with our unreflective common sense.

This suggestion relies, of course, on the idea that you should get rid of what you do not need. Such an injunction is not logically forced upon us, but it would seem to be the only reasonable course to adopt. Once one has given up any belief that merely being inherited as part of one's conceptual resources bestows any authority upon a concept or a proposition then it seems nothing remains to justify using such resources but their contribution to one's cognitive tasks.

Once again we have a problem that schools seem not to face. While they are prepared to bemoan supposed moral decadence in the surrounding society, in most other respects schools like to think they are in harmony with their clients, or at least their clients' parents. But if they were to set out to inculcate the best understanding we have of the world, they would continually be running up against and denying beliefs which these parents hold, at least implicitly.

For the same kind of reasons as I mentioned in connection with conceptual revisions, school subjects often seem to underplay their differences from what their students think. Or they are distorted in related ways. Thus a lot of science teaching is bifurcated, with one aspect focussing on limited things that can be done in a school laboratory and observed without much sophistication, and another, apparently quite unrelated aspect, in which the teacher reports on what the theorists are saying (or rather, were saying some years before). Eddington's or Russell's success as popularizers of science owed a lot, I think, to their insistence on the challenge science offers to our ordinary views: you think you are sitting at a solid table, but really it is mostly empty space. I suspect our teaching of science in the schools could benefit from a return to such challenges, rather than pretending that they do not exist.

And of course it is not only science where the pursuit of knowledge undermines popular prejudice. History does not leave much of nationalistic or patriotic belief standing secure, and the social sciences rarely agree with the claims of politicians or the spokespeople of the status quo. (I am here talking simply about the facts of the matter, for example what a country's war aims were, or whether certain people are being discriminated against; not about disagreements on which policies to pursue.) And perhaps one of the most controversial cases is the problem of religion. But for the moment, the point has been made that explanatory knowledge challenges popular belief rather more than schools seem willing to acknowledge.

The Asymmetry of Verification and Falsification

We have been looking at the ramifications of the fact that our concepts can be seen to involve implicit generalizations. We can now move on to the second major logical point about our knowledge. There is a simple logical fact at the root of our reflections about human knowledge of the world: logically singular claims can falsify but not verify open universal generalizations. To take a popular and simple example, 'That swan is not white' (which is logically a singular statement) falsifies, is logically incompatible with the open universal generalization 'All swans are white'; but however many different claims like 'This swan is white', 'That swan is white', 'The next swan is white', etc., you collect, they do not serve to verify, they do not logically force upon you the universal generalization, 'All swans are white.' And a good thing too, since we know a few swans that are not white.

To put it in terms of logical entailments, 'This X is not Y' entails 'It is not the case that all Xs are Y' whereas any number of claims like 'This X is Y' do not entail 'All Xs are Y.' The singular claim entails the negation of a universal generalization, but no collection of singular claims ever entail a universal generalization itself. It is worth noting that the negation of a universal generalization only amounts to an existential generalization: 'It is not the case that all Xs are Y' is equivalent to 'At least one X is not Y.'

So much is a simple matter of logic. How does this relate to the human predicament? We have already noted that empirical knowledge connects in some way with observation and experience. We cannot discover or intuit how things are simply by thinking about them; we are forced to investigate; we are forced to rely at some point on the evidence of our senses or of our machines. And when we do so, what they reveal is something that is logically singular. We see that today is sunny or we smell that room 12 has got hydrogen sulphide in it or we hear that the baby is crying or the seismograph records a particularly strong tremor. The judgments we make on the basis of this sort of evidence are about what is happening at a particular place and time, or at least they would be if we were scrupulous about them. You might boil mercury and report what the boiling point of mercury is; your report would be logically a universal generalization to the effect that mercury always boils at such and such a temperature (in certain standard conditions of pressure, etc.); but strictly speaking what you are really in a position to report is that one sample of mercury boiled at such and such a temperature, and that is logically singular. And in other contexts, you would be less inclined to jump from your observations to general claims - if you see people demonstrating in the street, you are not likely to claim that all people, or even all people looking like those you can see spend their time demonstrating. So what observation or experience reveals is singular.

But as we saw above, what we want is explanatory understanding, and that requires strong generalizations. People argue whether proportional or statistical generalizations can really explain things, but I think everyone is agreed that some sorts of what I have called 'universal' generalization can do that job. If it is true that unlike magnetic poles attract each other, then, given various standing conditions, you have an explanation why a magnetic compass keeps pointing north, and why this one does, and that one, and the next one. We shall return to generalizations that are weaker than universal ones; for the time being let us simplify and sharpen our problem by restricting ourselves to universal generalizations.

We can now see the relevance of the logical fact to our knowledge. We want universal generalizations, but what we have to use can only produce singular claims. The logical fact tells us that no matter how many singular claims we can collect, they will never logically entail one of the universal generalizations we want. We cannot get a logical guarantee that our generalizations are as safe as our evidence. We cannot have a device for generating the true premises we referred to at the end of the previous chapter.

Methodological Responses

There are two things we can do in this predicament. We could try to find ways of making our generalizations as safe as possible, given our evidence. We might hope to discover some sort of 'inductive logic' that could guide us in moving from singular claims to the sorts of generalization we want. People differ on the cost effectiveness of this reaction. We know that any such inductive logic will be unable to guarantee success; we know also from the attempts made along these lines that it will be a pretty complicated affair; and we know, from a consideration I shall come on to later, that there are very important areas where it can hardly hope to guide us. But despite these difficulties, most people think there is something to be done here - some moves from evidence to conclusion seem a lot more sensible than others, and we ought to be able to say why and to organize such reasons into a fairly neat system.

But there is another reaction to the situation. So far we have been stressing what we logically cannot do - we cannot derive universal generalizations validly from our singular claims. But the logical fact had another side to it: we can validly deduce the negation of universal generalizations; we can throw out generalizations, even if we cannot rule them in. So instead of concentrating on arriving at generalizations, we could focus our attention on rejecting inadequate generalizations. We can then be as reckless as we like about coming up with generalizations, so long as we are careful to test them rigorously, that is to say, to test them in such a way that they are likely to be falsified if they are false.

If we follow this path, we will not need to seek any sort of guarantee for our generalizations; we will instead seek to be as harsh as we can with them so that we can weed out inadequate ones. In the terms that Popper has made famous (1959, or more generally accessible, 1969), we can see the history of knowledge as a matter of conjecture and refutation. Or in more familiar terms, we can see in this reaction an endorsement of the centrality of a kind of trial and error to our knowledge of the world.

It is worth noting that these two reactions are not mutually exclusive. We can think it worthwhile to take care about the formulation of new generalizations and we can hope for an inductive logic to help us in this task, while still acknowledging the comparative strength of refutations. But there is another consideration that makes the second, Popperian, reaction seem much more pertinent.

Depth of Understanding

I have already mentioned the fact that one of the most powerful means of achieving explanatory understanding has been to postulate new structures. This profoundly changes the logical situation we looked at above. In the simple case of the swans, the terms occurring in the singular claims and in the universal generalizations were the same ('swan' and 'white'); it is intuitively obvious how the universal claim is a generalization of the various singular claims we mentioned. But when, for instance, we explain the distribution of eye colour in different generations by reference to different genes, we now have quite different terms in the singular claims from those in the generalizations ('blue eyes', 'brown eyes' in the singular claims; 'gene 1', 'gene 2' and talk of dominant and recessive genes, etc. in the generalizations).

Putting it crudely, the two sides of the fence are now 'This X is Y' and 'All Z are W'; and there are no direct logical linkages left: neither verification nor falsification. There is simply no way to move logically from talk of X and Y to talk of Z and W, or vice versa. But of course, that is putting it too crudely. The explanations we offer do bridge this gap, and they do it in various and often complex ways; but in the simplified terms of our logical schema, what these links amount to are claims such as 'All Z are X', and 'All W are Y.' But while we can reinstate logical links between our singular claims and our explanatory generalizations, the main thing to realize is that talk about our Zs and Ws is not directly suggested to us by what we start from, the talk of Xs and Ys. Someone has to invent the idea, of molecules or genes or rules of transformational grammar, or whatever, and of how these 'new' things behave. None of this is given directly by the phenomenon to be explained, and it is very difficult to see how general formal rules, such as an inductive logic might aspire to offer, could ever begin to help in finding such ideas. But, as I've said, it is these ideas that have proven explanatorily powerful; and they can apparently only come as creative conjectures. So perhaps we should put most of our eggs in that basket.

One lesson of all this is that cognitive revision involves both creative conjectures - 'divergent' thinking in Hudson's (1967) terms - and critical testing, which requires careful 'convergent' thinking. To the extent that schools cultivate only one kind of thinking, people are not being helped to participate fully in our cognitive life. Again, while both sorts are necessary it would be salutary not to confuse them as so often happens. Students are asked to 'deduce' things in a comprehension exercise or in a laboratory experiment that can only be hypothesized, while there are other things that can in fact be deductively inferred. Not to see that different skills are involved can only hinder fruitful teaching here.

The issue we have just been looking at is often portrayed as the existence of a distinction between observational and theoretical terms. I have tried to explain it without making that connection explicit, though it may well have occurred to you: we observe blue eyes, we don't observe genes. But of course, there's a problem with this way of putting the issue: some people these days do observe genes, just as other people observe muons and other sub-atomic particles. I have mentioned already that in some cases you will be likely to claim that you have seen or measured the boiling point of mercury in general, when what you have actually seen or measured is only a singular occurrence of this sample of mercury boiling at this place and time. But since we think this measurement is reliable, your report incorporates a lot of these assumptions of reliability; you stick your neck out, without even noticing it on many occasions. And similarly with so-called theoretical terms: if someone grasps the theory and thinks it sufficiently reliable, he will report his observations in terms of that theory, rather than redescribe them in less ambitious ways. We can always move down towards more generally accepted claims, if we have to; but in general we try to stay with the most powerful and informative ways of expressing them, and these will be the ones incorporating what we currently deem to be reliable theory.

The Role of Background Assumptions

I have said that the most important point here is that the ideas incorporated in the generalizations are conjectures going beyond anything in the singular data. Another crucial fact is that the logical links are now between singular data and a set of generalizations, many of which are usually left pretty unspecific. When we only had 'This swan is not white' and 'All swans are white', the former directly refutes the latter, so if you accept the former you cannot accept the latter. But when schematically the situation is 'This X is not Y' on the one hand and 'All Z are W, and all X are Z, and all W are Y' on the other then the first claim only shows that there is error somewhere in the set of generalizations, in the theory; it doesn't tell us where. The only way we can now chalk up a refutation of the theoretical claim, 'All Z are W', is to assume that the other generalizations, the ones that link theory to observation, are all correct. And that is obviously a pretty large assumption.

We can in fact go back to the simple example and find similar difficulties. We took as a given of that elementary situation the singular claim, 'This swan is not white.' But, to be strict with ourselves, our evidence may be less ambitiously stated as 'Jim observed that this swan is not white.' The claim about the swan is itself based on visual evidence available to Jim. But to get from the visual evidence to the claim about the swan in the objective world, or vice versa, we similarly would need various linking claims. We assume things about the way light behaves, we assume things about Jim's normality as an observer. We certainly cannot infer validly from the report of what Jim saw that what he reported to have seen was in fact the case. (This claim may need some qualifications in the light of how we sometimes use our language here, but my point could have been made equally well by retreating yet further down the line to the very weak, but perhaps most honest claim that it appeared to Jim that he saw a swan that was not white: nothing follows from this about what colour the swan was; there might not even have been a swan at all.) So we can now see that even in the apparently simplest cases, we make a lot of assumptions in using singular claims to weed out our generalizations. Of course, we still cannot verify any of our universal generalizations, but we can now see how provisional even our refutations have become. They are at the mercy of these varied assumptions that bridge the various gaps we have uncovered.

It should be noted that in talking of assumptions or background knowledge here I am not suggesting that these items have any particularly privileged position. They are not immune from critical examination. Our position is that any claim to knowledge can be examined, but that it is not possible to test all of them at once. To examine a claim you must take other claims for granted, for the time being. Of course many of the assumptions referred to in the preceding discussion are so deep-rooted and inarticulate that it may be misleading to call them 'theory', but this does not affect the logic of the situation.

A different way of describing the situation, in terms of observational terms, is that our use of observational terms is logically no more secure than that of theoretical terms. The provisional, conjectural status we have seen in theoretical generalizations inheres equally in the singular observations we make, though in practice we rarely notice this, since we take a great deal of our knowledge as finally settled. Few people realize that they are making a theoretical leap in talking about the sun yesterday and today. But it is a theoretical, non-observational claim that there is one persistent object that 'rises' each day (or at least it was until very recently), and it is a theoretical claim that has occasionally been rejected, as in Xenophanes' cosmology. A much more pertinent area in which to see the provisional status of singular observations is that of scientific measuring techniques. One important achievement in the natural sciences (and one of their differences from much of our study of ourselves) is to have gained an understanding of how their measuring instruments work. But of course that understanding is a matter of conjectural theory, and there are occasions when we have had to revise singular measurements because we have come to revise the theories built into the measurement techniques. Thus when it was realized that the concentration of radio-carbon in the atmosphere has not been constant we had to revise a large number of radio-carbon dates, and with them our provisional theories of European prehistory (Renfrew, 1976).

We have only taken the first steps in complicating the account of cognitive revision. It might be more realistic to see change in terms of Lakatos' (1970) 'research programmes' rather than isolated claims; and one should acknowledge that there are several ways in which history has been a lot messier and a lot more lucky than my picture suggests. In making these admissions it is important, I believe, to tread a path between the tendency to think in terms of a framework within which people operate (which is a reasonable description of what we do) and the logical point that propositions belonging to such frameworks are no different from those within them, so that frameworks are not immune to criticism.

The Falsification of Less Than Universal Generalizations

In the preceding section we have seen that the neat asymmetry between verification and falsification with which we started has been radically modified, since falsification is now only relative to assumed background knowledge. While I have been avoiding the complexities of open proportional or statistical generalizations, it is worth noting here that they also involve a similar relativizing of falsification. (Verification remains as impossible as ever.) In this case, falsification is only possible given some decisions about sampling and about how unlikely one wants errors to be. The reason can be easily seen. While one black swan refutes the universal generalization, 'All swans are white', how many do we need to refute the weaker 'Most swans are white'? You have to make some choices about the kind of sample you will examine and what proportions in that sample you will allow to exclude the generalization in question. Given some such decisions, observing 73% non-white swans in a large random sample would be taken to rule out the generalization; but it cannot do so absolutely. Even if most swans are white, it is still possible to get a random sample that is mostly non-white; it is just rather unlikely.

Description or Recommendation?

I have presented a picture of our knowledge that may well seem unfamiliar to you. Even if you are prepared to accept it for some specialized and perhaps peripheral areas, you are not likely to think that it is a correct description of most of what we all claim to know. What reply can I make to such criticism?

In the first place, we have to distinguish here, as elsewhere, between what we think we know about our knowledge and what, if anything, we really do know about it. Knowledge is as hidden by ideology as the rest of our social being; you should not be surprised if knowledge turns out to be somewhat different from what you unreflectively imagine.

In the second place, I would claim to have delineated the fundamental logical structures inherent in the propositions we believe or claim to know. If I am right, it doesn't matter that we do not recognize these logical structures; they are there, and put their limits on what we can, as a matter of logic, achieve. Our untutored thinking bears often only a tenuous relation to what the science of logic reveals. So in saying that I am talking about the logical structuring of our knowledge, I am not committed to any claims about how in practice we think or go about revising our beliefs. Given the largely negative achievements logic allows us, I would not even claim that what I have said tells us how we ought to proceed, except perhaps in criticizing others. We might well come upon the truth by ignoring everything I have said. Since no method can guarantee success, to that extent Feyerabend's 'Anything goes!' strategy (Feyerabend, 1975) might do as well as any other; though few would think we should be so carefree in the pursuit of knowledge.

But in the third place, I would want to say that a lot of our actual thinking does in fact reflect the logical principles I have been discussing. You think that if it has rained recently the lawn will be wet and when you find the lawn bone dry you conclude that despite the dark clouds it didn't rain. You may not have arrived at this conclusion by any process that involves what logicians call modus tollens,-- the principle that if p then q and not q together entail not p -- but it is certainly as if you had. You are playing around with a new machine and find that pressing a particular button always seems to produce a recognizable result, a beep for instance, so you begin to call it a beep button: that description incorporates the hypothesis you have to some extent tested in your interactions with the machine. These are deliberately trivial examples, but they are meant only to show that some of the processes I have been discussing are in fact pervasive features of our everyday thinking.

Although our ordinary thinking does in fact mirror a lot of the epistemology I have been expounding, it may often, as I have admitted, not in any sense be based upon it. And of course your objection started from the fact that much of our thinking does not seem to fit the picture. This is one reason why the revision of knowledge is in fact so social a matter. Each one of us may have the most diverse and logically unsupportable views about what we believe, but the social institution of enquiry produces outcomes somewhat more in accordance with the picture I have sketched. So whatever individuals may think about their particular views, the institution by and large takes the tentativeness of claims for granted. It keeps rewriting its history, as Kuhn (1970) stressed, to make the present contenders look virtually inevitable; but in another generation the picture will be redrawn. (I must acknowledge that this discourse too is part of such a history - what epistemology can do for teachers will no doubt look very different to other philosophers or at other times.) Rivalry between scholars or scientists may often degenerate into mud-slinging or other unfortunate forms, but it can also serve as a public embodiment of logical canons, however partisan the individual participants might be.

Representation, Realism, and Scepticism

In the preceding discussions I have been taking one crucial matter for granted: that the whole enterprise I have described is justifiable. But as we saw in the first chapter and later, once one begins to ask simple questions about our knowledge and its reliability, it is very easy to think it all factitious, a social construction without solid foundations. And what I have said may not seem to make it any easier to discern such solidity. Indeed, many philosophers would say that my account invites extreme scepticism, since it allows that on the one hand there is our conceptualized experience of the world and on the other there is what the world itself is like. How can we successfully make the leap across from the experience to the world it represents?

Since these questions arise naturally, and have indeed already been mentioned, I ought perhaps to say a little about how I would answer them, though this answer may not have any direct repercussions on the conduct of schooling in the way that some of the other points we have looked at probably do. The account I have offered is indeed a 'representative' theory of perception: we have experiences of what presents itself directly as a three plus one dimensional world outside ourselves, but I have denied that we can simply regard the world as presented as in fact the world as it is. I have endorsed Locke's view that much of how the world appears to us is merely our contribution - whatever daffodils are like, they are not yellow-as-perceived. I have allowed that virtually all our experience comes imbued with concepts, but I deny any special authority to those concepts, and to the experiences themselves.

However we acquire the elements of experience, this account makes the fundamental question of justification the question of whether we should regard experience as revealing anything at all beyond itself. In other words, the question of scepticism about an external world. I do not pretend that such scepticism is easily answered, but I would say that once one seeks to move anywhere beyond the most attenuated interpretation of experience (for instance, filling in the gaps between blinking your eyes) then the only plausible explanatory account is the ordinary commonsensical one that there are independent objects 'out there'. It is possible to tell other stories - it is always possible to tell other stories - but none of them have the plausibility of the ordinary one.

Note that I have made explanation crucial, even at this very basic, taken for granted level. It is then not difficult for me to continue to insist on explanatory power with respect to the elements of the total picture presented to us by untutored perception. And so the Lockean account offered above fits smoothly into the whole strategy. That there are real daffodils out there explains our intermittent experiences of them; but we do not need to postulate that those real daffodils have the yellow-as-perceived that those same experiences endow them with. In both cases, what is needed for explanation is what counts.

To insist on explanatory power does not guarantee convergence on one theory of the world, but in fact at the levels and areas we have currently reached there is only one overall plausible theory, that embodied in the comparatively settled 'findings' of scientific inquiry. That is the picture technology unhesitatingly looks to.

I am thinking here of a fairly general picture of things. For reasons we shall note later, and no doubt for others too, some of the more detailed claims made in the sciences may deserve alternative interpretations. In some cases, we know that an account is only a useful picture that distorts almost as much as it reveals. It is not therefore to be taken literally. All I have claimed is that realism, the taking literally of what theories say, is the obvious line to take, the one that makes the fruitfulness of those theories intelligible.

I am also thinking of the more settled, larger-scale levels of the world. Fundamental physics operates with ideas that often seem incredible, and certainly upset the comparatively straightforward view we have at a more homely scale. While I do not pretend to understand such physical theories, what their popular interpreters say creates a problem for the view I am endorsing. There are various alternatives: we might concede that the more settled picture is in fact false on its own terms. Quantum indeterminacies may characterize everything, and the world may be a much stranger place than we imagine. Alternatively, we may allow that at the scale of microphysics things are indeed as strange as the popularizers suggest, but that the 'settled' picture remains true at a larger scale. This would presumably entail a certain incompleteness in the physical theory, but we certainly cannot rule out further developments of such theory. This possibility allows also for another scenario in which present theory is replaced by a somewhat less mysterious account, or given a less mysterious interpretation - just as we now recognize that a lot of the 'relativity' of Einstein's celebrated theories was exaggerated by early interpreters. Perhaps as much as anything else, the lesson to be learnt from the unintelligibility of popular versions of quantum mechanics is a general modesty regarding our achievements to date. Socrates, at least, would not have been surprised that we should learn how little we know about the cosmos, and indeed ourselves.

Pedagogical Relevance

We have been looking at the logical structure of our ordinary claims to knowledge of the world and of the theoretical explanations we offer of them. You might well think that by now there is not much structure left, since we have uncovered gaps all over the place. Be that as it may, the discussion so far has brought up several factors that would seem to have consequences for passing on our knowledge. Perhaps the most obvious is that all this knowledge is logically provisional. We simply cannot hope for items that will have a logical guarantee that they need never be revised. I have avoided saying much about what our concept of knowledge involves; if it does commit us to having logically unchangeable items then it is shot through with error. I don't think it does so commit us, but that may be because I am convinced of the truth of what I have said about the nature of our knowledge; I may have adjusted my concept of knowledge to fit. Some people certainly seem to think that knowledge must be final if it is the real thing. One way of taking this is fair enough: if you know some proposition then that proposition is true, and in that sense final. But our predicament is that whatever we know and whatever we know that we know could yet turn out to be false.

It cannot be denied, however, that the picture of conjectural knowledge I have presented does not sit easily with our linguistic habits. On my view we think we know a lot that we don't know. One could choose to speak differently, but any choice is likely to lead to awkwardness somewhere. Thus Bernard Williams (1985) prefers to speak of knowledge within a perspective, so that, to take an example from an earlier chapter, from one perspective we can know that daffodils are yellow and from another we can know that there is no such intrinsic property. But he then has to admit that cognitive advance can result in the loss of knowledge, which does not seem to me to be a very happy way of characterizing the situation. The general point is that the language we use has not been framed to express these sorts of truth and so it is not surprising that it does an inadequate job. That is one reason for not discussing the 'logic' of knowledge, and for recommending that you don't let yourself become preoccupied with verbal puzzles about knowledge.

It is important to note that I have been talking about logical guarantees, logical finality. The lack of this sort of logical conclusiveness does not necessarily mean that we cannot be sure, for all practical purposes, that we have got hold of some truths. I am as sure as I ever could be that I am now sitting in front of a word-processor in Mona on a fairly windy day. We are all equally certain that if we stepped out of a window on the 30th floor of a skyscraper we would fall to the ground rather disastrously. And of course there are a vast number of similar claims that we all claim to know and claim to know that we know. The logical gaps we have been looking at do not require us to deny any of this. All they show is that being to all intents and purposes sure of what we believe falls short of a perfect guarantee. It is logically possible that I am suffering from some massive hallucination; it is logically possible that things will stop falling towards the centre of the earth. But these are logical possibilities that we do not regard as worth considering in our normal dealings with the world. If all our supposed knowledge were as reliable as my beliefs about where I am or our acquaintance with what happens when we drop things, making all this fuss about logical gaps would be an impertinence to would-be teachers, at least. But our knowledge isn't all like that. We may well be pretty sure about what is going on around us (in a superficial way, at least) and about some general and unspecific facts about the world we live in, such as that it has got things and people in it. But in between these particularities and extreme generalities there are a host of general beliefs about how things work and what influences what, the business of the special sciences and human studies, which show very considerable historical divergences and where the future revision of our currently best-supported beliefs is virtually guaranteed, rather than the reverse. And it should be obvious that schools, at least at the secondary level, deal almost entirely with knowledge from this intermediate band - historical interpretations, chemical theories, biological taxonomies, etc., etc.

So in most of what we do in fact teach, the logical points are not just philosophical niceties. Rather they are reflected in the actual histories of the subjects concerned. So in those subjects, we simply cannot avoid facing the actual tentativeness of our best-supported claims to knowledge. To avoid passing on a false picture of its status, teachers should find ways of transmitting such knowledge that both concedes its impermanence and reveals why it is yet to be learnt. It is no good getting students to see only that our present views in history, science and so on are going to be superseded with the result that they think that they needn't bother to learn what those current views are. They must be brought to see also that it is only on the basis of some views that we can test others and thereby hope to make progress. They are being initiated into a social tradition of criticism which is aimed at weeding out inadequate views and testing new conjectures. Such a tradition has proven remarkably successful in some fields at coming up with views that work well enough - technology can be based on false views but successful theory-based technology suggests very strongly that the views it relies upon are at least approximations to the truth. The tradition can only do better by the students' active participation in the social dialogue of conjecture and refutation.

Teachers should therefore convey a feeling for theoretical impermanence, and for the way that provisional status can infect our reports of data. But as Freire (1978) notes, 'the impatient educator often transfers knowledge like a package while discoursing volubly on the dynamic nature of knowledge' (p. 64).

To put the issue in terms of authority, I have denied the relevance of most types of appeal to authority in evaluating claims to knowledge. What remains is a question of whether a claim explains what is to be explained and a question of how it fits logically with other claims we tentatively accept. Explanation may not be entirely a logical matter, but it involves logical relations, and the consistency of claims is totally a logical issue, so that a large component in what is authoritative regarding knowledge is logical. The authority the teacher can appeal to is then the authority of the logical 'rules of the game' - not so much the authority of facts but rather the authority of the consistency or inconsistency between one claim and another.

Evidence and Explanation

Connected with the preceding points is a very important re-orientation of one's view of the relation between evidence and explanation. Most people conceive the relation as one of evidence or testimony pointing towards a conclusion; the data give one reasons for a conclusion, or for accepting a theoretical explanation. Teachers who have been taught methods of educational research may well have been reinforced in such views by the typical procedures of feeding raw data into statistical programmes and coming out with theories of a kind: factor analyses or regression equations. Some writers have recognized how distant these procedures are from the kind of cognitive growth sketched here where ideas are formulated before evidence is collected to test them, but they seem still to be voices crying in a wilderness (cf. McDonald, 1985; Mulaik, 1985).

We have seen already that data cannot entail a theory. The Popperian view we have been sketching would suggest that we adopt a different approach and consequently different intellectual strategies. Instead of hoping to argue from the data to an explanation, we should look for alternative explanations of the data. We should not expect that the data will leave us with only one conclusion (unless we can rule out all but one of the explanations we can think of) but we can ask how well different explanations account for the data. This immediately gives us an important strategy that is not encouraged by our usual view of the matter, viz. searching for alternative explanations. You offer me some evidence for a particular conclusion; we could debate how well it supports that conclusion; but it might well be salutary to note that the existence of the data is much better accounted for in some quite different way. This could save us all a great deal of futile speculation.

It is important to note the phrase 'the existence of the data' in what I have just said. Connected with our tendency to try to reason from data to a conclusion is a willingness to treat a lot of data as 'transparent'. It says in a document that John Smith married Ann Baker so we immediately assume that John Smith married Ann Baker; or a dial reads '40 amps' and so we immediately assume that it is measuring 40 amps. But in all such cases, strictly speaking our evidence is the fact that this document or machine says something. (This is the same point we noted earlier in connection with the boiling point of mercury - we put the fullest interpretation we reasonably can on our evidence.) It requires further assumptions that things are as the document or machine says they are. If we are instead looking for alternative explanations of our data it is, I think, a little easier for us to step back from the usual trusting attitude and see the fundamental thing to be explained as the fact that the document or the machine says what it says. Once that shift is made, we need not be so inclined to take the evidence at face value.

While some teachers do now try to inculcate such attitudes to historical sources or to other kinds of data, the pressures of the normal presentation of school subjects do not make it easy for them to succeed. Evidence is in fact very rarely presented for any of the claims that are made. And for the kind of reasons sketched above relating to the cumulative build-up of theories it would in very many cases be incredibly difficult to present this evidence. So as we have seen already, school science is split between extremely simplified experimentation and the more sophisticated theorizing that has to be taken simply on trust. School history likewise spends most of its time offering 'facts' with very little leisure for the critical examination of documents or other sources.

For both, a large part of the problem is that real work in these subjects depends on a vast array of background assumptions. These have to be told to students if they are to participate properly in the enterprise, but that is going to take up too much time. So the rigged procedures of 'guided discovery' arise (see, for instance, Driver, 1975, or Atkinson and Delamont, 1976). Or history teachers tack on an unintegrated lesson about how to examine a document critically. While I do not pretend to have solutions to most pedagogical problems, it does seem to me that if we want to engage students in something approaching real work in these sorts of subject it might be better to take circumscribed topics in which they can be genuinely immersed rather than hoping to cover vast chunks of material as well as getting an inkling of real 'discovery'. Or it might be better to give up trying to include discovery (which involves an unteachable creativity) and concentrate on the other side of the enterprise: the critical testing of ideas. In many areas there is no real need for students to know most of the 'answers' if they have grasped how proposed answers are to be evaluated.

The Structure of Explanation

One of the main claims I have made is that one important motive behind our search for knowledge is to have explanations, or to be able to understand things, or to find them intelligible. It is not just a matter of data gathering. As I mentioned in passing earlier, these notions are not totally clear. People argue about how much a statistical generalization can explain anything, and there are many other problems in giving a clear account of the requirements for explanations. But for our purposes we do not need the final and complete story about explanation. There are several very important points we can see with the aid of a simplified model.

In terms of the logical distinction we met earlier, there are two main sorts of thing we might want explained: singular claims or generalizations. (As before I shall stick in this simplification to open universal generalizations.) We might want to know why Reagan was re-elected, for instance, or we might want an explanation of the fact that gold does not dissolve in water. Typically an explanation will involve some other claims of the same logical type. If you want to understand why Reagan was re-elected you will be told other singular facts about the situation at the time. If you seek understanding of the properties of gold you will be given generalizations about the particles that make up gold molecules. But in the case of singular claims it is also very common for people to invoke various generalizations as well. In our example, you may be told about how people, or people in the USA, react to certain facts, how they seek to achieve certain goals. Even if such general claims are not made explicit, it is very plausible to say that any explanation of a singular item must implicitly rely upon some such generalizations. And it is clear that a great deal is omitted from virtually all explanations, though it must be assumed to be there if the explanation is to work.

So, putting it very crudely, all explanations will involve generalizations, and explanations of singular items will also involve some other singular items. From these facts it is obvious that we will never be able to explain everything. In any explanation there will be some claims that are not explained in that explanation. We can of course always ask for another explanation, in which these things that have been taken for granted before are themselves explained. But even this process will always stop somewhere at any particular stage of our knowledge. We may be able to ask why sub-atomic particles obey the laws we think they obey, but we may not be able now to answer that question. We may be able to take some sequence of singular occurrences a long way back, but they will still be preceded by other occurrences, whose explanation we may be unable to give. This doesn't mean that we will never be able to give such explanations; I'm not saying that there is anything whose nature is such that it cannot be explained. I'm simply saying that we will never be able to explain everything all at once (cf. what we said earlier about criticizing claims to knowledge).

How Not To Complete Explanations

One common reaction to these thoughts is that while ordinary naturalistic explanations must stop somewhere religious explanations can carry on to give some more profound and final explanation. But this is a confusion. It is in the nature of explanation that we can always ask for an explanation of anything used previously as part of an explanation. So if a divine being or intention were offered as an explanation of anything, we could always ask for an explanation of that being or intention in its turn. Put otherwise, nothing is self-explanatory. It is true that some things may appear self-explanatory in that we wouldn't normally bother to ask for their explanation, but that is a matter simply of what happens to satisfy our curiosity; it doesn't reflect some intrinsic self-explanatoriness.

There is, then, a subtle but important difference between the modesty of naturalistic science and the presumption of religion. The former simply says that we can explain up to this point, but as of now we simply have to accept that these are how things work and that this was how the universe was at some earlier time; we do not currently have any explanations of why these things are as they are. The latter is willing to agree with most of this, but it tries to add that it does have a further explanation. My point is that while that further explanation might even be true there is no reason to suppose it has the finality its supporters usually pretend to. They tend to denigrate naturalistic explanation for its incompleteness, but any explanation is going to be incomplete in that manner, so we should stop hoping otherwise.

It is perhaps worth noting that our present scientific cosmology does talk about the beginning of the universe in the 'big bang'. As far as I understand this talk, it does not alter what I have said about the necessary incompleteness of explanations since any use of the equations for the big bang has to adopt certain values for their variables, and these values are not derived from any deeper theory. They are simply brute facts. In fact one way of estimating some of them is to work backwards from our present situation by asking what initial distribution would yield what we see around us. But what we see around just is what it is; it doesn't have any self-explanatoriness built into it. And of course, the equations used to work out how the big bang developed are equally non-self-explanatory. And equally obviously, most of our knowledge in other subjects stops a long way short of the beginning of the universe!

In rejecting the feasibility of religious explanations it might seem that I have overlooked one frequent feature of such explanations. They are often answers to the question 'Why?' understood, not as asking for an account of the mechanisms involved, but rather as asking for a reason, as one asks for a reason for performing a particular action. There is nothing intrinsically wrong with such questions, but it must be recognized that there is equally nothing wrong with answering by saying that there is no such reason for something. Typically such questions presuppose an agent with purposes, so one pertinent answer is to deny the presupposition, just as one hopes one can do in response to questions like 'When did you stop beating your wife?' This point is relevant when there are in fact purposes around, since at some point an explanation of the existence of the purposes will be unable to retreat to some further purposes: I may boil water to make coffee, and I might want to drink coffee to quench my thirst and get my daily dose of caffeine, but at about this point explanations in terms of intentions or purposes are likely to stop.

Coincidences and Closed Systems

I have argued that putative 'deeper' explanations offered by religion cannot do what they pretend to do. There is another area in which some religions seek to offer explanations that go beyond what a scientific naturalism would accept, an area that points up some very important aspects of scientific explanation. I have kept saying that understanding typically involves simplifications. In a lot of the sciences we deliberately simplify our problems; we abstract from the enormous complexity of the real world, and consider only what would happen in some isolated system. So in thinking about how unsupported objects behave, we start by ignoring the resistance of the air and we claim that all objects fall towards the centre of the earth with a constant acceleration. Having worked out how it would be in this fictitious situation, we can then move on to complicate the story. But the basic point is that this sort of abstraction from real complexity is typical of our attempts to understand the world. What results are a set of theories about idealized, simplified situations. One important and very common idealization is that nothing interferes with the system under consideration, that is to say, we have a theory for 'closed' systems. We can now calculate the motions of the planets to a very high degree of accuracy, on the assumption that there won't be any large and hitherto undetected object interfering with them. But of course that is an assumption, and one that may well not be true.

Since so much of our knowledge concerns the workings of closed systems, it is perhaps not so surprising that we can say very little about what will actually happen in the real world, outside of our laboratories. In that real world, we encounter open systems and so Popper's remark, 'we are very far from being able to predict, even in physics, the precise results of a concrete situation' (1961, p. 139), is not so exaggerated as it may sound. (Compare this remark by two physicists on a related theoretical deficiency: 'even the simplest system undergoing vigorous convective motion cannot be given an exact mathematical description' (Velarde and Normand, 1980, p. 79), though the difficulty in describing boiling water is due to interactions rather than to outside interferences.)

People are disheartened by the inability of economists to predict what is happening, but to expect that anyone could is perhaps to ask for much too much. What then often happens is that we can explain events in terms of their place in relatively closed sequences. But when two or more such sequences converge on one event, we cannot explain the 'coincidence', however fraught with significance it might be. So, to take a mundane example, we might explain why John Smith was standing under a particular building at a particular time by reference to his intentions, beliefs, etc. We might also be able to explain why that building was struck by lightning at a particular time (or at least we know the sort of factors necessary for such an explanation). But we have no further explanation of why John Smith should have been standing underneath it at the same time as it was struck. These two aspects of the event of his death are each explicable, but the coincidence is not; it is an irrelevance from the standpoint of the comparatively closed systems of explanation. But it is obviously the coincidence that has much greater significance for us, and that we would like to have explained. Here again many religious systems would offer explanations (witchcraft, perhaps, or inscrutable providence) where naturalistic science can only refrain from the attempt.

For our present purposes, the other important aspect of this kind of case is that it reveals a basis for some of the subject divisions we find around us. It has proved fruitful to isolate some properties of things or to attend to their constituents at some particular levels. Slicing up the world in these ways has allowed us to generate powerful theoretical explanations. We also need ways of linking the levels or aspects we have found it useful to distinguish, but these linkages may be very messy. Thriving subjects occupy the slices, less fruitful speculation, and perhaps philosophy, focus on the linkages. So, for example, historical linguistics has made progress without any serious attention to the questions of the physiology and psychology of speech production that presumably somehow underlie its results; thermodynamics is carried on to a considerable extent in isolation from the statistical mechanics that we know in principle explains it (Sklar, 1976). And a whole branch of philosophy concerns the relations between mind, or our conceptualizations of mental life, and our attempts to theorize the matter in which it is manifested.

I think it is important to stress these limitations on the power of our best supported explanations. People in general expect too much. They want knowledge that is more than the provisionally best account we can give; they want 'everything to be explained'. Instead, as we have seen, our knowledge of the world is inherently tentative, and a great deal of the most interesting is clearly provisional. Our best explanations typically involve simplifications or idealizations - they cannot then predict what normally happens in the rough and tumble of ordinary life. And simply as a consequence of their logical structure, our explanations will always leave some facts and laws of working unexplained, though not in principle inexplicable. To use two metaphors that are often invoked, Neurath's ship at sea (or Putnam's fleet of ships) any part of which can be repaired, but not all at once, or Popper's city built on piles in a swamp, we should remember that both circumstances are insecure and require makeshift expedients. I have said I wouldn't talk about mathematics, but it is worth saying that the picture people have of mathematics as secure and tightly bound knowledge can mislead them into expecting a similar security elsewhere. We simply don't have it and in many respects cannot achieve it.

Concluding Remarks

To sum up the pedagogical implications of this fairly lengthy survey of the nature of our knowledge, we can see that teachers should find ways of genuinely presenting its provisional status. They should seek also to acquire and pass on a re-oriented conception of the relation between evidence and explanation which stresses much more the way that explanations account for data rather than thinking of data as determining explanations. Further, they must recognize the way the development of knowledge depends on assumptions or background knowledge and a backbone of logical relations that are more permanent than the provisional data and theories we employ. They must also be prepared to recognize the ways in which knowledge revises our common sense picture of the world and so clashes with what people are otherwise brought up to believe. The enterprise of knowledge can easily lead to what Weber called the 'disenchantment' of the world, a loss of its cosiness and its apparent endorsement of our values: an issue we shall take up in the next chapter. The search for understanding reveals a lot of common sense as ideological obfuscation. Finally we have just been looking at some of the things our knowledge and our best explanations cannot do. These limitations are equally important for teachers to pass on; exaggerated expectations can lead to exaggerated disillusionment.

Footnote

3. 'Consider your race:

The original is from Dante's Inferno XXVI. The words are put in the mouth of Ulysses; Dante consigned him to Hell, but did not conceal a certain sympathy for his supposed motivation.

Preface

1 Introduction

2 Truth

3 Knowledge

4 Opinion

5 Further Reading and References

Originally published by Allen & Unwin, 1987, this version last revised June 17th, 2000.

URL http://www.uwichill.edu.bb/bnccde/epb/KNOW.html