How Deep is the Distinction between a priori and a posteriori Knowledge?




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How Deep is the Distinction between A Priori and A Posteriori Knowledge?1

[in a volume on the a priori edited by Albert Casullo for OUP]


Timothy Williamson

Abstract: The paper argues that, although a distinction between a priori and a posteriori knowledge (or justification) can be drawn, it is a superficial one, of little theoretical significance. The point is not that the distinction has borderline cases, for virtually all useful distinctions have such cases. Rather, it is argued by means of an example, the differences even between a clear case of a priori knowledge and a clear case of a posteriori knowledge may be superficial ones. In both cases, experience plays a role that is more than purely enabling but less than strictly evidential. It is also argued that the cases at issue are not special, but typical of a wide range of others, including knowledge of axioms of set theory and of elementary logical truths. Attempts by Quine and others to make all knowledge a posteriori (‘empirical’) are repudiated. The paper ends with a call for a new framework to be developed for analysing the epistemology of cognitive uses of the imagination.


Keywords: apriori, aposteriori, knowledge, imagination, logic, mathematics

1. The distinction between a priori and a posteriori knowledge can be introduced the bottom-up way, by examples. I know a posteriori whether it is sunny. I know a priori that if it is sunny then it is sunny. Such examples are projectible. We learn from them how to go on in the same way, achieving fair levels of agreement in classifying new cases without collusion. Of course, as well as clear cases on each side there are unclear cases, which elicit uncertainty or disagreement when we try to classify them as a priori or as a posteriori. But virtually all useful distinctions are like that. If we want to be more precise, we can stipulate a sharper boundary with the clear cases of the a priori on one side and the clear cases of the a posteriori on the other. How could such a distinction be problematic? If some philosopher’s theory puts a clear case on the wrong side of the line, surely that is a problem for the theory, not for the distinction.

The risk for the bottom-up method of introduction is that it may make a distinction of no special significance. On that scenario, our classifications follow similarities and differences that, although genuine, are largely superficial, like a taxonomy of plants and animals based only on colour. If so, epistemologists would do better to avoid the distinction between the a priori and the a posteriori in their theorizing, because it distracts them from deeper similarities and differences.

The alternative method of introduction is top-down, by a direct statement of the difference between the a priori and the a posteriori in epistemologically significant theoretical terms. For instance: a priori knowledge is independent of experience; a posteriori knowledge depends on experience. The risk for the top-down method is that it may turn out that everything is on the same side of the theoretically drawn line. If Quine (1951) is right, all knowledge depends at least indirectly on experience. As with the risk for the bottom-up method, that would make the distinction epistemologically useless, but for a different reason.

Friends of the distinction typically assume that the bottom-up and top-down methods yield equivalent results and so are mutually supporting, each averting the risk to the other. Of course, the risk here is that the two methods may have incompatible results. If so, assuming otherwise leads us into epistemological error.

But are any of the risks realized? In this paper I will suggest two ways in which reliance on a distinction between the a priori and the a posteriori does more harm than good in epistemology. Neither of them is exactly Quine’s. As a framework for discussion, I first sketch the top-down distinction as contemporary philosophers tend to conceive it.


2. The distinction between the a priori and the a posteriori is primarily a classification of specific ways of knowing.2 A way of knowing is a priori if and only if it is independent of experience. It is a posteriori if and only if it depends on experience. The relevant senses of ‘independent’ and ‘experience’ are discussed below. Every specific way of knowing is either a priori or a posteriori, and not both. One knows p a priori if and only if one knows p in an a priori way. One knows p a posteriori if and only if one knows p in an a posteriori way. Thus if one knows p, one knows it either a priori or a posteriori.

One may know p both a priori and a posteriori, if one knows it in several ways, some a priori, some a posteriori. Tradition excluded that case on the grounds that only necessities (truths that could not have been otherwise) are known a priori whereas only contingencies (truths that could have been otherwise) are known a posteriori. But that was a mistake. Here is a simple counterexample. Suppose that Mary is good at mathematics but bad at geography, while John is bad at mathematics but good at geography. Both of them can perform elementary deductions. Mary knows a priori by the usual standards that 289 + 365 = 654 and does not know at all that there are cable cars in Switzerland. John knows a posteriori by the usual standards that there are cable cars in Switzerland but does not know at all that 289 + 365 = 654. From the premise that 289 + 365 = 654, Mary competently deduces the disjunctive conclusion that either 289 + 365 = 654 or there are cable cars in Switzerland (since a disjunction follows from either disjunct), and thereby comes to know the disjunction a priori by the usual standards, since the logical deduction introduces no dependence on experience. Meanwhile, from the premise that there are cable cars in Switzerland, John competently deduces the same disjunctive conclusion, and thereby comes to know it a posteriori by the usual standards, for although the disjunction itself is a priori, his knowledge of the conclusion inherits the dependence on experience of his knowledge of the premise. Thus John and Mary know the same disjunctive truth, but Mary knows it a priori while John knows it a posteriori. Since the disjunction that either 289 + 365 = 654 or there are cable cars in Switzerland inherits necessity from its first disjunct, John knows a necessary truth a posteriori.

The primary distinction between ways of knowing can be used to effect a secondary classification of things known. A truth p is a priori if and only if p can be known a priori. If we stipulate analogously that p is a posteriori if and only if p can be known a posteriori, then a truth may be both a priori and a posteriori, as with the disjunction that either 289 + 365 = 654 or there are cable cars in Switzerland. Alternatively, if we stipulate that a truth is a posteriori if and only if it is not a priori, then a truth that cannot be known a posteriori counts as a posteriori if it also cannot be known a priori. There is no presumption that a truth can be known at all. Perhaps the best fit to current practice with the term is to stipulate that a truth is a posteriori if and only if it can be known a posteriori but cannot be known a priori. Even this way of drawing the distinction is subject to Kripke’s forceful case (1980) that there are both contingent a priori truths and necessary a posteriori ones. Although some philosophers reject Kripke’s arguments, even they now usually accept the burden of proof to show why the epistemological distinction between the a priori and the a posteriori should coincide with the metaphysical distinction between the necessary and the contingent.

To go further, we must clarify the terms ‘independent’ and ‘experience’. One issue is that even paradigms of a priori knowledge depend in a sense on experience. For example, we supposedly know a priori that if it is sunny then it is sunny. But if our community had no direct or indirect experience of the sun or sunny weather, how could we understand what it is to be sunny, as we must if we are so much as to entertain the thought that it is sunny, let alone know that it is so?

The standard response is to distinguish between two roles that experience plays in cognition, one evidential, the other enabling. Experience is held to play an evidential role in our perceptual knowledge that it is sunny, but a merely enabling role in our knowledge that if it is sunny then it is sunny: we needed it only to acquire the concept sunny in the first place, not once we had done so to determine whether it currently applies. Experience provides our evidence that it is sunny, but not our evidence that if it is sunny then it is sunny; it merely enables us to raise the question. The idea is that an a priori way of knowing may depend on experience in its enabling role but must not depend on experience in its evidential role.

Another issue is how widely to apply the term ‘experience’. It is mainly associated with ‘outer’ experience, involving perception by the usual five senses, but why should it exclude ‘inner’ experience, involving introspection or reflection? After all, one’s knowledge that one is in pain is presumably a posteriori, even though the experience on which it depends is inner. Excluding inner experience by mere stipulation, without reference to any deeper epistemological difference, is liable to make the distinction between a priori and a posteriori knowledge epistemologically superficial. Inner and outer experience will therefore provisionally be treated on an equal footing.

One might worry that if inner experience is included, our experience of reflecting on the proposition that if it is sunny then it is sunny will play an evidential role in our knowledge that if it is sunny then it is sunny, and that Mary’s experience of calculating that 289 + 365 = 654, on paper or in her head, will play an evidential role in her knowledge that 289 + 365 = 654. Presumably, the response is that the role is purely enabling. The relevant evidence is not the psychological process of reflecting or calculating, but rather in some sense the non-psychological logical or mathematical facts to which that process enables one to have access.

On further thought, however, that response causes more problems than it solves. For what prevents it from generalizing to outer experience? For example, part of the evidence that a massive comet or asteroid collided with the Earth about 250 million years ago is said to be that certain sediment samples from China and Japan contain certain clusters of carbon atoms. That those samples contained those clusters of atoms is a non-psychological fact. Of course, in some sense scientists’ outer experience played a role in their access to the fact. But, by analogy with the logical and mathematical cases, the relevant evidence is not the psychological process of undergoing those outer experiences, but rather the non-psychological physical facts to which that process enables us to have access. The role of the outer experience is purely enabling, not evidential. If so, what would usually be regarded as paradigm cases of a posteriori knowledge risk reclassification as a priori.

The threat is not confined to theoretical knowledge in the natural sciences. Even for everyday observational knowledge, it is a highly controversial move to put the psychological process of undergoing the outer experience into the content of the perceptual evidence we thereby gain. What we observe is typically a non-psychological fact about our external environment, not a psychological fact about ourselves.

One obstacle to resolving the problem is the unclarity of the terms ‘experience’ and ‘evidence’ as many philosophers use them. The top-down way of introducing the distinction between the a priori and the a posteriori promised to put it on a firm theoretical footing, but in practice relies on other terms (such as ‘experience’ and ‘evidence’) understood at least partly bottom-up, through examples and prototypes. Although bottom-up understanding often serves us well enough, in the present case it leaves us puzzled all too soon.

Of course, many attempts have been made to explicate the a priori – a posteriori distinction by introducing new theoretical apparatus. My aim here is not to discuss those attempts separately. Instead, I will address the distinction more directly, by comparing what would usually be regarded as a clear case of a priori knowledge with what would usually be regarded as a clear case of a posteriori knowledge. I will argue that the epistemological differences between the two cases are more superficial than they first appear. The conclusion is not that the cases are borderline. I do not deny that they really are clear cases of a priori and a posteriori knowledge respectively, at least by bottom-up standards. In any case, showing that a distinction has borderline cases does not show that it is unhelpful for theoretical purposes. Rather, the appropriate conclusion is that the a priori – a posteriori distinction does not cut at the epistemological joints.

An analogy may be helpful with an argument of the same kind about a political distinction. If one aims to criticize the distinction between liberal and non-liberal policies, one achieves little by producing examples of policies that are neither clearly liberal nor clearly non-liberal. Every useful political distinction has borderline cases. But if one can produce an example of a clearly liberal policy that is politically only superficially different from a clearly non-liberal policy, then one has gone some way towards showing that the liberal – non-liberal distinction does not cut at the political joints.3


3. Here are two truths:
(1) All crimson things are red.
(2) All recent volumes of Who’s Who are red.
On the standard view, normal cases of knowledge of (1) are clearly a priori, because by definition crimson is just a specific type of red, whereas normal cases of knowledge of (2) are clearly a posteriori, because it takes direct or indirect experience of recent volumes of the British work of reference Who’s Who to determine their colour (that is, the predominant colour of their official cover). But let us describe two cases in more detail.

Suppose that Norman acquires the words ‘crimson’ and ‘red’ independently of each other, by ostensive means. He learns ‘crimson’ by being shown samples to which it applies and samples to which it does not apply, and told which are which. He learns ‘red’ in a parallel but causally independent way. He is not taught any rule like (1), connecting ‘crimson’ and ‘red’. Through practice and feedback, he becomes very skilful in judging by eye whether something is crimson, and whether something is red. Now Norman is asked whether (1) holds. He has not previously considered any such question. Nevertheless, he can quite easily come to know (1), without looking at any crimson things to check whether they are red, or even remembering any crimson things to check whether they were red, or making any other new exercise of perception or memory of particular coloured things. Rather, he assents to (1) after brief reflection on the colours crimson and red, along something like the following lines. First, Norman uses his skill in making visual judgments with ‘crimson’ to visually imagine a sample of crimson. Then he uses his skill in making visual judgments with ‘red’ to judge, within the imaginative supposition, ‘It is red’. This involves a general human capacity to transpose ‘online’ cognitive skills originally developed in perception into corresponding ‘offline’ cognitive skills subsequently applied in imagination. That capacity is essential to much of our thinking, for instance when we reflectively assess conditionals in making contingency plans.4 No episodic memories of prior experiences, for example of crimson things, play any role. As a result of the process, Norman accepts (1). Since his performance was sufficiently skilful, background conditions were normal, and so on, he thereby comes to know (1).

Naturally, that broad-brush description neglects many issues. For instance, what prevents Norman from imagining a peripheral shade of crimson? If one shade of crimson is red, it does not follow that all are. The relevant cognitive skills must be taken to include sensitivity to such matters. If normal speakers associate colour terms with central prototypes, as many psychologists believe, their use in the imaginative exercise may enhance its reliability. The proximity in colour space of prototypical crimson to prototypical red is one indicator, but does not suffice by itself, since it does not discriminate between ‘All crimson things are red’ (true) and ‘All red things are crimson’ (false). Various cognitive mechanisms can be postulated to do the job. We need not fill in the details, since for present purposes what matters is the overall picture. So far, we may accept it as a sketch of the cognitive processes underlying Norman’s a priori knowledge of (1).

Now compare the case of (2). Norman is as already described. He learns the complex phrase ‘recent volumes of Who’s Who’ by learning ‘recent’, ‘volume’, ‘Who’s Who’ and so on. He is not taught any rule like (2), connecting ‘recent volume of Who’s Who’ and ‘red’. Through practice and feedback, he becomes very skilful in judging by eye whether something is a recent volume of Who’s Who (by reading the title), and whether something is red. Now Norman is asked whether (2) holds. He has not previously considered any such question. Nevertheless, he can quite easily come to know (2), without looking at any recent volumes of Who’s Who to check whether they are red, or even remembering any recent volumes of Who’s Who to check whether they were red, or any other new exercise of perception or memory. Rather, he assents to (2) after brief reflection along something like the following lines. First, Norman uses his skill in making visual judgments with ‘recent volume of Who’s Who’ to visually imagine a recent volume of Who’s Who. Then he uses his skill in making visual judgments with ‘red’ to judge, within the imaginative supposition, ‘It is red’. This involves the same general human capacity as before to transpose ‘online’ cognitive skills originally developed in perception into corresponding ‘offline’ cognitive skills subsequently applied in imagination. No episodic memories of prior experiences, for example of recent volumes of Who’s Who, play any role. As a result of the process, Norman accepts (2). Since his performance was sufficiently skilful, background conditions were normal, and so on, he thereby comes to know (2).

As before, the broad-brush description neglects many issues. For instance, what prevents Norman from imagining an untypical recent volume of Who’s Who? If one recent volume of Who’s Who is red, it does not follow that all are. The relevant cognitive skills must be taken to include sensitivity to such matters. As before, Norman must use his visual recognitional capacities offline in ways that respect untypical as well as typical cases. We may accept that as a sketch of the cognitive processes underlying Norman’s a posteriori knowledge of (2).

The problem is obvious. As characterized above, the cognitive processes underlying Norman’s clearly a priori knowledge of (1) and his clearly a posteriori knowledge of (2) are almost exactly similar. If so, how can there be a deep epistemological difference between them? But if there is none, then the a priori – a posteriori distinction is epistemologically shallow.

One response is to argue that at least one of the cases has been mislocated in relation to the a priori – a posteriori boundary. Perhaps Norman’s knowledge of (1) is really a posteriori, or his knowledge of (2) is really a priori (although presumably we did not make both mistakes). The risks of such a strategy are also obvious. If we reclassify Norman as knowing (1) a posteriori, we may have to do the same for all or most supposed cases of a priori knowledge, perhaps even of basic principles in logic and mathematics (such as standard axioms of set theory). For Norman’s knowledge of (1) did not initially seem atypical as a supposed case of a priori knowledge. On the other hand, if we reclassify Norman as knowing (2) a priori, we may still lose the distinction between a priori disciplines such as logic and mathematics and a posteriori disciplines such as physics and geography. Either way, we end up with an a priori – a posteriori distinction that cannot do much theoretical work.

Another response to the descriptions of how Norman knows (1) and (2) is more sceptical: it may be suggested that if the cognitive processes are really as described, then they are too unreliable to constitute genuine knowledge at all. However, this option is also unpromising for friends of the a priori – a posteriori distinction, for at least two reasons. First, it imposes an idealized epistemological standard for knowledge that human cognition cannot be expected to meet. None of our cognitive faculties is even close to being globally infallible. More local forms of reliability may suffice for knowledge; sceptics have not shown otherwise. Second, even if we are sceptical about knowledge in such cases, we should still assign belief in (1) and (2) some other sort of positive epistemic status, such as reasonableness, to which the a priori – a posteriori distinction should still apply in some form. Norman’s a priori reasonable belief in (1) and his a posteriori reasonable belief in (2) could still be used in a similar way to argue against the depth of the new distinction. For purposes of argument, we may as well accept that Norman knows (1) and (2).

In the terms used in section 2, the question is whether Norman’s experience plays an evidential or a merely enabling role in his knowledge of (1) and (2). Even in the case of (1), the role seems more than purely enabling. Consider Norbert, an otherwise competent native speaker of English who acquired the words ‘crimson’ and ‘red’ as colour terms in a fairly ordinary way, but has not had very much practice with feedback at classifying visually presented samples as ‘crimson’ or ‘not crimson’. He usually makes the right calls when applying ‘crimson’ as well as ‘red’ online. By normal standards he is linguistically competent with both words. He grasps proposition (1). However, his inexperience with ‘crimson’ makes him less skilful than Norman in imagining a crimson sample. As a result, Norbert’s reflection on whether crimson things are red comes to no definite conclusion, and he fails to know (1). Thus Norman’s past experience did more than enable him to grasp proposition (1). It honed and calibrated his skills in applying the terms ‘crimson’ and ‘red’ to the point where he could carry out the imaginative exercise successfully. If Norman’s experience plays a more than purely enabling role in his knowledge of (1), a fortiori it also plays a more than purely enabling role in his knowledge of (2).5

If the role of Norman’s experience in his knowledge of (1) is more than purely enabling, is it strictly evidential? One interpretation of the example is that, although Norman’s knowledge of (1) does not depend on episodic memory, and he may even lack all episodic memory of any relevant particular colour experiences, he nevertheless retains from such experiences generic factual memories of what crimson things look like and of what red things look like, on which his knowledge of (1) depends. By contrast, Norbert fails to know (1) because his generic memories of what crimson things look like and of what red things look like are insufficiently clear. On this interpretation, Norman’s colour experience plays an evidential role in his knowledge of (1), thereby making that knowledge a posteriori. But we have already seen that such reclassification is a risky strategy for defenders of the a priori – a posteriori distinction. Instead, it may be proposed, although colour experience can play an evidential role in a posteriori knowledge of what crimson things look like, and so indirectly in a posteriori knowledge of (1), we need not develop the example that way. The only residue of Norman’s colour experience active in his knowledge of (1) may be his skill in recognizing and imagining colours.6 Such a role for experience, it may be held, is less than strictly evidential. Let us provisionally interpret the example the latter way. In section 5 we will reconsider, but reject, the idea that even supposed paradigms of a priori knowledge are really a posteriori.

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