سکوت

DEDUCTIVE CLOSURE, SCEPTICISM AND THE PARADOXES  OF  CO NFIR MAT IO N1

Hamid Vahid

Abstract

To undermine much of what we ordinarily claim to know, sceptics  have often appealed to a principle  (known as the principle of closure) according to which knowledge (justification) is closed under  known  entailment.  In this paper after expounding the views of Stein, Klein and others, I shall argue that they all fail to take note of different contexts in which  the principle of closure is applied. The relevance of the principle of closure for scepticism is then analyzed in the light of, what I call, the 'infectious' character of epistemic contexts. I shall also highlight the similarities in the behaviour of the concepts of justification and confirmation and appeal to certain solutions to the paradoxes of confirmation to provide a comprehensive account of the different  instances of the principle of closure.

Much recent work on scepticism has been concerned with a principle according to which knowledge (justification) is closed under known entailment. That is to say if one knows  that  p and that p entails q then one knows that q. Suppose, for example, you claim to know that you are reading this  paper. You  presumably also know that if you are reading this paper then you are  not a brain in a vat. But, the sceptic argues, you do not know that you are not a brain in a vat and thus, by modus tollens, you do not know that you are reading this paper. In this way the sceptic is able to undermine  much of what we ordinarily claim to know.

The issues surrounding the principle of closure are not, however, so straightforward. Because of the contextual character of the concepts involved, there is a great deal of ambiguity regarding the context where the principle is applied. This contextual ambiguity, as will be shown, gives rise to different instances of the principle each requiring a distinct treatment. The failure to distinguish this variety has led to serious misunderstandings regarding both the nature and function of the principle of closure. Another point that will  be stressed  concerns  the similarities in the  behaviour of  the

1 I am indebted to Professors William Alston  and Susan  Haack  for  comments  on  an earlier draft of this paper. I have also greatly benefited from from discussion with Dr Gary Legenhausen.

concepts of justification and confirmation.2 Ther are both context­ dependent and both give rise to similar puzzles. Indeed, as I shall explain, the solution to certain paradoxes of confirmation can be exploited to clarify and sharpen our understanding of the principle of closure. I begin, however, with Dretske's presentation of the problem  of deductive closure.

Discussing this question in an early article Dretske posed the problem  in  terms of penetratability of certain  operators.4  Some of these operators such as 'it is true that' and 'it is necessary that' penetrate to every necessary consequence of the propositions upon which they operate; if it is true that p and p entails q then  it is true that q. At the other end of the spectrum lies the  class  of nonpenetrating operators such as 'it is a mistake that' or 'it is accidental that'. They fail to penetrate to some of  the  most elementary logical consequences  of a  proposition.  It may have been a mistake that Bill and Susan married each other but not a mistake that Bill married. Dretske also identifies a class of semi-penetrating operators which falls between these two extremes. These include epistemic operators such as  'knows  that',  'has  reason to  believe that', etc. One of the examples that Dretske gives is the following. Suppose you take your son to the zoo where you  see several striped animals. When asked about the identity  of the animals you tell him that they are zebras. You know  this  because  you  know what zebras look like and the animals are kept in a cage marked "zebras". Being a zebra, however, implies that the animal is not a disguised mule. According to the principle of closure by knowing these facts you also know that the animals are not disguised mules. Dretske denies that such a  consequence  follows  and, thus,  rejects the closure principle. He argues that your adequate  evidence  for 'That is a zebra' ceases to be adequate for 'That is not a mule with stripes painted on it'. That evidence has  been effectively neutralized. It no longer supports the  claim  that  these  are  not mules disguised to look  like zebras. The  evidence did  not  involve checking  with the

zoo authorities  or examining  the animals closely enough.

Dretske also offers an  explanation  for  the failure  of the  principle of closure. Knowing that this animal is a zebra, he claims, does not necessarily involve knowing that it is not a disguised mule. This is because knowing that x is A always involves knowing it within a framework of relevant alternatives B, C, D. So that by changing the set of contrasts one also changes what a person is  said  to  know. Epistemic operators, thus, penetrate only to those  contrast consequences which belong to the set of relevant alternatives. This explains why we can claim to know that this  animal  is  a  zebra without knowing that it is  not  a disguised  mule. The  possibility  of the animal being a disguised mule is not relevant in the (ordinary) context of the example and, thus, need  not  be ruled  out  in order for us  to know.

I shall now turn to two different sorts of responses to the above claim. Gail Stine has  offered  another  account  of this  example,  on the relevant alternative view, in which one need not give up the principle of closure.5 On the relevant alternative view to say that S knows that p is  to  presuppose  that  not-p is  a  relevant  alternative. On Stine's account, however, this is viewed  as  a  pragmatic  rather than  a  semanticf.resupposition   thus  having  the  Gricean   property of cancellability. It is therefore possible for 'S knows  that  p'  to be true while 'not-p is a relevant  alternative'  is  false.  So  although  it may sound misleading to claim to know that the animal is not a disguised mule but this does not  mean  that  the claim  is false. One can cancel, says Stine, the implication that one has investigated the questions raised by the  consequence.  Stine  offers  no  direct argument in support of her  account.  She only  suggests  that  we  do not need evidence to know a proposition and  that we do not even  have evidence for the claim that the animal  is a zebra.  As she does  not elaborate on these claims it is very difficult to  evaluate  her defence of the principle of closure. I thus turn a different style of defence  proposed  by  Peter Klein.7

Klein thinks that the hostility towards the principle of closure is the result of confusing it with a different but false epistemic principle according to which justification is transitive i.e., if the evidence ejustifies p for Sand p entails q, then ejustifies q for S. But,

Klein claims, justification is not transitive and the principle of closure does not depend on that, for the justification of q is already secured via p itself. If e justifies p for S, then p, on its own, would provide adequate justification for anything that is entailed by it. So in the zebra example, ifl  am justified in believing that the animal is a zebra, I would, a fortiori, be justified in believing anything entailed by this including the consequence that the animal is not a disguised mule.

As the preceding remarks show there are conflicting intuitions regarding the principle of deductive closure. On the one hand there is Dretske's claim that certain consequences of a proposition raise possibilities that affect the epistemic status of the evidence, thus undermining our claims to know those consequences. On the other hand there is the intuition expressed by Stine and Klein in their claim that entailments are justification (knowledge) preserving - though each has a different style of explaining away the contrary intuition. The whole issue has, however, been obscured by the fact that the parties involved have all failed to take note of different contexts in which the principle of closure may apply. They only consider the application of the principle across contexts i.e., where the initial context is allowed to change as a result of taking new information into account. There is, however, another instance of  the principle of closure when it is considered relative to a fixed context where no additional information besides the evidence is admitted. Once the context is so fixed then, relative to this context, the principle of closure may hold. This seems to be what Lehrer has in mind  in  the following remarks.

The closure principle is intended to concern what one knows at a single time, however, and then it seems correct. The principle says that if one knows that p and that if p then q all at once, then one knows that q as well.8

How are we to account for these conflicting intuitions? Is there a uniform account  that  could  explain  them  all?  Before  presenting such an account it will be useful to consider a recent analysis of the principle of closure developed by James White.9 He  sides  with Dretske on this issue and rejects the principle of  closure.  His intention, however, is to undermine  Klein's defence of the  principle of closure.  I  believe his analysis eventually fails to bring out  what is

really at issue but it is instructive to see why it does. To set the stage for discussion however, it should be noted that I shall only be concerned with the application of the principle of closure in justification contexts since this is what is at issue between  White and Klein. Any conclusion we reach, of course, also will be true of the application of the principle to those conceptions of knowledge which include a justification clause. So our intended principle of closure states that if the evidence e justifies S in believing that p and that  p entails q,  then  e justifies S in  believing that q.

As we just saw, Klein claimed that when the evidence justifies p for S then p is justified simpliciter, thus, providing adequate justification for q when q is entailed  by p. White's main contention is that Klein has missed the 'context-relativity' of  epistemic concepts. Knowledge claims, he argues, can only make sense against the background of the cognizer's credentials for making those claims in response to the existing questions and doubts. That is to say whether something is evidence for something is not just a matter oflogical relations between believed propositions but is also a function of the questions and concerns existing in  that context. He echoes Dretske's claim that changing the context,  by adding  new information, may neutralize the evidence. So when, in the zebra example, you are asked about the identity of black and white striped animals in a cage marked 'zebra', you are well qualified to answer that they are zebras. But, argues White, when a stranger, having heard a rumour that the zoo is covering up its lack of zebras by using disguised mules, asks you whether the animals are zebras or diguised mules the evidence you had earlier relied on ceases to have any weight. What constitutes evidence for an animal's being a zebra depends on the kind of questions and concerns people have about the animal. By taking a different set of questions  and concerns into account we may need different evidence to justify our claims.

To have reason for thinking that the animals are zebras is not thereby to have reasons for thinking that they are not disguised mules, because the latter possibility may raise  questions  which were not considered when arriving at the belief that the animals were zebras. Given this fact, argues White, the belief that the animals are zebras fails to justify believing they are not disguised mules and Klein's argument is thus proven ineffective. White concludes that the principle of closure is false. He also voices the common wisdom that this neutralizes one important sceptical argument. With  the failure of the  principle of closure the sceptic is

no longer able to undermine much of what we ordinarily claim to know. Even though we cannot rule out the sceptical possibilities, he says, we can still claim to know much of what we ordinarily regard ourselves as knowing because those possibilities are not relevant to the ordinary situations in which  those  claims  are made.

I agree with much of what White  says  about  the  context­ relativity of epistemic concepts, but he fails, I think,  to carry  this point  to its logical  conclusion,  and  in particular,  he fails to provide a comprehensive  account  of the different  instances  of the  principle of closure. In what follows I shall elaborate on each of these claims beginning with the context-relativity of the concept of epistemic justifica tion.1 0

Justification is a matter of degree. The degree of justification of one's belief is dependent on the position of the cognizer within the epistemic web of evidence and on the amount and strength of that evidence. In short, justification, unlike truth, is perspectival i.e., it is measured relative to subject's evidence. This means that a proposition that is justified for one person can be unjustified for another. Because of the context-dependent character of epistemic justification it is always possible that the cognizer's belief is not adequately grounded so that further evidence may show his initial evidence was misleading.

To see the point consider a case in which, although the subject's evidence in itself provides an adequate  basis for  his  belief that  p, that would not have been the case had he been conducting himself properly. It is quite possible that if he had  looked  into  the  matter more carefully, he would be in possession of overriders for his evidence for p. In that case we will not want to count the subject's belief as justified, even though the grounds which gave rise to  the belief were themselves adequate. Their adequacy is undermined  by the larger  pcrspcctival  context  in  which  they arc  considered. 11

Evidence, however, may provide ground for a belief only against the background of one's other beliefs. This means that by changing the background beliefs the evidence may cease to support the belief in question.  Not only does  the  context  of the  background beliefs

determine the adequacy of the evidence but it also  determines which evidence is to be regarded as misleading in the sense of inclining us to form a false belief. That is to say, whether evidence is misleading can only be determined relative to a set  of background beliefs. We might come to form a belief as a result of some evidence but the obtaining of a new evidence might oblige us to revise our judgement and make new adjustments. In the light of this new evidence the initial evidence might be judged as misleading. This process may be iterated indefinitely. What this shows is that both the adequacy and epistemic relevance of the evidence is measured relative to the doxastic situation of the cognizer.

Let me illustrate these points by considering an extended version of Dretske's zebra story. ( l) Smith takes his son to the zoo. They come across a cage marked 'zebras' in which there are a number of black and white striped animals. When asked by his son about the identity of the animals he tells him that they are zebras. (2) One of the zoo keepers, Jones, who is a friend of Smith and from whom he has not heard for some time tells him that the manager is using cleverly  disguised   mules  to  cover  up  the  zoo's  lack  of zebras.

3. Jones's doctor who is around at that time tells Smith that Jones is currently undergoing treatment for mental disorder which manifests itself in the form of a disposition to spread false rumours.
4. An acquaintance of both Smith and Jones informs Smith  that the doctor is actually Joncs's nephew who has been trying  to deprive him of some inheritance on  the ground  of mental illness. (5) ...

Let us go through the story step by step to sec how  the  change  in the context of the story affects  the  epistemic status of the  evidence. At stage (1) of the story, the evidence supports, but fails to prove beyond any  doubt,  the claim  that  the animals  are  zebras.  So Smith is  (defeasibly)  justified  in  believing  that  they  are  zebras.  At stage

(2) the evidence undermines, but does not conclusively rule out, the claim that the animals are zebras. The evidence of stage ( l) is now regarded to have  been  misleading.  At  stage  (3)  the  evidence justifies, but fails to prove beyond any doubt, the claim that  the animals are zebras. The evidence of stage  (2)  is  now  regarded  to have been misleading. At stage (4)  the  evidence  undermines,  but does not conclusively rule  out,  the  claim  that  the  animals  are zebras. The evidence of stage (3) is now regarded to have been misleading  and  so on.

This example  illustrates  that  both  the adequacy  and epistemic

relevance of evidence are measured against the context of the background beliefs of the cognizer. What was justification­ conferring evidence in one context may cease to be adequate in another context and indeed may be regarded  as misleading. This, as we have seen, can only be determined from the perspective of a more comprehensive evidential context. Note that it is irrelevant if the cognizer persists in placing credence in his belief. That does not change the justificatory status of his belief or of the evidence. Suppose at stage (2) of the story Jones falsely (but honestly) tells Smith of the cover up by the zoo's manager and suppose Smith is persuaded that all the appearances are fake  but  nevertheless persists, on the basis of visual evidence, in believing that the animals are zebras. In such a case he is not permitted to hold this belief because this permission is undermined  by his cognitive state at that time. It is sufficient for undermining that a cognizer believes that certain conditions obtain which, if they did  obtain,  would imply that the target beliefs are no longer justified. 12 I shall now consider the consequences of the preceding remarks for the principle of deductive closure and the problem of scepticism respectively.

It is obvious from our  example  that  what  underlies  the  shift  in the epistemic status of the evidence at each stage in  the story  is the fact that we are assuming that one may be justified in believing something even though one's evidence fails to be conclusive. At no stage in our story did the evidence conclusively establish or rule out that the animals are zebras. In each case the possibility of defeat by further evidence was left open. At each stage  the  change  in  the  belief context introduced possibilities which affected the epistemic status of the earlier evidence. If this is correct then the failure of the principle of closure in such circumstances  is  no  more  than  a triviality. For if one and the same proposition  can  change  its epistemic status as the context undergoes change, it would be no wonder if its consequences turned out  to  have  different  status  in those  contexts.

Going back to our story, at stage (1) the evidence based on visual appearances justifies the proposition 'These animals are zebras'. At stage (2) however, the same piece of evidence fails to justify the very same proposition and also its consequence 'These animals are not disguised  mules'. In general if evidence e can provide adequate

justification for a proposition in one context but fails to do so for that very proposition in another context,  then it  may, afortiori, fail to provide justification for the consequences of that proposition. Indeed since p entails p, and p, justified for S in one context, may fail to be justified for him in another context we already have a counter-example to the principle of deductive closure.13 This also explains why Klein's defence is unsuccessful. He claims that when evidence e justifies p and p entails q, we already have a perfectly sufficient reason for q, namely p itself. But if the above remarks are correct and we are considering the principle of closure across contexts then, as we have seen, as the context undergoes change we may no longer be justified in believing p and, consequently, the belief in p would fail to justify  the belief in q.

So as long as we are evaluating  the  principle  of  closure  across contexts (i.e., we allow the context to change) its failure is a trivial consequence of the context-dependence of the concept of epistemic justification. This is not, of course, to claim that any  change  in  the context of the belief will necessarily upset the  transmission  of justification to  its  consequences.  There  may,  indeed,  be  situations where the change in the context is of such a nature as to leave the transmission  undistur  bed.1 4  This is most evident in cases where pis taken to  be  its  own  consequence.  Here,  as  elsewhere,  the  change  in the context need not necessarily involve information which has an undermining  character.   The   subject's   justification   for   p   may actually  be  increased  as  the  result  of  the  added  information.   But  as I am concerned with the possibility of the failure of the principle  of closure,  the  sort  of  situations  that  are  considered  are  those   that involve   cases   where   the   change   has   an   undermining   character i.e.,  it   neutralizes   the   evidence  for  p  and   its consequences.

While rightly emphasising the context-dependent character of epistemic concepts, White does not attack the principle of closure exactly in the way I have done and, indeed, our conclusions regarding the significance of its failure for scepticism (as we shall shortly see) are quite different. He also evaluates the principle of closure only across contexts i.e., in a setting where the context is allowed   to  change  as  a  result  of  taking  new  information  into

account. Referring to the entailments of p and the possibilities they introduce   he says

... bringing up such entailments may introduce a new context with regard to which one may no longer have evidence, a context with  regard  to which  one's  reasons  are  no longer  reasons.1 5

But this is not the end of the matter for the principle of closure. Its (possible) failure across contexts does not imply that it also fails relative to a single context, and White ignores this possibility. We have as yet to evaluate the principle in a setting where the context  is fixed and no additional information besides the evidence is admitted. Recall Lehrer's remark that the principle of closure 'is intended to concern what one knows at a single time'. Before turning to this problem, however, let us consider the implications of the failure of the principle of closure for the sceptical arguments which exploit it.

The sceptic, we recall, argues that if you know (are justified in believing) that you are, for example, reading this paper then you know (are justified in believing)  what it entails, e.g., that you are not a brain in a vat. But since you do not know that you are not a brain in a vat, by modus to/lens, you do not know that you are reading this paper. Now if the principle of closure is false the sceptic's move is automatically blocked. It seems to me, however, that so long as we are allowing the context to change, the failure of the principle of closure cannot stop the sceptic from undermining much of what we ordinarily regard ourselves as knowing. To clarify this point  the following remarks are in  order.

We saw that our argument against the principle of closure exploited the context-dependent character of the concept of epistemic justification. It was argued that if we are evaluating the principle of closure across contexts, then we need to take into account the possibilities that these contexts introduce; and we saw that the epistemic status of evidence might undergo radical change once these possibilities are taken into account. It is true that by rejecting the principle of closure the sceptic can no longer appeal to modus tollens to deny that, for example, you know you are reading this paper despite the fact that you do not know you are not a brain in a vat. But he can still deny that you are reading this paper via the new context  which is introduced  as a result of taking the sceptical

possibilities into account. This is possible because epistemic contexts are, as I call them, 'infectious', in the sense that they affect the epistemic status of the propositions  involved  with them.

Consider, once  again,  the  story  of Smith  and  the  zebras.  At  stage (1) of  the  story  he  is  justified  in  believing  that  the  animals  are zebras. At stage (2) the initial context changes as a result of the new information (about the fraud) that.Jones adds to it.  In  the  resulting context Jones is not justified in believing that the animals  arc  not disguised mules but he is equally unjustified  in  believing  that  the animals  are  zebras.  So  when  the  context   decides   the   epistemic status  of some  consequences  of  p,  the  verdict  may  change  the  status of p  itself.  When  additional  information  is  injected  into  the  context, the  epistemic  status  of  all  the  propositions  are  affected.   A   belief once  justified  in  the  context  may  lose  its  status  when  new information  is  added  to   that   context.   This   also   explains   why Klein's attack on scepticism  is  unsuccessful.  He  thinks  that  the principle  of  closure  can  actually  be  exploited  to  defeat  scepticism. For  if  you  are  justified  in  believing  that  you  are   reading  this  paper ( p), then, by modus ponens, you  are  justified  in  believing that you  are not a brain in a vat (q). But if  what  we  have  said  is  correct  the infectious character of epistemic contexts undermines the subject's justification in believing p. The mere  entertaining  of  sceptical possibilities is enough to create a setting where  the  evidence  is neutralized both for p  and  q.  As  indicated  earlier,  another  context which  shares  the  same  property  with  epistemic  contexts  is   the context of the confirmatin  of  theories.  It  is  well  known  that  adding new information to the background knowledge affects the degree of confirmation of the theories involved. I shall say more on  this  point shortly.

Being infectious is not, however, the same thing as being semi­ penetrating. First of all, 'being infectious' is a predicate of  a context, whereas it is an operator which is semi-penetrating. Operators introduce and operate in contexts. They are not identical with the contexts. Secondly, the direction of penetration is always from propositions to their necessary consequences whereas an infectious context may affect propositions in both, and indeed  in all, directions. Moreover, it is not the case that all semi-penetrating operators introduce infectious contexts. There are semi-penetrating operators whose contexts are not infectious. It  is, I believe, failure to take note of this point that has led philosophers like Dretske to claim that epistemic operators 'share the same logic with such operators as "explains why" '. He exploits this claim  to  undermine

the principle of closure. Let us consider the example which he gives in support  of his claim.

Suppose we want to explain why  Brenda  did  not  order  any dessert. To do this,  Dretske  claims,  we  have  to  take  into account the  range of relevant  alternatives.  He  admits  that  he does  not  have a theory about what  a  relevant  alternative  is,  but  the  case,  he hopes, is intuitive enough. When we explain why  Brenda  did  not order any dessert by saying that  she was full,  we explain  why she  did not order any dessert as opposed to ordering some dessert and eating it. If we consider a different set of alternatives, for example, Brenda ordering some  dessert  and  throwing  it  at  the  waiter,  etc., the above explanation fails to be an  explanation  of why Brenda  did not order any dessert even though  the  fact  that  Brenda  did  not order some dessert entails the fact that Brenda did not order some dessert and throw it  at  the waiter. Dretske concludes  that  to say  we do not know the animals in the zoo are zebras because we do  not  know that they are not disguised mules is 'as much a mistake  as arguing that we have not explained why Brenda did not order any dessert (within the original, normal, setting) because we did  not explain why she did not order some dessert and  throw  it  at  the waiter'. 1 6

But there is a difference. Introducing the competing possibility of Brenda ordering some dessert and throwing it  at  the waiter does not invalidate the original explanation of why Brenda did not order some dessert because explanatory contexts are not infectious.  When we explain why Brenda did not order some dessert by saying that she was full, our contrast class consists of Brenda ordering some dessert and eating it. This explanation, of course, fails when a new contrast class consisting of Brenda ordering some dessert and throwing it at the waiter, etc., is introduced. But that does not diminish its effectiveness when it is considered against the earlier contrast class. This is because explanatory contexts can be, as it were, partitioned into different sets of alternatives with each partition being completely autonomous. The failure of a certain explanation in one partition does not imply its failure in another partition.

But this is not the case with epistemic contexts. Introducing a novel alternative may neutralize a certain bit of evidence, thus affecting the epistemic status of the proposition whose consequence it is. In  the  zebra example,  the visual evidence justified  'This is a

zebra' but failed to support 'This is not a disguised mule' when the possibility of fraud was raised. The raising of this possibility, however, also affected its ability to justify 'This is a zebra'. In other words, the failure of the evidence to support one possibility infected its ability to support another. This is because in epistemic contexts we are concerned with the cognitive state of the cognizer and cognitive states cannot be divided  into  autonomous  partitions. They are of such a nature that a slight change in their belief repertoire may affect the epistemic status of all the propositions involved. They are, in other words, infectious. Dretske's conclusions regarding the analogy with explanatory contexts are thus, un­ warranted.

I turn now to the validity of the principle of closure itself. We have already argued that the principle may fail to hold when it is applied across contexts. We saw that in the context in which the possibility of fraud was raised not only the evidence was effectively neutralized for the proposition that the animals are not disguised mules but it also ceased to support the proposition that the animals are zebras. Nothing, however, has so far been said about the principle of closure when it is considered relative to a single context i.e., when no additional information other than the  evidence  is taken into account, or as Lehrer put it, when the principle is intended to concern what one knows at a single time. So, the question we need to address is whether the  principle  is correct when the context is fixed. Does, for example, the evidence for 'These animals are zebras' lend any support to 'These animals are not disguised mules' when that evidence is the only piece of information that is admitted? There seems to be an intuition in favour of a positive reply. How are we to account for these conflicting intuitions? To answer these questions we need a comprehensive theory which explains those intuitions. To do this I shall propose an analogy with one of the paradoxes of confirmation known also as the paradox of the ravens. We have already noticed the concepts of justification and confirmation  resemble each other to some degree. The analogy can be taken still further by the following observations.

The ravens paradox, in its simple form, has the following structure. Observations of black ravens  confirm  the  proposition 'All ravens are black' while observations of black pens, white swans, etc., are neutral to it. But the proposition 'All non-black things are non-ravens' is equivalent to (and, a.fortiori, entails) 'All ravens are  black'  and  since a  white swan confirms  the  former it

should also confirm the latter which seems paradoxical. To resolve the paradox a number of solutions have been proposed. The difference between these solutions can be traced to the amount of background information that is taken into account. Here I rely on Mackie's account of the paradox.17 I believe a theory along  the lines Mackie proposes to account for the conflicting intuitions in the case of the ravens  paradox  can  also  be exploited to account for the conflicting intuitions in the case of the  principle  of  closure.

One  early  solution  to  the  paradox  was  proposed by  H empel.1 8 Consider the proposition (p)  'All  sodium  salts  burn  yellow'. Suppose we hold a  piece of ice into a colourless  flame and  it  does not turn the flame yellow. This  confirms  the  proposition  (q) 'Whatever does not burn  yellow  contains  no  sodium  salt'  and because the two propositions  are  equivalent  it  also  confirms  p which is paradoxical. But, Hempel argues, the paradox is  only apparent. For if we take an  object  whose chemical  structure  we do not know and hold it into a flame and it fails to burn yellow and subsequent investigation proves  that  it  contains  no  sodium  salt, then this observation would confirm p. The difference between the  two cases is only that in the first one we already knew that the substance is ice and ice contains no sodium salt  whereas  in  the second case we did not know this. So, Hempel concludes, the seemingly paradoxical nature of the first case is only due to our allowing  the  additional  information  that  the  object  is ice.

So as long as we ignore any additional information we can solve the paradox in the way Hempel suggests. In a setting where no reference is made to additional knowledge we can say the observation of a non-black non-raven confirms 'All non-black things are non-ravens' and its equivalent (and, a fortiori, its consequence) 'All ravens are black'. But, as Mackie  points out, given some additional information, for example about the relative size of the classes of ravens and black things, we can solve the paradox in another way. We can still affirm that the observation of white swans confirm 'All ravens are black' but argue that its degree of confirmation is much less than that provided  by the observation of black  ravens.  The  admission  of more information,  as Mackie

shows, might further reduce the degree of confirmation provided by the observation  of black pens and  white swans.

So if we completely ignore the background knowledge we can follow Hempel and say that the observation of black pens and white swans confirms 'All ravens are black'. But Hempel's solution is unrealistic for it requires the theorist to ignore much of what he knows and regard the evidence ('This thing is non-black and non­ raven') as all the information  that is available. This is very odd for it involves a concept of confirmation which is totally at odds with how this concept  is understood  in ordinary contexts.

I think we are faced with more or less the same situation in the case of the principle of closure. Here we are concerned to know whether the same piece of evidence (visual appearances) which justifies p (These animals are zebras) equally justifies q (These are not disguised mules)  when  p entails q. Imitating Mackie's account of the paradox, we may now explain the apparent  plausibility  of this principle when it is considered relative to a context where no information besides the evidence is admitted and its apparent implausibility when the context is allowed to change (in an undermining  wa y).1 9    The   principle  of  closure  seems  to  hold  at  a single time (or in a fixed context) because no reference is made to the background information.Just as the observation of a non-black non-raven confirmed 'All ravens are black' in a context where the evidence was the only information available, similarly the evidence (based on visual appearances) which justified 'These animals arc zebras' equally confers justification on 'These animals are not disguised mules' in a context where no additional information besides the evidence is taken into account. So as long as the principle of closure is considered relative to a context where no additional information besides the evidence is admitted and we ignore all the possibilities and concerns, then it holds. That is to say the evidence also provides adequate ground for asserting, say, 'These animals are not disguised mules'. But once we take into account additional information and background knowledge, the evidence might cease to provide justification for 'These animals are not  disguised   mules'  just   as  the  consideration   of  background

knowledge led to a sharp reduction of the degree of confirmation of the observation of black pens, etc., for 'AH ravens are black'. The principle of closure, thus, fails to hold when the context is allowed to change as a  result  of  taking  sceptical  possibilities  into account.

Although the above account seems to do justice to the conflicting intuitions regarding the principle of deductive closure, there is something odd about the context where the principle holds. Just as speaking of confirmation in a context where the background was completely ignored seemed wholly unrealistic, it is equally un­ realistic to talk of justification in a context where no reference is made to additional information. It is, of course, possible to talk of justification in a context as bare as this where the evidence is all the information available, but this is far removed from the ordinary contexts where there is always some additional information lurking in the background. It would be very odd to require the cognizer to ignore most of what he knows and take some bit of evidence as the only piece of relevant information. One's judgement as to what is relevant is, of course, a matter of his background beliefs, concerns, etc. To limit the range of relevant information only to the piece of evidence is possible but quite unrealistic. In an epistemic situation to judge whether a certain body of evidence provides adequate justification for some proposition, the cognizer cannot pretend that the evidence is the only relevant piece of information available. On the contrary, since  he approaches  the situation  in  the  light of his

background knowledge, judgements as  to what is relevant  cannot be limited to the evidence alone. It is only by assuming that the cognizer's mind is completely blank that the evidence can be regarded  as the only piece of information.

Indeed mere reflection on the consequences of a justified belief p (together with one's background beliefs) may create a setting where the evidence is neutralized for those consequences and even the justification for p is undermined - this is in fact the case with the application of the principle of closure to the sceptical consequences of our ordinary beliefs. In such contexts the mere entertaining of sceptical thoughts (against the background of one's beliefs) affects the range of relevant information. I conclude,  thus, that although the principle of closure may always hold in contexts where the background information is reduced to a bare minimum, this is not what happens in ordinary epistemic situations where the cognizer's beliefs are always formed against the background of some information.  Under such  circumstances  it is always  possible that