Is knowledge closed under known implication?

 

The "closure principle" or the "principle of known entailment" (or known implication) posits that -

If (1) S knows that P

and (2) S knows that P entails (implies) Q

then (3) S knows that Q.

It is easy to see the intuitive appeal of this principle.  If I know that this tomato is red, and I know that a red tomato implies that it is ripe, then I know that this tomato is ripe.  Logical implication is a form of deductive argument.  And as such, given true premises it guarantees that the conclusion is also true.  So -

If (1) if it is true that P

and (2) it is true that P entails (implies) Q

then (3) it is guaranteed to be true that Q.

This seems to be a very good justification for believing that Q.  We use reasoning of this sort all the time.  Thus it is reasonable to assume that for a "Justified True Belief" model of knowledge, the principle of known implication is simply documenting the fact that logical entailment, and of deductive arguments generally, is a sufficient reason for justifying the conclusion.  This is perhaps easier to see if we focus in on the justification aspect of knowing.  A more interesting variation of the title question (for reasons you will see in a moment) is whether justified belief is closed under known implication.  In this form, the "closure principle" posits that -

If (1) S has a justified belief that P

and (2) S has a justified belief that P entails (implies) Q

then (3) S is justified in believing that Q.

Or alternatively, drawing upon the JTB notion of knowledge one can, without losing the sense of the above, mix the two forms as -

If (1) S knows that P

and (2) S knows that P entails (implies) Q

then (3) S is justified in believing that Q.

And it really seems quite counter intuitive to not believe something that one is fully justified in believing.  When phrased in terms of justified beliefs, it would initially seem as if all of the JTB theories of knowledge would have to agree that justified belief is indeed closed under known entailment.  Foundationalism, Coherentism, and Contextualism all agree that deductive implication is sufficient justification to qualify "a belief that Q" as knowledge.  For the externalist alternatives to the JTB family of theories, the situation is a little more complex.  For each particular theory it would depend on whether or not logical entailment is encompassed within the details of whatever reliable, causative, or law-like connection might be posited.  Nozick's "truth tracking" or "subjunctive conditional" theory of knowledge is, however, an example of an externalist theory of knowledge that would specifically deny that justified belief is closed under known implication -- mostly because Nozick's theory does not treat knowledge as a question of a belief justified by other beliefs.

The closure principle has gained its significance because of its employment in the arguments of Cartesian Scepticism.  The most famous example is the "Brain in a Vat" argument.  This argument proceeds as -

                      (a)   If   (I know that I have two hands), and
                                              (I know that knowing that I have two hands implies that I am not a BIV),
                                              then   -   (I know that I am not a BIV).

                      (b)   Things may not really be as they appear.

                      (c)   Despite appearances, I may really be a BIV.  [From (b)]

                      (d)   I cannot prove (I do not know) that I am not a BIV.  [From (c)]

                      Therefore (e) I do not know that I have two hands.  [From (a) and (d)]

In order to reason from the very reasonable sounding premise (d) that "I cannot prove that I am not a BIV" to the problematic conclusion (e) that "I do not know that I have two hands", the sceptic is drawing upon premise (a) that knowledge is closed under known implication.  The sceptic is drawing upon a modus tollens deductive argument that proceeds from the premise "I know that P" (I know that I have two hands) through the closure principle (if I know that P, and I know that P implies Q, then I know that Q), to the interim conclusion "I know that Q" (I know that I am not a BIV), and then denying this interim conclusion (I do not know that I am not a BIV) to reach the ultimate objective of denying P (I do not know that I have two hands).

Unfortunately for the Cartesian sceptic, and fortunately for we who have two hands, the closure principle does not in fact hold for most theories of knowledge.  So the sceptic's argument fails at step(a).  And if framed instead in terms of justified belief, then it is step (d) that trips up the sceptic.  In terms of justified belief, step (d) would have to be phrased as "I do not have a justified belief that I am not a BIV".  But of course, it is now obvious that I do indeed have a very well justified belief that I am not a BIV.  My belief might in fact be false (if the sceptic's hypothesis is true and I am indeed a BIV).  But I none-the-less have a justified belief that I am not a BIV.

Just why this closure principle does not hold will depend, or course, on just how one conceives "knowledge".  Under the traditional model of knowledge as a "justified true belief", one's knowing P is based on one's justification for P.  And that justification is presumably a collection of evidence and associated reasons for believing that P is the case. 

S knows that P iff   (1) P is true, and

                               (2) S believes that P, and

                               (3) S is justified in believing that P.

It is obvious that within the closure principle, conditions (1) and (3) are satisfied.  It is stipulated that P is true.  And the deductive reasoning of logical implication therefore ensures that Q is true.  But it is condition (2) in the JTB concept of knowledge that trips matters up.  Knowledge as a justified true belief is not closed under known implication because belief is not necessarily closed under known implication.  Assuredly it often is closed as a matter of contingent fact.  We do often believe what we have adequate justification to believe.  But it is also entirely possible that I might know that P, and know that P implies Q and yet refuse (for whatever reason) to believe that Q.  Perhaps I also believe (falsely) that Q is inconsistent with some other (also possibly false) beliefs that I hold.  Or perhaps I have just been cognitively lazy, and while knowing when I think about it that P entails Q, have not taken the trouble to acknowledge, and hence believe, that Q.  Advising me that I have very good justification for believing that Q does not necessitate that I will believe that Q.  Belief as a mental state does not follow logical rules out of necessity.

Most theories of knowledge, because they rely on the condition that S believe that Q before S can know that Q, would therefore fall into that group of theories for which knowledge is not closed under known entailment.  The group would include Foundationalism, Coherentism, Contextualism, Reliablism, the Causal and Law-like connection theories, and Nozick's "truth-tracking" or subjunctive-conditional theory to mention just the more readily recognizable. 

But there are also a few theories that do not incorporate this second condition on knowledge that S believe that Q.  The performative theory, as one example, would however also deny that knowledge is closed under known implication because it maintains that "to know" is performance verb that has nothing to do with beliefs or justification.  And while most deontological theories fall into the JTB family of theories (and hence deny the closure principle for knowledge) it is possible to frame a deontological theory in terms of S having a right or duty to believe that Q (when properly justified), rather than in terms of S actually believing that Q.  And for this variation, knowledge would indeed be closed under known implication. 

By dropping the requirement that S believes that Q, similar variations can be created out of some of the other theories as well.  The "diffident scholar" example has been proposed to argue that knowledge might not necessarily involve belief.  The diffident scholar studies some subject thoroughly, and answers questions on that subject correctly -- say that the mass of the tau neutrino is 18.2 MeV.  However, because of nerves or pressure, he does not believe that he knows the answer and thinks that he is only guessing.  He does not believe that the mass of the tau neutrino is 18.2 MeV.  He does believe he is only guessing when he picks that answer of the multiple choice list.  It is argued from this example that there are adequate grounds to claim he does know despite the missing belief.  If this reasoning is accepted, then it might be argued that a proper theory of knowledge would admit that knowledge is closed under known implication because logical inference would guarantee that justification is closed.

This consideration brings us to the reason proposed by Fred Dretske for denying that knowledge is closed under known implication.  Dretske's argument is that accepting Q could nullify all the justification one has for knowing that P.  And hence one could not accept the extension of knowledge from P to Q, even if one accepted that P implied Q. 

The example that Dretske provides is that of the zebras at the city zoo.  If one visits the city zoo and sees what looks to all appearances as zebras in a pen, along with a posted sign explaining that these are zebras in the pen, and one knows of no reasons for doubting this information, then one is justified in believing that there are zebras in that pen.  But there being zebras in the pen logically implies that what you are seeing are not mules cleverly disguised as zebras.  Dretske argues that you are not justified in believing that what you are seeing in the pen is not mules cleverly disguised as zebras because you are no longer justified in believing they are zebras.

The problem with Dretske's reasoning as a basis of denying the closure principle, is that it simply doesn't work at the level he intends.  Dretske's reasoning is based on the traditional justified-true-belief model of knowledge.  But the logical extension through the implication to Q only works if P is true.  If S knows that P, then by definition, P is true.  In other words, if I know that I am seeing zebras, then it is true that I am seeing zebras.  (If it is not true that I am seeing zebras, I cannot by definition know that I am seeing zebras.)   Then if I know that P (I am seeing zebras) implies Q (they are not cleverly disguised mules), I am completely justified in believing that they are not cleverly disguised mules.  Dretske's argument that the possibility that they might be cleverly disguised mules is certainly a sceptical challenge to my knowing that P.  But it fails as a demonstration that knowledge is not closed under known implication.

If looked at in terms of justified belief, on the other hand, Dretske's argument makes more sense.  Having a justified belief that I am seeing zebras does not entail that it is true I am seeing zebras.  Therefore, it is now readily apparent that my consideration of the alternative that they are disguised mules might nullify all the justification I have for my belief that they are zebras.  In other words, Dretske's reasoning, while not demonstrating that knowledge is not closed under known implication, does demonstrate that even justified belief is not necessarily closed under known implication.

The conclusion is that under most theories of knowledge, neither knowledge nor justified belief is closed under known implication.  The only exceptions to this conclusion would be some of the more obscure versions of reliablist, causal, and law-like theories, and the non-JTB variation of a deontological theory I outlined above (because they do not incorporate the condition that S believe that Q). 

Interestingly, we loose nothing by denying the closure principle.  We are, after all, not maintaining that logical inference is never an adequate justification for believing that Q.  We are only denying the argument that logical inference is always a completely sufficient reason for believing that Q.  The principle of known implication is a good prima facie reason for believing that Q -- ceteris paribus.  But there are necessarily other considerations that might come into play to govern whether S actually does believe that Q.  So we can suggest that knowledge is usually closed under most practical circumstances.  It is a useful rule of thumb.  But we cannot say that knowledge is closed -- period.  It is not a necessary consequence of what knowledge is.

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