p, if one knows that p, then one knows that one knows it. More complex
formulations say that if one knows that p, then one is in a position
to know that one knows it, and this is fleshed out in a variety of
ways. One reason why philosophers are interested in the KK principle
is its relevance to the question of whether epistemic logic is a
branch of modal logic. An important issue in modal logic is whether
necessary truths are necessarily necessary; the corresponding issue in
modal epistemic logic is whether the KK principle holds. Another
reason for interest in the principle is its relevance to the debate
between internalists and externalists about knowledge. It is natural
for internalists to endorse something like the KK principle, and for
externalists to reject it. A third reason for interest in the KK
principle is its connection to the paradox of the Surprise
Examination. The reasoning which generates this paradox seems to
assume that certain kinds of knowledge can be repeatedly iterated, and
hence that something like the KK principle holds. A final reason for
studying the principle is its relevance to recent debates about the
luminosity of mental states (where a mental state is luminous iff,
roughly, one cannot be in that state without being in a position to
know that one is in it). If the KK principle holds, then knowledge is
a luminous mental state; but there are powerful arguments against the
luminosity of other mental states which seem to show that this cannot
be the case.
1. Hintikka on the KK principle.
In his 1951, G.H. von Wright suggested that epistemic logic— the logic
of the term "knows"— is a branch of modal logic— that is to say, the
logic of possibility and necessity. Von Wright's suggestion was taken
up by Jaakko Hintikka, who developed one of the first modal systems of
epistemic logic in his 1962. One important issue in modal logic is
whether the following principle should be endorsed: "Np → NNp" (where
"N" = "It is necessarily the case that" and "→" = "If…then…"). The
corresponding issue in modal epistemic logic is whether the following
principle should be endorsed: "Kp → KKp" (where "K" = "One knows
that"). In chapter 5 of his 1962, Hintikka argues that it should.
Hintikka's arguments for this "KK principle" are hard to follow; but
the gist of them (as clarified in his 1970) seems to be this:
Suppose we say that evidence for a proposition, P, is conclusive
iff it is so strong that, once one discovers it, further inquiry
cannot give one reason to stop believing P. The concept of knowledge
used by many philosophers seems to be a strong one on which one knows
P only if one's evidence for P is conclusive in this sense. It is
plausible that the KK principle holds for this strong concept of
knowledge. For it is plausible that one's evidence for P is conclusive
in the above sense only if it rules out the possibility that one does
not know P, and thus only if it allows one to know that one knows P.
To see this, suppose one has evidence, E, for a proposition P, and
that E does not rule out the possibility that one does not know P. If
E does not rule out this possibility, then, after one has discovered
E, further inquiry can, in principle, reveal to one that one does not
know P. But if further inquiry were to reveal this, then it would
surely give one reason to stop believing P (since one should not
believe things that one does not know). So it is plausible that, if E
does not rule out the possibility that one does not know P, then it is
not conclusive in the sense just defined, and hence plausible that, if
knowledge requires evidence that is conclusive in this sense, the KK
principle holds. (cf. Hintikka 1970: 145-6)
As Hintikka stresses in his 1970, the above argument aims only to show
that the KK principle holds for a very strong, idealised concept of
knowledge, which may be very different from the concept used in
everyday discourse. Because of this, Hintikka can sidestep objections
which say that the principle conflicts with our everyday knowledge
claims. One such objection says that, when the claim is made that
someone knows that p, it cannot usually be claimed that they know that
they know that p, that they know that they know that they know that p,
and so on (cf. Rynin 1967: 29). The fact that one is not prepared to
claim these things may show that the KK principle fails for our
ordinary concept of knowledge, but it does not show that the principle
fails for the strong concept that Hintikka has in mind. Similarly, the
objection that the KK principle prevents knowledge from being ascribed
to animals and young children (who lack the concept of knowledge and
so cannot know that they know) is not problematic for Hintikka. For he
can say that, when knowledge is ascribed to such subjects, the
everyday concept of knowledge is being used rather than his strong
concept.
If the KK principle only holds for a concept of knowledge that is very
different from our everyday concept, then why should one be interested
in it? According to Hintikka, its interest derives from the fact that
(in spite of the differences between our everyday concept and the
strong concept) there are "many philosophers, traditional as well as
contemporary" who use the strong concept of knowledge for which the
principle holds (1970: 148). Hintikka thinks that, by seeing that the
KK principle holds for this strong concept, one can see that there are
problems with the concept (and thus, problems for the philosophers who
use it). He argues for this by appealing to some ideas about the
purpose of philosophical and scientific inquiry that are suggested by
the work of Karl Popper.
According to these Popperian ideas, philosophers and scientists should
always aim to encourage inquiry and discussion; they should never try
to bring it to an end. Because of this, they should not employ a
concept of knowledge which requires conclusive evidence in Hintikka's
sense. For evidence for P which is conclusive in this sense renders
further inquiry into P pointless, and so acts as a "discussion
stopper." And what philosophers and scientists should be aiming for is
evidence that encourages further inquiry and discussion, rather than
evidence that stops it. (Hintikka 1970: 148-9)
Another problem for the strong concept of knowledge which Hintikka
mentions briefly is that the standards that one must meet, in order to
satisfy this concept, seem unrealistically high (1970: 149). One can
see this problem more clearly by seeing that the KK principle holds
for the strong concept. For, as shall be seen in section 3, there is
reason to think that each iteration of one's knowledge requires an
improvement in one's epistemic position. Because of this, the KK
principle can seem to imply, implausibly, that one must be in a
maximally strong epistemic position in order to know.
2. Internalism, Externalism and the KK principle.
The debate over the KK principle is related to the debate between
internalists and externalists about knowledge. The connection between
the two debates can be illustrated by focusing on some examples of
internalist and externalist theories.
A good example of an internalist theory of knowledge is the classical
"justified true belief" or JTB theory that was the target of Edmund
Gettier's 1963 article. According to the JTB theory, knowledge is true
belief that is based on adequate evidence or reasons, where the
adequacy of our evidence or reasons is something that one can
determine by introspection and reflection.
A good example of an externalist theory of knowledge is the
reliabilist theory defended by Goldman (1979) and others on which
knowledge is, roughly, true belief that is produced by a reliable
process. The reliability of the processes that produce our beliefs is
not something that one can determine by introspection and reflection;
it is a matter for empirical investigation.
In general, internalist theories of knowledge say that the property
which distinguishes knowledge from mere true belief (which property,
following Plantinga 1993a, can be called warrant) is internal to our
cognitive perspective. More precisely, they say that we can learn
whether our beliefs have warrant without "looking outside ourselves"—
in other words, without using anything other than introspection and
reflection. Externalist theories say that warrant may be external to
our cognitive perspective, and that empirical investigation may be
needed to ascertain which of our beliefs have it. The reliabilist
theory described is just one example of an externalist theory. Others
include the causal theory of knowledge defended by Goldman (1967) and
the counterfactual theories defended by Dretske (1971) and Nozick
(1981).
It is natural for internalists to endorse something like the KK
principle. For knowing that one knows that p is primarily a matter of
knowing that one's belief that p is warranted, and it is natural for
internalists to say that one is always in a position to know whether
one's beliefs are warranted. Of course, to know that one knows that p,
one must also know that one's belief that p is true. But it seems
clear that anyone who knows that p is in a position to know that their
belief that p is true; so it is natural for internalists to endorse
the KK principle.
It is also natural for externalists to reject this principle. For, if
warrant may be external to our cognitive perspective, then there is no
special reason to expect those who know that p to be in a position to
know that their belief that p is warranted. This can be seen this more
clearly by focusing on the reliabilist theory of knowledge. If one's
belief that p is produced by a reliable process that one knows nothing
about, then one may have no way of knowing that this belief
constitutes knowledge, and thus no way of knowing that one knows that
p.
In light of the above points, it is natural to think that arguments
for internalist theories of knowledge support the KK principle, and
that arguments for externalist theories threaten it. Arguments for
externalist theories are given by Goldman (1967, 1976), Armstrong
(1973), Dretske (1971, 1981), Nozick (1981) and Plantinga (1993a and
1993b), and arguments for internalist theories by Chisholm (1966,
1988), Lehrer (1974, 1986) and BonJour (1985). Externalist theories
are often motivated by a desire to understand knowledge in terms of
scientific concepts, like causation and counterfactual dependence (cf.
Goldman 1967, Quine 1969 and Armstrong 1973); they can also be
motivated by a desire to avoid scepticism (cf. Nozick 1981).
Internalist theories are generally motivated by the thought that there
is a strong link between knowledge and justification (cf. Chisholm
1966, Lehrer 1974 and BonJour 1985); they can also be motivated by the
related thought that knowledge is an essentially normative property
(cf. BonJour 1985, Chisholm 1988 and Kim 1988). Whether these
motivations for the two kinds of theory are good ones remains to be
seen; but it is useful to see that they have a bearing not just on
these theories, but also on the issue of whether the KK principle
holds.
However, it is important to realise that, while it is natural for
internalists to endorse and externalists to reject the KK principle,
it is not necessary for them to do so. Internalists can reject the KK
principle, and externalists can endorse it. To see that internalists
can reject the KK principle, note that it is possible to adopt a
position on which one is not always in a position to know about the
internal, mental properties that are normally accessible to
introspection and reflection. Timothy Williamson holds a position of
this kind; his arguments for it are described in section 4. To see
that externalists can endorse the KK principle, note that one can say
that the property that externalists identify with warrant— such as
being caused in the right way, or being produced by a reliable
process— is one that has to be known about in order to have knowledge.
Alvin Goldman comes close to adopting a position of this kind in his
1967, when he argues that, in cases of inferential knowledge, a
subject must "correctly reconstruct" important elements of the causal
chain leading from the fact that p to their belief that p in order to
have knowledge.
Overall, it seems clear that, while the internalism/externalism debate
is relevant to the KK principle, there are other issues that bear on
its status. Some of these issues are described in the next two
sections.
3. The Surprise Examination and the KK principle.
There are a number of thinkers who hold that the KK principle, or
something very like it, plays a crucial role in the Surprise
Examination paradox (see Harrison 1969, McLelland and Chihara 1975 and
Williamson 1992: 226-32 and 2000:135-146 for examples). Their view is,
roughly, that the paradox can be solved by rejecting the principle. In
what follows, a brief outline will be given of the paradox and the way
in which the principle seems to be related to it. (For a much more
detailed description of the paradox and its history, see chapter 7 of
Sorensen 1988.)
Suppose that a teacher announces to her pupils that she intends to
give them a surprise examination at some point in the following term.
The pupils can argue, as follows, that she will not be able to do
this:
If you want the exam to be a surprise, then you cannot give it on
the last day of term; for if you do, then we will know, on the
second-to-last day, that it will be on the last day, and the exam
won't be a surprise. You also cannot give the exam on the
second-to-last day of term. For if you do, then we will know, on the
third-to-last day, that it will be on either the last day or the
second-to-last day, and will know, by the reasoning just described,
that it will not be on the last day; so again the exam won't be a
surprise. Parallel reasoning shows that you cannot give the exam on
the third-to-last day, or the fourth-to-last day, or on any of the
other days of term. Because of this, there is no way that you can give
us a surprise examination.
It is natural to think there must be something wrong with the pupils'
reasoning; but it is hard to see where the reasoning goes wrong. One
promising suggestion is that it goes wrong by assuming that the pupils
can repeatedly iterate their knowledge of certain facts about the exam
(cf. Williamson 2000: 140-1). To see that this suggestion is
promising, the pupils' reasoning needs to be divided into parts.
Let part 1 of the pupils' reasoning be the part that rules out the
last day, let part 2 be the part that rules out the second-to-last
day, and so on. Since part 2 of the pupils' reasoning rests on the
assumption that part 1 works, it is natural to say that part 2 works
only if they know that part 1 works. And since part 3 rests on the
assumption that part 2 works, it is natural to say that part 3 works
only if they know that part 2 works, and thus, only if they are in a
position to know that they know that part 1 works. Similar reasoning
seems to show that part 4 works only if they are in a position to know
that they know that they know that part 1 works, and so on. So the
pupils' reasoning seems to assume that they are in a position to
repeatedly iterate their knowledge of the fact that part 1 works, and
it is not at all clear that this assumption is correct.
To see that the assumption is implausible, imagine that the teacher
asks the pupils whether they know that part 1 of their reasoning
works, and then asks them whether they know that they know this, and
so on. It is plausible that, at some stage of this interrogation, the
pupils should stop saying "Yes" to the teacher's questions. For it is
plausible that the epistemic standard that the pupils have to meet in
order to appropriately say "Yes" goes up with each new question. If
someone is asked whether it is the case that p, and when they say
"Yes," they are asked whether they know that it is the case that p,
they are generally being asked to check their original assertion
against higher standards (cf. DeRose 2002: 184-5).
Because of this, it is plausible that the pupils cannot go on
iterating their knowledge of part 1's success forever. And if that is
so, then there is a limit to the number of possible examination days
that their reasoning can rule out. If there is such a limit, it can be
used to explain why the pupils' reasoning fails to show that the
teacher cannot give them a surprise examination. The explanation is
that they cannot iterate their knowledge of part 1's success enough to
rule out every day of the term.
In defense of this explanation, note that the pupils' reasoning does
seem to rule out later days of the term as possible days for the exam.
It is very plausible that part 1 of the reasoning rules out the last
day of term as a possible date for the exam, and quite plausible that
part 2 rules out the second-to-last day. But parts 3 and 4 seem more
questionable, and by the time part 10 is gotten to, it is clear that
something has gone wrong. The above explanation can account for this
gradual loss of power in the pupils' reasoning, by appealing to the
gradual increase in the number of iterations of knowledge required to
make the reasoning work (cf. Williamson 2000: 142).
If the failure of the pupils' reasoning is best explained in terms of
limits on their ability to iterate their knowledge, then one seems
obliged to say that their knowledge does not satisfy the KK principle.
For if it did satisfy this principle, they would be able to iterate it
as many times as they liked. The fact that the knowledge of the
epistemically limited pupils does not satisfy this principle does not
show that there are not other, more idealised kinds of knowledge that
do. But it does suggest that the principle fails to hold for our
everyday concept of knowledge, and hence that the best strategy for
defending it is to follow Hintikka in arguing that it holds only for a
strengthened version of this concept.
4. Williamson's Anti-Luminosity Argument.
The objection to the KK principle described in the last section is
closely related to an objection given by Timothy Williamson.
Williamson's objection uses the concept of luminosity; for him, a
condition, C, is luminous iff the following claim holds:
(L) For every case α, if in α C obtains, then in α one is in a
position to know that C obtains (2000: 95).
If the KK principle holds, then the condition of knowing that p is
luminous in Williamson's sense. In chapter 4 of his 2000, Williamson
argues that any condition that can be gradually gained or lost is not
luminous, and that, since knowing that p is a condition that can be
gradually gained or lost, the KK principle fails.
Williamson argues against the luminosity of conditions that can be
gradually gained or lost by focusing on the condition of feeling cold,
which seems to stand a very good chance of being luminous. His
argument is focused on a case in which:
(i) One feels freezing cold at dawn, very slowly warms up and
feels hot by noon.
(ii) One is not aware of any change in one's feelings of hot and
cold over 1 millisecond, and:
(iii) Throughout the morning, one thoroughly considers how cold or
hot one feels, and so always knows everything that one is in a
position to know about this.
Using t0, t1… tn for times at 1 millisecond intervals between dawn and
noon, and αi for the case that holds at ti (where 0 ≤ i ≤ n),
Williamson argues that the following principle holds for all values of
i:
(1i) If in αi one knows that one feels cold, then in αi+1 one feels cold.
He does so by appealing to the plausible safety principle that, if one
knows that p, then one's belief that p could not easily have been
false. When this principle is formulated in terms of possible cases,
it says: one knows that p in case α only if one's belief that p is
true in every possible case that is sufficiently similar to α. Since
αi+1 is extremely similar to αi for every value of i, it is natural to
infer from this principle that (1i) holds for all such values.
After arguing that (1i) holds for all such values, Williamson points
out that, if feeling cold is luminous, then this principle holds for
all values of i:
(2i) If in αi one feels cold, then in αi one knows that one feels
cold. (2000: 97)
He then attacks the luminosity of feeling cold by giving a reductio
argument against the assumption that (1i) and (2i) hold for all values
of i. One way of giving this argument (used in Neta and Rohrbaugh
2004) is to note that, by hypothetical syllogism, (2i) and (1i)
together entail:
(3i) If in αi one feels cold, then in α i+1 one feels cold.
If (1i) and (2i) hold for all values of i, then (3i) also holds for
all such values. And if it does, then this principle, which is clearly
true:
(40) In α0 one feels cold.
(since α0 is at dawn and at dawn one is freezing) implies this
principle, which is clearly false:
(4n) In αn one feels cold.
(since αn is at noon and at noon one is hot). No true principle can
imply a false principle. So (3i) cannot hold for all values of i,
which means that (1i) and (2i) cannot hold for all such values. It has
been argued that (1i) holds for all such values; so it seems that (2i)
must fail to hold for some of them. But if feeling cold were luminous
then (2i) would hold for all values of i. So it seems that feeling
cold cannot be luminous.
If the above argument shows that the condition of feeling cold is not
luminous, then parallel arguments will show the same thing for every
condition that can be gradually gained or lost. Since the condition of
knowing that p seems to be a condition of this kind, the above
argument threatens to show that it is not luminous, and hence that the
KK principle fails. But there are ways in which advocates of the KK
principle, or of luminosity more generally, can respond to the
argument. The next section describes two responses of this kind.
5. Replies to Williamson.
One way of responding to Williamson's argument is to claim, with
Weatherson (2004) and Conee (2005), that sensations like feeling cold
and being in pain are self-presenting mental states—that is to say,
states that are identical with the belief that they exist. If a state
is self-presenting, then the belief that it exists satisfies
Williamson's safety constraint; so if feeling cold is self-presenting,
then Williamson's defense of (1i) fails. However it seems clear that
the state of knowing that p is not a self-presenting mental state; for
one can believe that one knows that p without actually knowing it. So
while this line of response may show that states like feeling cold and
being in pain can be luminous, it seems unlikely to save the KK
principle (as Weatherson and Conee both grant).
Another way of responding to Williamson's argument is to claim, with
Brueckner and Fiocco (2002) and Neta and Rohrbaugh (2004), that the
safety principle to which Williamson appeals is false. This line of
response seems more likely to save the KK principle; one way of
developing it is to focus on the following example (taken from Neta
and Rohrbaugh):
"I am drinking a glass of water which I have just poured from the
bottle. Standing next to me is a happy person who has just won the
lottery. Had this person lost the lottery, she would have maliciously
polluted my water with a tasteless, odorless, colorless toxin. But
since she won the lottery, she does no such thing. Nonetheless, she
almost lost the lottery. Now, I drink the pure, unadulterated water,
and judge, truly and knowingly, that I am drinking pure, unadulterated
water. But the toxin would not have flavored the water, and so had the
toxin gone in, I would still have believed falsely that I was drinking
pure, unadulterated water. The actual case and the envisaged possible
case are extremely similar in all past and present phenomenological
and physical respects, as well as nomologically indistinguishable.
(Furthermore, we can stipulate that, in each case, I am killed by a
sniper a few minutes after drinking the water, and so the cases do not
differ in future respects.)" [Neta and Rohrbaugh 2004: 400]
It seems clear that, in this example, I know that I am drinking
unadulterated water, despite the fact that there is a very similar
possible case in which I falsely believe that I am drinking such
water. So the example conflicts with the safety principle's claim that
beliefs constitute knowledge only if they are true in all sufficiently
similar cases.
Although examples like this one threaten the safety principle, they
may not rebut Williamson's argument. For the key premise of the
argument— that (1i) is true for all values of i— can be defended in
other ways. To see this, consider the following claim, which is the
contrapositive of (1i):
(1i') If in αi+1 one does not feel cold, then in αi one does not
know that one feels cold.
It is plausible independently of the safety principle that (1i'), and
thus (1i), holds for all values of i. For if one does not feel cold in
αi+1 and one is not aware of any change in ones feelings of hot and
cold between αi and αi+1, then how could one possibly know that in αi
one feels cold?
Even if it turns out that (1i) cannot be adequately defended, it may
still turn out that the KK principle is rebutted by reasoning like
Williamson's. For it is possible to give an argument against the KK
principle which closely resembles the anti-luminosity argument
described above, but which does not appeal to (1i). This argument
focuses on cases of inexact knowledge— that is to say, of the sort of
knowledge that one gains by looking at a distant tree and estimating
its height, or by looking at a crowd and estimating the number of
people that it contains. In chapter 5 of his 2000, Williamson argues
that such knowledge satisfies margin for error principles like the
following:
(M1) If I know that the tree is not n inches tall, then it is not
n+1 inches tall.
(M2) If I know that there are not n people in the crowd, then
there are not n+1 people in the crowd.
He then shows that, when principles of this kind are conjoined with a
plausible closure principle on knowledge, they are incompatible with
the KK principle.
Although Williamson's arguments against the KK principle are powerful,
they can be resisted at a price. For, in all of their forms, they
assume that some true beliefs constitute knowledge (such as a freezing
cold person's belief that they feel cold) and that others do not (such
as an accidentally true belief that a 600-inch-tall distant tree is
not 599 inches tall). The first of these assumptions can be denied by
endorsing a skeptical theory on which no true belief constitutes
knowledge and the second can be denied by endorsing a "universalist"
theory on which every true belief constitutes knowledge. Although both
theories have implausible consequences, recent work (such as Goldman
2002: 164 on weak senses of knowledge and Hawthorne 2004: 113-141 on
skepticism) has revealed that both have attractive features. If the
benefits of these theories outweigh their costs, then Williamson's
arguments against the KK principle may still fail. In any case, it
seems fair to conclude that the KK principle, and the arguments for
and against it, remain important subjects of philosophical debate.
6. References and Further Reading.
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Cambridge University Press.
BonJour, L. 1985. The Structure of Empirical Knowledge. Cambridge,
Mass.: Harvard University Press.
Brueckner, A. and Fiocco, M.O. 2002. "Williamson's Anti-Luminosity
Argument," Philosophical Studies 110: 285-293.
Castaneda, H.N. 1970. "On Knowing (Or Believing) That One Knows (Or
Believes)," Synthese 21: 187-203.
Chisholm, R. 1966. Theory of Knowledge. Englewood Cliffs: Prentice-Hall.
Chisholm, R. 1988. "The Indispensability of Internal Justification,"
Synthese 74: 285-96.
Conee, E. 2005. "The Comforts of Home," Philosophy and
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Craig, E.J. 1990. Knowledge and the State of Nature. Oxford: Clarenden Press.
Danto, A.C. 1967. "On Knowing That We Know," in A. Stroll ed.,
Epistemology, New York: Harper and Rowe, pp.32-53.
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Dretske, F. 1971. "Conclusive Reasons," Australasian Journal of
Philosophy, 49: 1-22.
Dretske 1981. Knowledge and the Flow of Information. Oxford: Blackwell.
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Goldman 1967. "A Causal Theory of Knowing," Journal of Philosophy 64: 357-72.
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Goldman 1979. "What is Justified Belief?" In Justification and
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Hintikka, J. 1970. "Knowing that One Knows" reviewed. Synthese 21: 141-62.
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Quine, W.V.O. 1969. "Epistemology Naturalized," in his Ontological
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Unger, P. 1975. Ignorance: A Case for Scepticism. Oxford: Oxford
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Von Wright, G. 1951. An Essay in Modal Logic. Amsterdam: North-Holland
Publishing Co.
Weatherson, B. 2004. "Luminous Margins," Australasian Journal of
Philosophy 82: 373-83.
Williamson, T. 1992. "Inexact Knowledge," Mind, 101: 217-42.
Williamson, T. 2000. Knowledge and its Limits. New York: Oxford
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