Sensitivity and induction: the
discussion so far
Nozick suggests that if S knows that \(p\), then S’s belief that \(p\) tracks truth. He thinks that
subjunctive conditionals can best capture this truth-tracking relation.
Moreover, he argues that we have to take the belief-forming method into
account. Nozick (1981,
179) provides the following definition of knowing via a
method:
S knows, via method (or way of believing) M, that \(p\) iff (1) \(p\) is true (2) S believes, via method or
way of coming to believe M, that \(p\)
(3) In the nearest possible worlds where \(p\) is false and where S uses M to arrive
at a belief whether (or not) \(p\), S
does not believe, via M, that \(p\) (4)
In the nearest possible worlds where \(p\) is true and where S uses M to arrive at
a belief whether (or not) \(p\), S
believes, via M, that \(p\)
Condition (3) is the sensitivity condition, which I will focus on
here, and condition (4) is the adherence condition.
Nozick formulates these modal conditions on knowledge as subjunctive
conditionals, but he analyzes their truth conditions in terms of
possible worlds. For the sake of convenience, and in accordance with the
literature, I will use possible world terminology for formulating
conditions (3) and (4). For the purposes of this paper, nothing hinges
on this decision, as we acquire the same results for sensitivity and
induction when talking in terms of subjunctive conditionals.
Sensitivity accounts of knowledge face several major problems. First,
it has been claimed that they preclude us from having inductive
knowledge, as Vogel (1987, 1999) and
Sosa (1999)
contend. Second, they lead to implausible instances of closure failure
as Kripke
(2011) argues. Third, Sosa (1999) and Vogel (2000) argued that sensitivity
faces severe problems concerning higher-order knowledge about the truth
of one’s own beliefs. In this paper, I will focus on the first
objection. However, we will see in the last
section that the problem of inductive knowledge has structural
similarities to the problem of higher-order knowledge.
Despite the well-known challenges that sensitivity accounts of
knowledge face, the sensitivity principle is intuitively appealing,
leading to a “second wave” of sensitivity accounts, as Becker and Black
(2012) label it. These accounts aim at defending a
sensitivity-based theory of knowledge that avoids the problems that have
been raised for Nozick’s (1981) original account.
Accordingly, the results about sensitivity and induction are not only
relevant for Nozick’s original theory but also for these descendants.
Vogel and Sosa argue for the claim that making sensitivity a
necessary condition on knowledge rules out inductive knowledge by means
of examples; they provide cases where a subject plausibly knows via
induction although her belief is insensitive. Here are two cases:
Chute. On his way to the elevator, Ernie
releases a trash bag down the chute from his high-rise condo. Walking
along the street Ernie thinks about the trash and forms the belief that
the trash is in the basement. Plausibly, Ernie knows that his bag is in
the basement. But what if, having been released, it still (incredibly)
were not to arrive there? That presumably would be because it had been
snagged somehow in the chute on the way down (an incredibly rare
occurrence), or some such happenstance. But none of these would affect
Ernie’s belief, so he would still believe that the bag has arrived in
the basement. His belief seems not to be sensitive, therefore, but
constitutes knowledge anyhow, and can correctly be said to do so. (See Sosa 1999,
145–146)
Heartbreaker. Sixty golfers are entered in the Wealth
and Privilege Invitational Tournament. The course has a short but
difficult hole, known as the “Heartbreaker.” Before the round begins,
Jonathan thinks that, surely, not all sixty players will get a
hole-in-one on the “Heartbreaker.” (See Vogel 1999, 165)
These are cases of beliefs that are based on inductive reasoning,
more specifically, inductive reasoning about particulars, as Vogel puts
it. He argues that knowledge about
particulars via inductive reasoning is highly plausible. Intuitively,
Ernie knows that the trash is in the basement, and Jonathan knows that
not all sixty players will get a hole-in-one. However, in each case the
target beliefs are insensitive. Consequently, sensitivity is not
necessary for knowledge.
Sosa and Vogel argue against sensitivity accounts of knowledge by
presenting examples of insensitive inductive beliefs that plausibly
constitute knowledge. They need not argue for the stronger claim that
any belief formed via induction is insensitive to make their
point. The weaker claim that there are some plausible cases of inductive
knowledge that involve insensitive beliefs is sufficient for their
purpose. Nevertheless, the stronger view that any belief formed via
induction is insensitive is the dominant one in the current debate.
Sosa’s and Vogel’s line of argumentation against sensitivity accounts
of knowledge is not unopposed. One standard defense of sensitivity is
proposed by Becker
(2007). He accepts the view that induction yields insensitive
beliefs, but he argues that this does not create a devastating objection
to sensitivity accounts of knowledge. He admits that if we know
propositions \(p\)1…\(p\)n, then we do not have
inductive knowledge that \(p\)n+1 is true. However, we
still have knowledge about the probability of \(p\)n+1.
Thus, our view about knowledge via induction rests on a confusion
according to Becker. We cannot have knowledge via induction that \(p\)n+1; what we do know are
propositions in the neighborhood of this proposition. Becker’s account
not only rejects inductive knowledge but also provides an explanation of
our mistaken intuition that we can have this kind of knowledge. However,
knowledge via induction is widely accepted. Accordingly, most
philosophers are presumably not willing to bite the bullet of rejecting
inductive knowledge for the gain of acquiring a sensitivity-based
account of knowledge. We will take up Becker’s account later and see
that his solution faces additional problems.
Vogel, Sosa, and Becker agree that the subjects’ beliefs in cases
like Chute and Heartbreaker are insensitive, but draw
conflicting conclusions as to whether this claim creates a serious
problem for sensitivity accounts of knowledge. Until recently, the view
that induction yields insensitive beliefs has remained unchallenged.
Wallbridge
(2018) takes up this objection to sensitivity accounts of
knowledge and argues that properly understood, the purported
counterexamples fail to succeed because the beliefs formed via induction
are actually sensitive, not insensitive. Focusing on Sosa’s chute case,
Wallbridge argues that Ernie sensitively believes that the rubbish is in
the basement. He claims that in some cases, in order to avoid
“miracles”, i.e. events that would not easily have happened,
counterfactuals have to be interpreted as backtracking. According to a
backtracking interpretation, counterfactual conditionals can be
evaluated without keeping the past fixed until the time at which the
counterfactual antecedent obtains. Wallbridge argues that, according to
this backtracking analysis, Ernie’s belief is sensitive.
He suggests that other examples presented by Vogel (1987, 1999) and
Pritchard
(2012) can be analyzed analogously. Wallbridge is not
particularly clear about his conclusion. In the abstract, he claims to
show that inductive knowledge is sensitive. In the conclusion, Wallbridge (2018,
8) makes the weaker claim that “there are cases of sensitive
inductive knowledge” and leaves the reader with a challenge, concluding
that “if there are cases of insensitive inductive knowledge then they
have yet to be pointed out.”
In Section 2, I will show that the
situation concerning induction and sensitivity is more subtle than
opponents and defenders of sensitivity accounts of knowledge claim it to
be. Some inductive processes yield sensitive beliefs, others yield
insensitive beliefs, regardless of whether we opt for a backtracking or
a non-backtracking interpretation of counterfactual conditionals. In
Section 3, I will argue that this is
problematic since the subjects in the cases presented are concerning
inductive reasoning intuitively in similarly good epistemic situations.
Hence, sensitivity accounts of knowledge are committed to making
implausibly heterogeneous predictions about the knowledge status of
subjects who believe via induction.
Sensitive and insensitive
induction
In this section, I will discuss instances of enumerative and temporal
induction and backtracking and non-backtracking interpretations of
counterfactual conditionals. First, let me make some preliminary remarks
about backtracking and non-backtracking counterfactuals. Lewis (1973)
distinguishes between backtracking and non-backtracking counterfactuals.
Non-backtracking counterfactuals keep the past fixed until the time at
which the counterfactual antecedent obtains, whereas backtracking
counterfactuals do not. He argues that only
non-backtracking counterfactuals can be used for analyzing causal
dependencies. For example, in order to determine whether event \(c\) caused event \(e\), we consider those possible worlds that
are identical with the actual world until the time where \(c\) does not obtain.
In this paper, I will remain neutral about whether counterfactuals
are correctly interpreted as backtracking or non-backtracking. Rather, I
will investigate the consequences of these two interpretations for
sensitivity accounts of knowledge. Let me emphasize the point of
considering backtracking counterfactuals. We can say that whether S’s
induction-based belief is sensitive depends on whether “the minimal
‘change’ from truth to falsity of \(p\)
keeps the inductive evidence for \(p\)
intact.” In terms of possible worlds, the
sensitivity of S’s belief depends on whether the inductive evidence is
available to S in the nearest possible worlds where \(p\) is false. If we interpret
counterfactuals exclusively as non-backtracking, then we only consider
possible worlds that do not differ from the actual world until the point
at which the counterfactual antecedent obtains. If we allow for
backtracking interpretations of counterfactuals, then we need not keep
the past fixed until that point. Hence, backtracking or non-backtracking
interpretations make a difference concerning which nearest possible
worlds are considered and consequently whether a belief is judged to be
sensitive or not.
In the following, I will present and analyze further cases of
induction. We will see that some cases yield insensitive beliefs whereas
others yield sensitive beliefs, regardless of whether counterfactuals
can be backtracking or not. I will distinguish between enumerative
induction where we draw an inference from objects \(o\)1-\(o\)n to \(o\)n+1 and temporal induction
where we draw an inference about an object \(o\) from time \(t\)1-\(t\)n to \(t\)n+1.
In each of these cases, the method of belief formation in question is
induction. Moreover, the cases have to be
understood in a way such that the subjects are intuitively in equally
good epistemic positions concerning the inductive conclusion in that (1)
the evidence for believing the premises is equally strong; (2) the
numbers of cases \(n\) observed is
equally large (or the time interval observed is equally long); (3) the
relevant similarity between the induced case cn+1 and
observed cases c1 to cn is equally strong (or the
similarity between the basic conditions for \(o\) of the induced time point \(t\) and the observed interval \(i\)); (4) there are no rebutting or
undercutting defeaters available to the subjects; and (5) the predicates
involved are equally projectible. Take, first, the following example of
enumerative induction that yields an insensitive belief:
Raven (enumerative induction). Carl observes that
raven1–ravenn is black and infers that
ravenn+1, which he has not observed, is black. Ravens are
typically black, though not necessarily, since there also exist rare
mutations like albino ravens. In the nearest possible worlds where
ravenn+1 is not black, it is such a rare mutation. However,
raven1-ravenn is black in these nearest possible
worlds and Carl believes via observation of
raven1-ravenn and induction that
ravenn+1 is black. Thus, his belief that ravenn+1
is black is insensitive.
This analysis holds independently of whether counterfactuals are
allowed to be backtracking or not. In both cases, the nearest possible
worlds where ravenn+1 is not black are such that it is an
albino raven but where raven1–ravenn is black. In
these possible worlds, Carl still believes via induction that
ravenn+1 is black. Hence, his belief is insensitive. Thus, in
RAVEN, a case of enumerative induction, the subject believes
insensitively, regardless of whether we allow backtracking
interpretations of counterfactuals or not.
Notably, a similar case of temporal induction yields a
different outcome:
Blackbird (temporal induction). Miles observes that blackbirdn
has been black until yesterday and believes via induction that
blackbirdn is black right now.
Non-backtracking: We only consider the nearest possible worlds where
blackbirdn is not black right now, and, hence,
worlds where blackbirdn has been black until yesterday. We
ignore possible worlds where blackbirdn changed its color
earlier and worlds where it has never been black. In the nearest
possible worlds considered Miles believes via observation and induction
that blackbirdn is black right now. Hence, his belief is
insensitive.
Backtracking: If counterfactuals are backtracking the situation is
different. In this case, the nearest possible worlds where
blackbirdn is not black right now are presumably such that it
is an albino blackbirdn that has been white all the time.
They are not worlds where it changed the color since yesterday.
Accordingly, in the nearest possible worlds where blackbirdn
is not black right now Miles does not believe via observation and
induction that it is black right now. Thus, Miles’ inductive belief that
blackbirdn is black right now is sensitive.
So far we have seen that in the enumerative induction case of Raven, Carl’s belief is insensitive
no matter whether counterfactuals can be backtracking or not. However,
in Blackbird, a case of
temporal induction, Miles’ belief is insensitive if counterfactuals are
non-backtracking but sensitive if they are backtracking. At this point,
one might suppose that enumerative induction is typically insensitive
whereas the sensitivity of temporal induction depends on whether we opt
for a non-backtracking interpretation or a backtracking one. However,
this generalization is incorrect as the following cases will show. Take
a second instance of enumerative induction that delivers
sensitive beliefs in case of non-backtracking and backtracking
counterfactuals:
Examiner (enumerative induction). Ina is a lazy examiner. When she has
received all the exams she throws a dice and all the examinees get the
same grade. For a particular test, she throws a 2 and, accordingly,
marks all exams with B. Rachel is an examinee and does not know Ina’s
habits. Rachel asks numerous peers about their grades. Among them are
peers of whom she knows that they were better prepared than herself and
peers of whom she knows that they were worse prepared. All peers report
that they got a B. Rachel forms the belief that she also got a B. The
nearest possible worlds where Rachel does not get a B are such that
Ina’s dice throw delivered a different result than 2 and all students
got a different grade than B, but the same one. In these possible
worlds, Rachel does not believe via testimony and induction that she got
a B. Thus, her belief that she got a B on the exam is sensitive.
The grades of all students are determined at the same time. Thus, no
matter whether counterfactuals can be backtracking or not, the nearest
possible worlds where Rachel does not get a B are such that all the
other students do not get a B. In these possible worlds, Rachel does not
believe via testimony and induction that she got a B. Thus, Rachel’s
belief is sensitive, regardless of whether counterfactuals can be
backtracking or not. Raven and Examiner are both cases of
enumerative induction. In Raven,
the target belief is insensitive, no matter whether counterfactuals can
be backtracking or not, and in Examiner, it is sensitive in both cases.
So far we have reflected on one case of temporal induction, Blackbird, where the belief is
insensitive with a non-backtracking interpretation of counterfactuals
and sensitive with a backtracking interpretation. We will now see that
temporal induction can deliver different sensitivity results in
different cases. Let’s sketch a further case:
T –
shirt (temporal
induction). Sarah has
seen Tim wearing a red T-shirt the whole day until 30 minutes ago and
forms the inductive belief that Tim is wearing a red T-shirt right
now.
Is Sarah’s belief that Tim is wearing a red T-shirt right now
sensitive? This depends on how we fill in the details. Let us consider
two different scenarios:
Scenario 1: Tim and Sarah are on a hiking trail and they split thirty
minutes ago. Sarah has seen Tim wearing a red T-shirt the whole day and
forms the inductive belief that Tim is wearing a red T-shirt right now.
Tim does not have another T-shirt with him. Thus, he could not easily
get a fresh T-shirt. Suppose further that Tim accidentally grabbed a red
T-shirt in the morning, but that he might easily have grabbed a T-shirt
of a different color. If counterfactuals can be backtracking, then the
nearest possible worlds where it is false that Tim is wearing a red
T-shirt right now are such that he grabbed a T-shirt of any other color
in the morning. In these possible worlds, Sarah does not believe via
observation that Tim was wearing a red T-shirt until thirty minutes ago
and, therefore, does not believe via induction that he is wearing a red
T-shirt right now. Thus, her belief is sensitive. If counterfactuals can
only be non-backtracking, then we only consider possible worlds where
Tim recently changed his T-shirt. In this case, Sarah’s belief that Tim
is wearing a T-shirt right now formed via observation and induction is
insensitive. Hence, for Scenario 1, we acquire the same result as for Blackbird—a non-backtracking
interpretation of counterfactuals implies insensitive beliefs, and a
backtracking interpretation sensitive beliefs.
Scenario 2: Sarah and Tim were on a hiking trail until 30 minutes ago
where Tim was wearing a red T-shirt. In fact, for security concerns, Tim
only wears red T-shirts for hiking. Thus, it is not easily possible that
he had a non-red T-shirt for hiking. After the hiking trail, Sarah and
Tim split and Tim walks downtown for a drink. Sarah has seen Tim wearing
a red T-shirt the whole day until 30 minutes ago and forms the inductive
belief that Tim is wearing a red T-shirt right now. If counterfactuals
can only be non-backtracking, then we only consider possible worlds
where Tim recently changed his T-shirt. In these possible worlds, Sarah
believes via observation and temporal induction that he is wearing a red
T-shirt right now and, consequently, her belief is insensitive. However,
even if we allow for backtracking counterfactuals, then the nearest
possible worlds where Tim is not wearing a red T-shirt right now are
such that he changed it recently downtown, given Tim’s strict habit of
only wearing red T-shirts for hiking.
Again, Sarah believes that Tim is wearing a red T-shirt right now and
her inductive belief turns out to be insensitive.
In both scenarios, a non-backtracking interpretation of
counterfactuals yields insensitive beliefs, but if we allow for
backtracking interpretations, then Scenario 1 yields a sensitive belief
whereas Scenario 2 yields an insensitive belief. In this respect,
whether one’s belief is sensitive in cases of temporal induction depends
on how the cases are spelled out in detail.
We can now summarize the acquired results about sensitivity and
induction: Raven is a case of
enumerative induction. Carl’s belief that ravenn+1 is black
is insensitive regardless of whether counterfactual
conditionals can be backtracking or not. Examiner is a further case of enumerative
induction. However, Sarah’s belief that she got a B for the exam is
sensitive regardless of whether counterfactuals can be
backtracking or not. Blackbird
is a case of temporal induction. Miles’s belief that
blackbirdn is black is insensitive if
counterfactuals can only be non-backtracking, but it is sensitive if
they can be backtracking. As for T-shirt, a further case of temporal induction,
sensitivity depends on how we fill in the details. In Scenario 1 and 2,
Sarah’s belief that Tim is wearing a red T-shirt right now is
insensitive if counterfactuals can only be non-backtracking. If they can
be backtracking, then her belief is sensitive in Scenario 1 but
insensitive in Scenario 2. These results are captured in table 1:
Table 1: Whether the belief is sensitive in non-backtracking
and backtracking variants of the four cases.
| Induction type |
Case |
Non-backtracking |
Backtracking |
| Enumerative |
Raven |
\(-\) |
\(-\) |
| Examiner |
\(+\) |
\(+\) |
| Temporal |
Blackbird |
\(-\) |
\(+\) |
| T-shirt |
\(-\) |
\(+/-\) |
Let me provide a more systematic analysis: Suppose S observes that
object \(o\) has property F from
t1 to tn and believes via temporal induction that
\(o\) has property F at
tn+1. If we generally accept that counterfactuals can be
backtracking, then S’s inductive belief is sensitive only if, in the
nearest possible worlds where \(o\) is
not F at time tn+1, \(o\) is
not F from t1 to tn.
This is the case if worlds where \(o\)
lost property F from tn to tn+1 are more remote
than worlds where \(o\) does not have
property F from t1 to tn, e.g. if it is more
crucial for \(o\) to constantly be F
(or to constantly be not-F) from t1 to tn+1 than
to be F from t1 to tn, as in Blackbird and Scenario 1 of T-shirt, where Tim does not walk downtown. In
Scenario 2, where Tim walks downtown after the hiking trail, the worlds
where Tim changed his T-shirt downtown are closer than the worlds where
he was not wearing a red T-shirt on the hiking trail. Here, it is more
crucial for \(o\) to be F from
t1 to tn than it is to be constantly F or
constantly not be F from t1 to tn+1. Consequently,
Sarah’s belief is insensitive. In contrast, if counterfactuals can only
be non-backtracking, then we can only consider possible worlds where
\(o\) changed property F from
tn to tn+1. In this case, beliefs formed via
temporal induction are always insensitive. Thus, despite the
heterogeneity of the overall results, at least we can say that temporal
induction always yields insensitive beliefs if counterfactuals can only
be non-backtracking.
We obtain slightly different results concerning enumerative
induction. If the nearest possible worlds where \(o\)n+1 does not have property F
are such that \(o\)1-\(o\)n does not have property F,
then S’s belief that \(o\)n+1 is F formed via
observation of \(o\)1- \(o\)n and enumerative induction
is presumably sensitive. This condition is
fulfilled if it is rather accidental that \(o\)n+1 is F but characteristic
for \(o\)1- \(o\)n+1 that they have the same
status of being F (or not being F), as in Examiner. However, if the nearest possible worlds
where \(o\)n+1 does not have
property F are such that \(o\)1- \(o\)n still has property
F, then S’s belief formed via observation and induction is insensitive.
This holds for Raven.
Notably, theories of counterfactuals that allow for non-backtracking
counterfactuals and theories that do not deliver the same results for
each individual case of enumerative induction, i.e. both types of
theories imply that the target belief is sensitive or both theory types
imply that it is insensitive. Perhaps we can construct cases of
enumerative induction such that backtracking and non-backtracking
theories deliver different results with respect to sensitivity, but I
suspect that these instances of induction also involve a temporal
element.
Heterogeneity: The problem for
sensitivity and induction
Let me now diagnose what I regard as the real problem of sensitivity
and induction. There is at least a tendency among proponents and critics
of sensitivity that there is a homogeneous picture of sensitivity and
induction. Critics of sensitivity accounts of knowledge, but also some
adherents such as Becker, tend to think that induction yields
insensitive beliefs whereas Wallbridge suggests that it yields sensitive
beliefs. However, none of these opposing views is correct, since some
instances of induction yield sensitive beliefs whereas some others yield
insensitive ones.
I developed various cases of enumerative and temporal induction. In
each of these cases, the subjects make an empirical observation and draw
an inductive inference. Importantly, we intuitively judge that the
subjects in these cases are in equally good epistemic positions. This
view about the equality of the epistemic positions is also supported
when applying plausible parameters for induction. Let me briefly
explain. The epistemic force of induction comes in degrees. The
epistemic strength of inductive reasoning from cases
c1-cn to case cn+1 (or from time
interval \(i\) to point in time \(t\)) and whether it can yield justification
and knowledge intuitively depends on various factors. The strength of
induction depends first on the number \(n\) of cases observed (or on the length of
the observed time interval). All else being equal, the larger \(n\) is, the greater the epistemic strength
of a particular induction; Secondly, the epistemic strength of induction
varies with the relevant similarity between the cases observed and the
case induced (or on the relevant similarity between the observed time
interval and the point in time induced).
The more similar cn+1 is to c1-cn in
the relevant sense, the stronger the inductive reasoning is. Third, the
epistemic strength of inductive reasoning depends on whether there
exists a defeater \(d\) for the
inductive conclusion, either rebutting or undercutting, such that S is
propositionally justified in believing \(d\) and this justification undermines S’s
justification in holding an inductive belief about cn+1.
Finally, the predicate involved has to be projectible.
Moreover, the amount of justification for the conclusion of an
induction is also affected by the strength of the justification for
believing the premises. All else being equal, the stronger the
justification for believing the premises, the stronger the justification
for believing the inductive conclusion.
This paper aims at showing that sensitivity accounts of knowledge
have highly implausible consequences when it comes to inductive
knowledge. In order to make this point, it suffices to refer to
intuitively plausible criteria for inductive knowledge. We need not
develop a detailed theory of induction and confirmation, involving
Bayesianism or alternative conceptions.
For the purposes of this paper, it suffices to accept that in the cases
discussed, the inductive reasoning is intuitively of the same epistemic
strength according to the plausible parameters specified. That means
that the subjects in the cases have equally good evidence about an
equally high number of \(n\) cases (or
a sufficiently long time interval \(i\)), case cn+1 is equally
similar to the observed cases c1-cn, there is no
defeater \(d\) for S, rebutting or
undercutting, such that S is justified to believe that \(d\) and this justification undermines her
inductive justification, and the predicates involved are equally
projectible.
Since the subjects are intuitively all in equally good epistemic
positions, the minimal standards that a theory of knowledge has to
fulfill is that it delivers the same outcome with respect to knowledge
in all cases discussed. Here there are two options, first, that the
subjects know in all cases of induction presented and, second, that they
are precluded from knowing in all cases. I
assume that there is a wide agreement among epistemologists that we can
have knowledge via induction. Accordingly, the first, positive option is
far more popular than the second, negative one. However, sensitivity
accounts of knowledge cannot deliver any of these two uniform
pictures.
Let me explain in more detail. I regard it as an open question
whether counterfactual conditionals can only be correctly interpreted as
backtracking or also as non-backtracking. However, in any case we
acquire an unsatisfactorily heterogeneous picture. Suppose first that
counterfactuals can only be non-backtracking. Presumably, temporal
induction always provides insensitive beliefs, given a non-backtracking
analysis of counterfactuals as in Blackbird and T-shirt. However, some instances of enumerative
induction can yield sensitive beliefs, e.g. Examiner, but some others not, e.g. Raven. Thus, according to a
sensitivity account of knowledge, S does not know in Raven, Blackbird, and T-shirt but knows in Examiner, given that counterfactuals can only be
non-backtracking.
This result is counterintuitive since the epistemic position of the
subject is intuitively equally good in all four cases. Suppose now that
counterfactuals can be backtracking. In this case, Examiner, Blackbird, and Scenario 1 of T-shirt yield sensitive beliefs, but Raven and Scenario 2 of T-shirt yield insensitive beliefs.
Again, sensitivity accounts of knowledge are committed to accept that
the subjects know in the first three cases but not in the latter
two.
Thus, in both cases of backtracking and non-backtracking theories of
counterfactuals, some processes of induction yield sensitive beliefs but
some others insensitive beliefs. Hence, sensitivity accounts of
knowledge deliver in both cases an implausibly heterogeneous picture of
inductive knowledge. In both cases, we know via some
instances of induction but do not know via some other instances. This
heterogeneous picture is no less problematic than the orthodox view,
dominant so far, that sensitivity accounts of knowledge preclude us from
any kind of inductive knowledge.
These results affect extant pessimistic and optimistic accounts of
sensitivity in various ways. The orthodox view about sensitivity and
induction is based on cases of insensitive inductive beliefs that
plausibly constitute knowledge, as presented by Vogel (1987, 1999) and
Sosa (1999). The
popular generalization of these cases has it that any instance of
induction yields insensitive beliefs and that we cannot have any
inductive knowledge according to sensitivity accounts of knowledge. This
generalization is incorrect. However, Vogel and Sosa mainly aim at
arguing against sensitivity accounts of knowledge by presenting
counterexamples of insensitive knowledge via induction. This goal can
still be reached by pointing out that sensitivity accounts of knowledge
imply that we do not know in some (paradigmatic) instances of
induction that plausibly yield knowledge.
Becker
(2007) accepts that induction yields insensitive beliefs but
argues that this does not pose a serious problem since we can still
acquire knowledge about the probability of the target proposition. This is already problematic since
knowledge via induction seems highly plausible. Becker suggests that any
instance of induction provides insensitive beliefs. What he should say
is that in some cases of induction, we have knowledge of the target
proposition, but in some very similar cases, we only have knowledge
about the probability of the target proposition. This outcome is too
heterogeneous to be plausible and, thus, not more convincing than
Becker’s original conclusion.
Wallbridge
(2018) claims that inductive knowledge is sensitive, given that
we accept backtracking counterfactuals in some contexts, or at least he
leaves the reader with the challenge of presenting cases of insensitive
induction. He suggests that we should not exclude backtracking
counterfactuals in evaluating modal knowledge conditions like
sensitivity or safety. In this respect, Wallbridge’s analysis advances
the existing debate about sensitivity and induction. However, he does
not tell the whole story about sensitivity and induction since his
challenge of finding instances of insensitive induction can be easily
met. Moreover, sensitivity is not a matter of backtracking or
non-backtracking interpretations of counterfactuals, as he suggests,
since there are cases of sensitive induction and cases of insensitive
induction for backtracking and non-backtracking interpretations.
Thus, sensitivity accounts of knowledge do not face the problem of
precluding us from any inductive knowledge, as the orthodox view
suggests, nor is it true that induction typically provides sensitive
beliefs, as Wallbridge argues. Rather, some processes of induction yield
sensitive beliefs whereas some very similar processes yield insensitive
beliefs. Given this heterogeneous outcome, I do not see how a
sensitivity account of knowledge can plausibly integrate a theory of
inductive knowledge.
At this point, adherents of sensitivity might stick to their guns and
claim that the acquired results about inductive knowledge are correct,
since a sensitivity account of knowledge is correct, even though these
results seem implausible at first sight. Nozick (1981) himself frequently
endorses a similar line of argumentation, as when he argues that
knowledge does not transmit via conjunction elimination, a principle
that is highly plausible. However, such lines of
argumentation are usually regarded as a vice of Nozick’s account rather
than virtue. Even adherents of sensitivity usually do not choose this
strategy when defending sensitivity accounts of knowledge. For example,
DeRose (1995)
and Roush (2005)
develop sensitivity accounts that avoid Nozick’s implausible
consequences of closure failure and Adams and Clarke (2005) defend
Nozick’s account against Kripke’s (2011) objection by arguing that in
Kripke’s particular case knowledge closure is not violated. None of
these defenses of sensitivity simply claim that the reductio arguments
against sensitivity accounts fail because their highly counterintuitive
consequences are the correct ones. This strategy is not more plausible
in the case of induction.
The state of the discussion about sensitivity and induction has
evolved as follows. Sosa and Vogel started the discussion by arguing
that sensitivity precludes us from any kind of inductive knowledge, or
at least from paradigmatic instances of inductive knowledge. Wallbridge
objected that inductive beliefs are typically sensitive, providing a
rejoinder to the cases presented by Sosa and Vogel. We have seen that
neither of these positions is correct, pointing out instead that the
relationship between sensitivity and induction is actually quite
heterogeneous.
Interestingly, this development resembles the development of the
discussion concerning sensitivity and higher-level knowledge, another
purported challenge to sensitivity accounts of knowledge. Sosa (1999) and Vogel (2000) pointed
out that one’s beliefs that one does not falsely believe that \(p\) are insensitive. From this, Vogel
concludes that sensitivity accounts of knowledge preclude us from any
kind of higher-level knowledge while Sosa argues that this fact leads to
implausible instances of closure failure since one can know that \(p\) without knowing that one does not
falsely believe that \(p\). Becker (2007) and
Salerno (2010)
respond to these concerns, pointing out that beliefs in weaker
propositions with the formal structure \(\lnot\)(B(\(p\)) \(\wedge
\lnot p\)) are insensitive but beliefs in the stronger
propositions with the formal structure B(\(p\)) \(\wedge
p\) or B(\(p\)) \(\wedge \lnot \lnot p\) can be sensitive.
They conclude that we can have the relevant kind of higher-level
knowledge according to sensitivity accounts. In Melchior (2015), I argue that the
outcome that we know stronger higher-level propositions but fail to know
weaker higher-level propositions is too heterogeneous to be plausible,
calling this the heterogeneity problem for sensitivity accounts. Sensitivity does not preclude us
from all inductive knowledge, nor does every instance of induction yield
sensitive beliefs. In fact, some instances of induction yield sensitive
beliefs, but very similar processes of induction lead to insensitive
beliefs. We face a further instance of the heterogeneity problem for
sensitivity accounts of knowledge when it comes to sensitivity and
induction. This supports the view that heterogeneity, along different
dimensions, is a characteristic feature of sensitivity and a more
systematic problem for sensitivity accounts of knowledge.
Conclusion
The orthodox view about sensitivity and induction has it that
induction always delivers insensitive beliefs. Critics conclude that
sensitivity accounts of knowledge are mistaken. Adherents of sensitivity
accounts also assume that induction is homogeneous with respect to
sensitivity. Becker accepts that any instance of induction is
insensitive but argues that we still can have knowledge about the
probability of the target proposition via induction. Wallbridge, in
contrast, claims that induction yields sensitive beliefs. A careful
analysis reveals more differentiated results. Some instances of
induction yield sensitive beliefs but some instances in the neighborhood
yield insensitive ones, regardless of whether we interpret
counterfactuals as backtracking or non-backtracking. Sensitivity
accounts of knowledge must, therefore, accept that we can know in some
instances of induction but in very similar ones we cannot, although the
epistemic situations of the believing subjects are intuitively equally
good. These results are too heterogeneous to provide a plausible picture
of inductive knowledge in terms of sensitivity.
