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The Foundations of Skeptical Theism:
On the Importance of Being Sensitive1
Stephen Wykstra
Calvin College
Timothy Perrine
Calvin College / Indiana University
PART I: A BAYESIAN DEFENSE OF CORNEA AGAINST MCBRAYER CASES
Against inductive arguments from evil, so-called “skeptical theists” like to
ask: “But if the God of traditional theism were to exist, are not features of evil
pretty much what one should expect? And if that’s so, how can they count
against theism?” But to have teeth, this question must rest on some general
principle. One proposal is CORNEA—the Condition of ReasoNable Epistemic
Access. Recently, however, Justin McBrayer2 notes that CORNEA is close kin3
to the sensitivity constraints found in Nozick’s truth-tracking accounts of
knowledge and evidence.4 Many think Nozickean sensitivity falls to
counterexamples involving—especially—inductive knowledge.5 CORNEA,
McBrayer thinks, falls to similar counterexamples.6
Our first aim here is to show that McBrayer-type counterexamples do not
bring CORNEA down. Our second aim is to illuminate what gives them their
apparent force, by examining the relationship between counterfactual
1
We have benefitted from comments by many readers on ancestors of this paper. These include:
the participants—including Justin McBrayer—in two of Jon Kvanvig’s earlier annual Baylor
Research Conferences; the students in my 2008 and 2009 Modal Logic Interim, especially Ru
Ye, Lu Teng, Luis Oliveira, and C.J. Majeski; all of the members of the Tuesday Departmental
Colloquia at Calvin (but especially Del Ratzsch and Greg Mellema); the criticisms of Glenn
Ross and Paul Draper at a 2010 Pacific Division APA Symposium; and several discussions by
“Alvin and the Chipmunks”—a intermittent gathering of dedicated Calvin philosophy majors
(including Emi Okayasu, Noah Cawley, Eddy Chen, Scott Wolcott, Chad McIntosh, Cameron
Gibbs, and Joshua Schendel) and Alvin Plantinga.
2 Justin P. McBrayer, “CORNEA and Inductive Evidence,” Faith and Philosophy. 26.1 2009, p7787.
3 This kinship was noted early on by Richard Swinburne. See his 1987 article “Does Theism Need
a Theodicy?” Canadian Journal of Philosophy 1987.
4 McBrayer has in mind Nozick’s truth-tracking account of knowledge (Robert Nozick,
Philosophical Explanations, Belknap Press, Cambridge; 1981) pp. 172 ff. It is worth noting
that in addition, Nozick also offers necessary condition on evidence that virtually identical with
this test: pp. 248 ff.
5 See Jonathan Vogel’s essay “Tracking, Closure, and Inductive Knowledge,” in The Possibility of
Knowledge, ed. Steven Luper-Foy. (Rowman & Littlefield, Totowa, New Jersey; 1989). esp.
204-13. There are, however, signs of a new ground-swell of resistance against the charge that
Nozickean sensitivity cannot handle inductive evidence.
6
McBrayer, “CORNEA and Inductive Evidence,” p. 77.
conditionals and conditional probabilities. CORNEA expresses its core
constraint using English “were… would” sentence: evidence E can’t be strong
evidence against H, because if H were false, E would still be about what one
should expect.7 Now such “were…would” sentences often express
“counterfactual conditionals,” which many philosophers (including McBrayer)
interpret using a Lewis-Stalnaker closest-possible-worlds semantics. But
“were…would” sentences are also used to express conditional probabilities. (“If
Special Creationism were true, it is unlikely that pandas would have a jerryrigged “thumb.”) This suggests that there are two ways of interpreting
CORNEA. As friends of CORNEA, we’ll of course plump for the interpretation
on which it fares better. But why do the two interpretations fare differently? We
seek to track this mystery into its lair.
1. CORNEA AND NOZICKEAN SENSITIVITY
1.1. WHAT CORNEA IS A CONSTRAINT ON
CORNEA proposes a necessary condition for regarding E as “levering
evidence”—for or against some hypothesis H.8 The notion of “levering
evidence” is here meant to reflect the type of evidence that Rowe-style
arguments9 purport to provide against theism.10 Rowe-style arguments seek to
adduce evidence that that has two features.
First, it is dynamic. A Rowe-style argument seeks to show that the
probability or credibility of theism is changed (changed, of course, downwards)
by this evidence—by, that is adding those facts of evil to our previous evidence,
our background information. Second, it seeks to show the probability is
changed very sharply. To give this content, we (following Wykstra 1996)
distinguish between three “square” doxastic states: square disbelief, square
suspended belief (the “agnostic” state), and square belief. A “sharp” decrease is
from one square doxastic state to another square state. We also adopt the
simplifying assumption11 that “squarely suspended belief” can be associated
“E counts as evidence against H only if it’s the case that were H false, E would likely be
different than it is.”
8 As will be discussed below, the evidence is serious evidence against the hypothesis if it
“seriously supports” the denial of the hypothesis.
9 For both a survey and last phase of the Rowe-Wykstra dialectic, see Rowe’s “Friendly Atheism,
Skeptical Theism, and the Problem of Evil,” in International Journal for the Philosophy of
Religion (2006), 59: 79-92; and Wykstra’s “Suffering, Evidence, and Analogy: Noseeum
Arguments versus Skeptical Gambits,” in Ryan Nichols, Nicholas D. Smith, and Fred Miller
(eds.), Philosophy Through Science Fiction: A Coursebook with Readings, Routledge, 2009.
10 It could also be called “Rowean” evidence.
11 This is, we stress, a simplifying assumption, in which the term “associate” is used advisedly.
On one hand, identifying probabilistic thresholds with belief and disbelief leads quickly into
familiar lottery paradoxes. On the other, as Rowe argues, “belief” can be used in a weaker sense
7
2
with giving the proposition a probability or degree of credulity of around .5, and
square belief and disbelief as associated with probabilities fairly close to 1 (say,
.99 or greater) or fairly close to 0 (say, .01 or less). 12
1.2. WHAT CORNEA PROPOSES
Rowe finds levering evidence against theism in that much suffering in the
world serves no God-justifying purpose that we are able to see.13 Consider,
Rowe says, a fawn burned in a forest fire, lying suffering for days. Try as we
may, we see no God-justifying good served by this suffering. Rowe argues that
this “noseeum” feature (as Wykstra terms it) entitles one to claim that there
appears to be no such good served by the fawn’s suffering, and by way of this,
is levering evidence against theism. Wykstra’ argued that this is rather like
supposing that seeing (by naked eye inspection) no HIV-viruses on a
hypodermic needle entitles one to claim that there appear to be no HIV-viruses
on the needle. Here enters the earliest version of his “Condition of Reasonable
Epistemic Access” (CORNEA):
On the basis of cognized situation s, human H is entitled to
claim ‘It appears that p’ only if it is reasonable for H to
believe that, given her cognitive faculties and the use she has
made of them, if p were not the case, s would likely be
different14 than it is in some way discernible by her.
CORNEA thus says that evidence can be reasonably deemed to be levering
evidence only if it satisfies the key constraint here put in bold-face font.15
compatible with a probabilty something like ‘.7.’ The particular illumination we offer will, we
hope, be sustainable however these tricky issues are eventually negotiated. In an earlier version
of this paper given at the 2010 Pacific Division APA meeting, criticism by Glenn Ross was here
helpful to us.
12 Levering, in this way, is by no means an unrealistically high bar for evidence to meet. The
simplest testimonial evidence is, for example, able to exceed this bar without difficulty. A
colleague informing me that he has three children properly levers me to believing that he has
three children, even though I was in a state of square agnosticism prior to that.
13 Here a “God-justifying good” is some good that is outweighs the evil in question,such that the
obtaining of that good requires an all-powerful being to allow this or some comparably bad evil.
Wykstra 1984/1991, p. 140. Wykstra’s term has been widely adopted.
14 This formulation requires some Chisholming to handle counterexamples noted by Del Ratzsch.
Consider the case of a bomb squad worker who snips the right wire on a ticking bomb in the last
seconds, and when it does not go off, sighs “It appears that (p) I deactivated the bomb.” His If p
were false, his sensory input would not be different in any way discernible to him. The main
revision will change the words “would likely be different” to “would not likely be the same.”
15 Rowe 1979 initially put his argument in the “appears” idiom, and the initial 1984 formulation of
CORNEA was put similarly. Rowe 1986 (?) shifted to a plain inductive formulation, and in
Wykstra 1987 morphed CORNEA accordingly. For fuller treatment of the shifts see Wykstra
1996 and Wykstra 2009.
1.3. THE CORE OF CORNEA
CORNEA is a condition on when someone is reasonably entitled to regard
certain data or some cognized situation as levering evidence for some
hypothesis. But as McBrayer notes, CORNEA at its core seems to reflect a
deeper restriction “on when any ‘cognized situation’ counts as [levering]
evidence.”16 This constraint, arguably at the core of CORNEA, can thus be
called CORE17:
(CORE): Evidence E is levering evidence for hypothesis H only if, were
H false, E would likely be different.
Putting CORNEA and its CORE side by side:
CORNEA
CORE
When in cognitive situation S getting new
input E, person p is, on the basis of E,
entitled to claim that “E is Rowean evidence
for H” only if: it is reasonable for p to
believe that: if H were false, then one
would not likely get E in the situation.
In cognitive situation S giving new input
E, E is Rowean evidence for H only if it is
the case that: if H were false, then one
would not likely get E in the situation.
2. MCBRAYER’S FIRST LOTTO ARGUMENT (FLA)
McBrayer offers three scenarios18 in which which—he claims—CORE
entails that a cognizer lacks evidence, even though our best epistemic
intuitions say he has evidence. He focuses especially on his lottery case; so
shall we.19
2.1. THE FIRST LOTTO STORY & ARGUMENT
3. MCBRAYER’S SECOND LOTTO ARGUMENT (SLA): TWO CONSTRUALS
McBrayer, “CORNEA and Inductive Evidence,” p. 81.
With special thoughts of Matt Haltemann and Andrew Chignell, we advance CORE as an
acronym for Condition On leveRing Evidence.
18 The Lotto case has several versions which we explore here. His second counterexample
involves an inductive inference that his son is napping; the third involves an inductive inference
that all ravens are black.
19 He suggests that readers can easily see that the other two can be similarly developed in the same
way. We make the same suggestion regarding flaws in his first counterexample.
16
17
3
3.1. THE SECOND LOTTO ARGUMENT
Here is McBrayer’s his Second Lotto Story20:
[Holt] is given a lottery ticket in ignorance of how many
tickets are sold. Perhaps Holt has the only ticket, or perhaps
there are a million tickets. Being rational, Holt withholds
belief concerning the proposition that he will win the lottery.
Later Holt learns that the odds of winning are one in a
million. Based on this new information, Holt disbelieves that
he will win the lottery.
McBrayer then argues:
[Holt’s] cognitive situation in this case warrants a revision
from non-belief to disbelief. The evidence is therefore
levering evidence. However, it remains irrational for [Holt]
to believe the required subjunctive conditional: he knows
full well that [his] cognitive situation would be exactly the
same in the closest world in which he wins the lottery. And
so according to CORNEA, his cognitive situation does not
warrant the revision from non-belief to disbelief. [But it
does.] And so even on the restricted reading, CORNEA is
false.
We can put McBrayer’s Second Lotto Argument this way:
(SLA 1) If CORE is right, then in the Second Lotto Story, Holt does
not have levering evidence that his ticket is a losing ticket.
(SLA 2) In the Second Lottery Story, Holt does have levering
evidence that his ticket is a losing ticking.
(SLA 3) So CORE is false.
3.2. THE NATURAL CONSTRUAL OF THE STORY
Notice, in particular, how McBrayer describes the initial doxastic state:
I am given a lottery ticket in ignorance of how many tickets
are sold. Perhaps I have the only ticket, or perhaps there are
a million tickets. Being rational, I withhold belief
concerning the proposition that I will win the lottery.
Here we can read McBrayer’s word “perhaps” in two very different ways.
On one construal—the natural one, we think—the two “perhaps” clauses are
merely illustrative. The subject knows he holds a lottery ticket, but has no idea
20
Our rendition below adjusts McBrayer’s wording so as to incorporate, in brackets, our
preference for formulation in third-person rather than first- person
how large the lottery is: it may be a one-ticket lottery, or a two ticket one, or a
three-ticket one… up to a one-million ticket lottery.
But construed in this way, the story by no means warrants an initial state of
“square non-belief” 21 A state of square non-belief, we saw, is a confidence level
associated with a probability of around .5. And in the story as now told, the
probability of holding a losing ticket is not around .5, but rather around 0.999
986.22 The ticket-holder begins in a state of square belief that his ticket is a
losing ticket, and square disbelief that he holds a winning ticket. And his new
information—that the ticket comes from a million ticket lottery—only raises the
probability that his ticket is a losing ticket from .999 986 to .999999. This
means that SLA-2, claiming that Holt has levering evidence, is false.23 On the
natural construal of the story, the Second Lottery Argument thus fails.
3.3. THE ARTIFICIAL CONSTRUAL OF THE STORY
21
Instead of specifying a situation of sheer ignorance, we could set up the scenario as one where
Holt has been given a ticket that he knows has been randomly chosen from one of a million
distinct lotteries, one of which is a one-ticket lottery, another a two-ticket lottery, and so on.
This will give our argument in the next paragraph better traction. When we instead specify the
situation as one of sheer ignorance, as here, it might be plausibly objected that the ticket-holder
cannot meaningfully associate any probability at all with the proposition that the ticket in hand is
a winning ticket: the proposition may best be regarded as neither having a high, middle, or low
probability, but instead as being indeterminately probable, or “aprobable.” It will not then seem
to not count as a case of square non-belief in Wykstra’s sense. Indeed, Glenn Ross [and Harry
Plantinga] have suggested exactly this line of thought: that Holt should be in a state of nonbelief regarding [the probability of] whether he holds a winning ticket. On that understanding,
however, neither CORE nor CORNEA would apply to the scenario, so it again affords no
counterexample against them.
22 It’s easiest to see this if we imagine Holt knowing there are two million possible lotteries,
ranging incrementally from 1-ticket lottery to a 2-million ticket lottery, and knowing his ticket
comes from one of them. The probability that the ticket he holds is a losing ticket would simply
be a summation of weighted probabilities, namely, for each lottery in the series, the conditional
probability that a ticket is a losing ticket given that it is from that lottery, multiplied by the
probability that it is from that lottery. The probabilities here are done for the denial of the
hypothesis: The weighted summation is (1/1 * 10 -6) + (1/2 * 10 -6) + (1/3 * 10 -6) ...+ (1/10 -6)
* 10 -6). This is equal to [1/1 +1/2 + 1/3 + (1/10 -6 ]* 10 -6 . T he series in brackets, [1/1 +1/2 +
1/3 + 1/10 -6 ], is equal to about 14.39. The weighted summation is then 14.09 / 10 6 If the
terminal number is not one million but two million, then the series sums to about 15.08, and the
weighted summation is 15.08 / 2 * 10 -6 . We thank Harry Plantinga in Calvin’s Computer
Science Department for grinding out the 14.39 and 15.08 numbers using a quickly written and
executed Pascal algorithm.
23 It also means that SLA-1 is false, for it does not come under the scope of CORE: CORE yields
no implications for the scenario on the natural construal. While CORE is compatible with the
new data not being levering evidence, it does not entail this, for it offers only a necessary
condition for something’s being levering evidence: the ticket-holder does not gain levering
evidence.
4
We can, however, make the Second Lottery Story relevant to CORE by
taking a different construal of McBrayer’s two “perhaps” clauses. The construal
is in a sense artificial; on it, we take McBrayer’s two “perhapses” not as
illustrative but as exhaustive. That is, the scenario imagines that there are only
two lotteries involved—a million ticket lottery and a single ticket lottery…
Let’s flesh this version into a story detailed enough to give our intuitions a
toe-hold.
Holt has purchased two tickets, one for a one-ticket raffle at a
Dutch church picnic, and the other from a million-ticket
Catholic raffle; he stores them in his room. He then finds one
ticket missing—his thieving brother Klep slipped into the room
and stole one of them. Holt knows this much, but he doesn’t
know which ticket Klep stole. In this initial situation, the odds
that he holds the Dutch ticket are thus 50/50. Holt’s initial state
is one of square non-belief about whether he holds the losing (or
winning) ticket.
But Holt now gets new input: Klep confesses that the ticket
he stole is the Dutch ticket. This new information properly
levers Holt from square suspended belief into squarely belief
that the ticket he holds is a losing (because Catholic) ticket.
Klep’s testimony thus qualifies as levering evidence.
Does Holt’s levering evidence in fact fail to meet CORE’s constraint on
levering evidence?
It certainly seems to fail, and this for exactly the reason McBrayer gives.
4. CORE AS A CONDITIONAL PROBABILITY : WHAT BAYES’ THEOREM
REQUIRES
But CORE’s constraint on levering evidence can also be interpreted as a
conditional-probability constraint. Our aim here is to show two things.
The first aim is to show that when the CORE requirement is so interpreted, it
is, as a requirement on levering evidence, in perfect conformity with—and is
entailed by—what is required by Bayes’ theorem.
The second aim is to show that, and further, why this CORE (and Bayesian)
requirement is indeed satisfied in the McBrayer scenario.
CORE, recall, says this:
In cognitive situation S giving new input E, E is Rowean
evidence for H only if it is the case that: if H were false, then
one would not likely get E in the situation.
If we take the bold-faced conditional as a conditional probability, then what
CORE says is that E is levering evidence for H only if E is not likely on H—that
is, only if the conditional probability of E on not-H is not above .5—that is (not
to put too fine a point on it) is below .5. 24
In cognitive situation S giving new input E, E is Rowean
evidence for H only if it is the case that: P (E / not-H and k)—
the conditional probability of E on not-H—is below .5.
Now this move in no obvious way gets CORE out of trouble. It’s not
immediately obvious that E meets the CORE Constraint, for it’s not obvious
how to think about the conditional probability of E on not-H.
To shed light on this, we view the question through the lens of a standard
application of Bayes’ theorem, in its familiar Swinburnean form:
𝑃(E/𝐻 & 𝑘)
𝑃(𝐻 /𝐸 & 𝑘) = 𝑃 (𝐻 / 𝑘) ∗ [
]
𝑃 (E/𝑘)
Here 𝑃(𝐻 /𝐸 & 𝑘)is the new probability of a hypothesis, once new evidence
E is “factored in”. This is equal to the old probability of H multiplied by a
beloved fraction that Keynes called “the relevance quotient,” and we call just
“The Quotient”:
𝑃(E/𝐻 & 𝑘)
[
]
𝑃 (E/𝑘)
The denominator, P(e/k), has an expansion:
P (E/k) = [P(E/H&k)*P(H/k)]+[P(E/~H&k) *P(~H/k)]
4.1. THE BAYESIAN REQUIREMENT : STEP ONE
In applying Bayes’ theorem to Holt’s scenario, we let “H” be the claim that
Holt has a losing ticket, and “e” be Holt’s new evidence—Klep’s confession to
stealing the Dutch ticket.
CORE says that E counts as levering evidence for H only on the condition
that if not-H were true, then E would likely be different. The condition, put as
conditional probability, is that P (E/ ~H & k) is low.
By working backwards from the hypothesis that E is levering evidence in
the SLA scenario, we first want to see what requirement Bayes’ theorem puts on
levering evidence, and compare it with the CORE requirement.
Suppose, then, that E is levering evidence for H. Evidence is levering
evidence only if it is able to boost a hypothesis from a probability of around .5
to a probability of .99 or better. Plugging these into Bayes theorem, we get
24
Our simplification dulls the point that a conditional probability that is exactly .5 or that is
indeterminate would still, technically, meet the CORE constraint.
5
𝑃(E/𝐻 & 𝑘)
. 99 = .5 ∗ [
]
𝑃 (E/𝑘)
But this shows by inspection that E’s being levering evidence for H requires
The Quotient to be nearly 2.
4.2. THE BAYESIAN REQUIREMENT: STEP TWO
But what, in turn, does this require?
We can make one more step by noting that in the SLA scenario, the
numerator of the Quotient (namely, the conditional probability of E on H & k) is
rather close to 1. H is, after all, the hypothesis that Holt holds a losing ticket.
Since our background information k tells us that the Dutch ticket is a sure
winner, H—together of course with k—entails that Holt holds the Catholic
ticket, which entails that the ticket Klep stole is the Dutch one, which entails
(given k of course) that Klep will confess to stealing the Dutch one. But this
means the Bayesian Formula now becomes
. 99 = .5 ∗ [
1
𝑃 (E/𝑘)
]
And this by inspection shows E’s being levering evidence for H requires that
The Quotient’s denominator—namely, P(E/k)—must be about .5.
4.3. THE BAYESIAN REQUIREMENT: VIOLA!
But what does this, in turn, require?
To answer this we must look closely at the expanded form of the
denominator, P(E/k):
P(E/~H&k) and .5, this requires that P (E/~H &k) be very, very low—close to
zero, comparatively speaking. 26
[ 𝑃(E/~H&k) ≅ 0 ]
In other words, E’s being levering evidence for H requires that the
conditional likelihood of E on H (along with our background knowledge) be
low. But this is just what CORE (taken as a conditional likelihood) requires.
This should renew our confidence that there is something fundamentally rightheaded about CORE.
But is this requirement satisfied in the McBrayer scenario? When treated as
a subjunctive, using standard possible world semantics, it seems not to be. But
what about when it is treated as a conditional probability?
4.4. BUT IS IT SATISFIED? THE QUOTIENT’S OBSCURE CORNER
The question, now, is whether, in the SLA scenario, P(E/ Hw & k) is indeed
low. So far we have let H stand for the hypothesis that the ticket Holt holds is a
losing ticket, and ~H stand for the hypothesis that it is a winning ticket. Let us
now re-label these as HL and Hw —keeping firmly in mind that in the scenario,
~HL is equivalent to Hw .
To evaluate whether P(E/ Hw & k) is indeed low, we must notice that there
are two ways in which Holt can possess the winning ticket: by possessing a
winning Dutch ticket, or by possessing a winning Catholic ticket. P(E/ Hw & k)
is thus a weighted average on each of these. A weighted average is of course a
sum, namely:
P(E/Hw & k) = [P(E/ Hwd & k) * P(Hwd /k)] + [P(E/ Hwc& k) * P(Hwc /k)]
𝑃(𝐸/𝑘) = 𝑃(𝐸 /H&k) ∗ 𝑃(𝐻/𝑘) + 𝑃(E/~H&k) ∗ 𝑃(~𝐻/𝑘)
In this expanded form, the denominator is a sum of two products. We can
quickly ascertain that in the SLA scenario, value of the first product—namely,
[P(e/H + k) · P(H/k)] is equal to about .5, for as we have seen, its first term is
1, and its second term is about .5. 25 And since P(H/k) is equal to about .5,
P(~H/k) will be as well.
. 5 ≅ [1 ∗ .5] + [ (𝑃(E/~H&k)) ∗ 𝑃(~𝐻/𝑘)]
.5
≅
.5 + [ 𝑷(E/~H&k) ∗ . 𝟓 ]
But this means that for E to be levering evidence, the second addend (here put
in boldface) must be very low. And since that addend is the product of
25
It is the conditional probability, on his background information, that Holt holds a non-winning
ticket, which as we have specified in the scenario is very close to .5
Clearly, for this sum to be very low requires that each addend must be very
low. So our question is now: in the actual scenario, are they both very low?
Yes, they are, but for interestingly different—indeed, opposite—reasons!
The first addend is very low because while its second factor is middling high,
the first factor is very low. The second factor is middling high because it is the
background probability that Holt holds the Dutch ticket (a sure winner), which is
.5 (as middling as one it gets). But the first factor is very low because it is the
conditional probability of getting E ( “I, Klep, stole your Dutch ticket”) on the
hypothesis that Holt holds the Dutch winning ticket.
The second addend is also very low, but for opposite reasons: while its first
factor is very high, its second factor is extremely low. The first factor is high
because it is the probability that Klep’s testimony will be that the ticket he stole
26
For the right hand side to be within some small amount of .5, P (E/~H) must be about 2Δ.
6
is the Dutch ticket, conditional on Holt’s holding the Catholic ticket: this
probability is close to 1. But the second factor is very low, for it is the
background probability that Holt holds a winning Catholic ticket, which is about
one in a two million.
The sum is low because both addends are low, though for opposite reasons.
And this means that CORE, on the conditional likelihood interpretation, does
not fall to McBrayer’s Second Lottery Argument. It does not fail because its first
premise—SLA-1—is false, due to the weighted average tucked into an obscure
corner of the Bayesian Formula.
PART II: SUBJUNCTIVES AND LIKELIHOODS—VIVA LA DIFFERENCE
CORE says or any hypothesis H and putative evidence E, coming from
cognitive input situation S, E is levering evidence for H only if a specific test
conditional is satisfied— roughly, that if H were false, one should not expect to
get E. But this Test Conditional can be interpreted in two ways:
Way 1 of putting the test conditional
(as a “conditional probability” 27 or
“inverse likelihood”):
The probability/ likelihood of
getting E in situation S, supposing
that H is false, is not high.
Way 2 of putting the test conditional
(as a subjunctive or counterfactual
conditional):
If H were false, then one would
likely not likely get E in situation
S.
Interpreted in Way 1, CORE does not fall to SLA. Interpreted in Way 2, it
seems vulnerable indeed. Friends of CORNEA could now say that Way 1 is the
right way, and walk away. But as philosophers, we are curious: why do the two
give such different results?
5. THE SUBJUNCTIVE TREATMENT OF CORE
27
The term “conditional probability” has a well-defined meaning for probability functions within
the probability calculus, but that meaning does not give one any “interpretation” of these
functions, and it is, in itself, mute as to whether such functions can be taken as properties of
collectives as von Mises thought, or can also be taken as partial entailments as Carnap thought,
and so on. I mean to be similarly mute here, and it should not be supposed that mere use of the
term ‘conditional probability’ sheds any great light on these mysteries. The term “likelihood”
was used by Keynes for inverse probabilities of the sort that figure in Bayes theorem, and I shall
thus regularly the term “conditional likelihood” (rather than “conditional probability”) as my
term of choice for the closely related conditionals in CORE and CORNEA. The terms are
interchangeable, and adopting a pet term for these crucial probabilities obviously does not (and
is not meant to) shed light on what exactly they are.
Let’s recap. Holt has purchased two lottery tickets, one for a Dutch 1-ticket
lottery and one for a Catholic 1-million-ticket lottery. Klep has pilfered one of
these tickets—Holt knows not which. Holt then gets new data E: Klep confesses
that the ticket he stole was the Dutch ticket. Our clear intuition is that E gives
Holt levering evidence that he holds a losing ticket. But CORE seems to entail
the contrary. For CORE says that E is levering evidence only if it’s true that
(TC) If Holt’s ticket were a winning ticket, then Holt’s evidence
would likely be different than it is.
But on the standard semantics, TC is—or so McBrayer persuasively argues—
just false. For when we go to the closest world where Holt holds a winning
ticket—a world where by a stroke of Lotto luck Holt’s Catholic ticket is a
winner—we find that E is still exactly as specified.
So what what’s gone awry here?
[To our readers: We’ve edited out the following sections in an effort to meet
time constraints. Here we will plunge right into the lair of the puzzle, and our
attempted solution to it.]
5.1. STARTER-THOUGHT #1: TWO CONCEPTIONS OF EVIDENCE
5.2. STARTER-THOUGHT #2: THE COAT-TAILS PRINCIPLE
6. PERRINE E-SEMANTICS AND THE DUTCH TICKET ARGUMENT
6.1. THE DUTCH TICKET ARGUMENT/EPISTEMICS
6.2. APPLYING PERRINE SEMANTICS TO HOLT’S SITUATION
7. WORRYING OUT THE PERRINE SEMANTICS
7.1. BACKGROUND: LEWIS ON CO-TENABILITY THEORIES
Consider a theory of conditionals discussed by Lewis in Counterfactuals, the
co-tenability theory.28 The theory can be taken as an account of the truth-value
of conditionals, or of their assertability conditions.29 Either way, we imagine
adding the state of affairs in the antecedent to our noetic structure, and then ask
if we would then reasonably conclude (or, regard as probable) the consequent; if
so, then the conditional is true. Of course, we cannot simply add the antecedent
to our noetic structure: we would get contradictions or probabilistic
inconsistencies. So we must “add and adjust” in an appropriate way, retaining
just those beliefs that are “co-tenable” with the antecedent, and then evaluate
whether the consequent is probable or certain.
28
29
See Counterfactuals, pp. 68-72.
Lewis considered the co-tenability account as an account of the warranted-assertability of
counterfactuals (pp. 70-1), but similar objections apply to the truth-conditions version as well.
7
Lewis thinks this theory gives wrong results for counterfactuals. He gives a
case involving, as we’ll call them, sanely moderate “Warrenites.”30 A
Warrenite is one who accepts Warren Commission’s findings that Lee Harvey
Oswald acted alone, with no group behind him, no second shooter from the
infamous “grassy knoll,” no conspiracy of any kind. A moderate Warrenite
grants that our evidence establishes this only with very high probability, leaving
some very small but not dismissible probability that some larger group was
involved. A sanely moderate Warrenites nevertheless dismiss certain logical
possibilities as, on our evidence, so wildly improbable as to be utterly beyond
the pale, (e.g. Kennedy was not killed but instead, by massive government
conspiracy, kidnapped). Now, says Lewis, consider a conditional like
If Oswald weren’t to have shot Kennedy, then Kennedy would (likely)
have served out his term.
For this, co-tenability and close-possible worlds accounts give different results,
and the co-tenability result is wrong. On it, we must take the state of affairs
represented in the antecedent (namely, Oswald’s not killing Kennedy), add it
(imaginatively) to our current beliefs about the actual world, make appropriate
adjustments, and then ask if we would be warranted in believing the state of
affairs represented in the consequent (namely, Kennedy’s not serving out his
term). The answer is not. We’d instead conclude that someone else fired the
bullets that killed JFK, so the above conditional comes out false. Yet, intuitively,
this counterfactual is clearly what we should embrace (as the closest-world
approach vindicates).
8. LEARNING FROM CO-TENABILITY
8.1. THE DIFFERENCE
A deeper lesson of Lewis’s argument is that asking how events would have
gone differently, that dark November day, had Oswald not shot JFK is not at all
the same as asking how we should appropriately revise our beliefs about that the
events of that day, were we told that Oswald did not shoot JFK after all. The
latter, however, seems to be one way of thinking about conditional likelihoods.
They express how an agent should evaluate a proposition (i.e. the consequent),
given the agent’s background beliefs and some new information or evidence (i.e.
the antecedent). Thus, suppose someone with the background beliefs of a
moderate Warrenite considers
If Oswald weren’t to have shot Kennedy, then Kennedy would (likely)
have served out his term
but considers it, not as a counterfactual, but as a conditional likelihood . Taken
this way, she’ll rank the conditional as false or very improbable. For one of her
firmest background beliefs is that JFK was shot and killed. So, on adding (in
imagination) the new information “Oswald did not shot Kennedy” and making
appropriate adjustments, she will not conclude that Kennedy would have served
out his term, but instead that there was some second shooter. A conversational
context may help us imagine this: in a group of sanely moderate Warrenites, our
moderate Warrenite will say:
Of course we firmly hold that Oswald shot Kennedy. But if it
were by to turn out true that Oswald didn’t shoot Kennedy, then
it would be true that some second shooter did so.
We thus get the following results:
as a counterfactual
If Oswald weren’t to have
shot Kennedy, then Kennedy
would (likely) have served
out his term
And
If Oswald weren’t to have
shot Kennedy, then it
(likely) would have been the
case that some second
shooter did
True
as a conditional
probability
False
as a counterfactual
as a conditional
probability
False
True
So understanding a conditional as a counterfactual or as a conditional
likelihood leads to divergences in truth value. For one, one keeps constant
certain salient features about the base world, making minimal modifications to
others. For the other, one keeps constant those propositions in the background
information that are best supported. When the salient features of the base world
are not the best-supported features of the background information, then
divergence happens.
8.2. HOLT’S TEST CONDITIONAL
Is this divergence at play in McBrayer Cases for the test conditional:
(TC) If Holt’s ticket were a winning ticket, then Holt’s evidence
would likely be different than it is?
8
Taken as a counterfactual, whether TC is true depends crucially upon which
ticket Holt actually holds. If Holt holds the winning Catholic ticket, it is false.
But if Holt holds the Dutch ticket, it is true. And this Key Option (let us call it)
directly bears on the assertability of TC for Holt. For prior to accepting Klep’s
testimony, Holt is up in the air about which ticket he actually holds. So—
assertability-wise—he cannot make any call as to whether TC is true or false.
Supposing that Holt actually does hold the winning Catholic ticket, TC
understood as a counterfactual is false; but since the proposition that he does is
not well supported by his background information (it is indeed .0000005), it has
very low warranted assertability for him.
But taken as a conditional likelihood,31 TC has, prior to Klep’s testimony,
very high assertability. To be sure, this is not (as in the JFK case) by virtue of
some single proposition that has is extremely well supported by our background
evidence. It is rather because—as we saw in Part I—the Key Option can be
handled by doing a weighted average over both possibilities. Nevertheless, the
divergence arises, arguably, because the feature of the world that determines the
falsity of TC (as a counterfactual) is not a feature best-supported on the
background evidence.
9. EVIDENTIAL CONTEXTS
McBrayer cases, we think, also get part their apparent power from conflating
two distinct evidential contexts, which we call the ‘pre-evidential context’ and
the ‘post-evidential’ context. The ‘pre-evidential context’ is the context in
which some data is being considered as potential evidence for some hypothesis
H of interest, in order to evaluate whether it is indeed evidence for H. In this
context, one applies evidential contrasts to this data to determine if it is evidence
or not. The ‘post-evidential context’ is the context in which this evaluation has
been made,32 E has been determined to be (or not be) evidence for H, and our
doxastic stance toward H has been appropriately “updated” in accord with this
evaluation. Put simply, the pre-evidential context is where data is being
evaluated; the post-evidential context is where it has been evaluated.
We can see—in retrospect—that our Bayesian analysis treated CORE in the
pre-evidential context. The crucial part of that analysis used a weighted average
“sum” to evaluate P(E/k). Since we used our (or Holt’s) background information
exclusive of E to weight the addends in that sum, we clearly kept in the preevidential context.
Readers will have to excuse the occasional fluctuation in this draft from Tim’s “conditional
probability” to Steve’s “conditional likelihood”: we use these to mean the same thing, but Steve’s
preference for the latter is recent and we haven’t had time to consistently incorporate it as our term
of choice.
32
Tim suggests “pre-evaluational” and “post-evaluational” as better terms here; I agree.
31
Using values from the post-evidential context would give a very different
result. But this is, in effect, exactly what happens in the SLA version of
McBrayer’s ‘counterexample.’ In answering the question “Would Holt’s
evidence be any different, were the hypothesis (I hold a losing ticket) false,”
McBrayer’s assumes that Holt holds a Catholic Lottery Ticket. That assumption
will ultimately be the conclusion drawn from a positive evaluation of Klep’s
testimony (Holt’s new evidence). But make use of this new evidence when
evaluating whether the test conditional is satisfied is inappropriate, in effect
begging the very question at issue.