Is consistency always epistemically virtuous? Is it possible for a set
of rational beliefs to be inconsistent? The traditional view has been that logical
consistency is a minimum requirement for rational belief. Recently, this traditional view
has been challenged, and is now in some disrepute. The issue is not only of interest in
its own right, but also has bearing on several other aspects of our conception of
justified belief. In particular, it is a critical issue for the coherence theories of
justification which have been so prominent of late, for coherence is normally understood
in such a way as to presuppose logical consistency. (1)
Three distinct lines of argument against consistency can be discerned
in the current discussion (although not always clearly distinguished there): (i) the
paradox of the lottery, (ii) the fallibility argument, the core of which is an inference
from my fallibility in the past to the claim that my present justified beliefs also
contain error, and (iii) what I call `the epistemic probability argument', which is based
on the rules of the classical probability calculus. I shall focus, in this paper, just on
what I take to be the most significant threat, the third argument. (2)
An examination of this argument will prove of interest both for its implications
concerning consistency and, as well, for what it reveals about epistemic probability.
The strength of one's evidence is surely a critical factor in assessing
warranted belief. So the epistemic probability of a proposition, understood as the degree
of confirmation or support provided by the total available evidence, would seem to be a
central notion in epistemology. Epistemic probability, of course, is to be sharply
distinguished from other types of probability, in particular from statistical probability;
the latter has to do with the relative frequency with
which a given property or event occurs within a certain reference
class. It is the concept of epistemic probability which bears directly on issues of
warranted belief. The degree of epistemic probability adequate for warranted belief is
commonly thought to be high, but less than certainty.
The argument for inconsistency I want to examine may be put informally
as follows. Consider a large set of statements which I am warranted in believing. My
evidence for each, let us suppose, is very strong, but not perfect; that is, these
statements are not certain on the basis of my evidence. So for each of my beliefs there is
some risk of error. If a number of these statements are conjoined, the risk of error
associated with the conjunction is still greater than that associated with any individual
belief. Given a sufficiently lengthy conjunction, the likelihood of error will increase to
the point that I am justified in believing that the conjunction is false. Hence, I am
justified in believing each member of the original set, as well as the negation of their
conjunction. Since this is an inconsistent set, it follows that I am justified in
believing inconsistent propositions. (3)
This is the core of the argument for the rationality of inconsistency.
The argument can be stated more formally, and more precisely, in terms of the rules of the
probability calculus. The multiplication rule for the epistemic probability of a
conjunction is:
Cr(p & q) = Cr(p) x Cr(q / p)
(where `Cr' designates degree of confirmation). It follows from this
rule that if a statement q is added as a conjunct to some statement p, the probability of
the conjunction will be less than that of p provided that Cr(p) _ 0 and Cr(q / p) < 1.
Thus, if p is a contingent statement and p does not logically imply q, then Cr (p & q)
must be less than Cr(p).
Suppose now that A1, ..., An are logically
independent contingent statements which I am justified in believing; (4)
they might be about what I ate for breakfast, what I read in the newspaper, and so on.
Their degree of confirmation is high (but less than 1). It follows from the above that the
probability of the conjunction A1 & ... & An will decline
with the addition of each conjunct; with enough conjuncts, the degree of confirmation of
the conjunction will be very low. But the probability of a statement and its negation are
complementary: Cr(~p) = 1 - Cr(p). So the degree of support for ~(A1 & ...
& An) will be very high; given sufficient components, it will be high
enough to warrant belief. Consequently, I will be justified in believing each of A1,
..., An, as well as ~(A1 & ... & An). That is, I
will be justified in holding inconsistent beliefs.
The standard presentation of this argument in the literature makes use
of the conjunction rule. It is worth noting, however, that another version of the argument
can be constructed focusing on the rule for the probability of a disjunction. The general
rule for disjunctions is:
Cr(p v q) = Cr(p) + Cr(q) - Cr (p & q)
We can thus see that if a statement q is added as a disjunct to some
statement p, the probability of the disjunction will be greater than that of p provided
that Cr(q) - Cr(p & q) > 0, which will be the case if Cr(q) > Cr(p & q). So
if p is a contingent statement and q does not logically imply p, the probability of p v q
must be greater than that of p. Now consider a large set of logically independent
contingent statements B1, ..., Bn, where each Bi is the
negation of a statement I am justified in believing. Given sufficient components, the
degree of support for the disjunction of Bi's will be high, and I may be
justified in believing it. It is thus possible that I be warranted in believing each of ~B1,
..., ~Bn, as well as B1 v ... v Bn. But this is an
inconsistent set of beliefs.
I want to show that these arguments for the rationality of inconsistent
belief are unsound. For purposes of illustration, I shall focus on the version which draws
on the multiplication rule for conjunctions. The plausibility of this argument rests, in
part, on the fact that the conclusion, ~(A1 & ... & An),
does not strike us as particularly informative or significant. My strategy will be to
apply analogous probabilistic reasoning to carefully tailored examples. The conclusions
which emerge are substantive, informative and, I shall argue, utterly implausible. To
accept the epistemic probability argument against consistency is thus to be forced to
grant absurdities.
Consider: A medical research laboratory is working on different aspects
of AIDS research. A list is prepared, extremely lengthy as it happens, of natural
medications and therapies which have at some time been claimed to have some impact on the
disease. Each item on the list is investigated by the research team as a potential cure. A1
is the claim that the first item on the list is not a natural cure for AIDS; and so
on. The total evidence amassed by the research team provides strong support for each Ai.
It follows from the now familiar reasoning that given sufficient Ai's,
the degree of confirmation of the negation, ~(A1 & ... & An),
will reach the level associated with warranted belief. Consequently, we will be warranted
in believing each of A1, ..., An, as well as the negation of their
conjunction. This much is analogous to the original argument. Note, however, that ~(A1
& ...&An) logically implies:
A: There is a natural cure for AIDS.
It follows that the degree of confirmation of A is at least as high as
that of the negation of the conjunction; and if we are warranted in accepting ~(A1
& ... & An), then clearly we are also warranted in accepting A.
But these are absurd results. Surely we are not prepared to grant that
belief in A, belief that there is a natural therapy which cures AIDS, could ever be
warranted on this basis. And it seems absurd to suggest that the results achieved
by the research team could, in principle, provide strong confirmation for A. Repeated failures
to establish that a particular natural regimen cures AIDS cannot provide strong
confirmation or adequate evidence for the positive thesis that there is a
natural cure for AIDS. The degree of support for A, it appears, is no higher than that of
the negation of the least probable Ai. The consequences of this analogous
argument, then, are simply unacceptable; and it is difficult to see how one might
nevertheless justify accepting the epistemic probability argument.
(Note that the same absurd consequences will flow from the disjunction
rule. Let B1 be the claim that the first item listed is a natural cure
for AIDS. Given sufficient components, the degree of support for the disjunction of Bi's
may reach the level associated with warranted belief. But the statement B1 v
... v Bn implies that there is a natural regimen which cures AIDS.)
The AIDS case provides a particularly striking example with which to
illustrate the absurdity of this form of reasoning; but other effective examples are
readily available. One might take each Ai to be the denial of a report that a
particular person has witnessed an event which constitutes a miracle (in which case A
would be the assertion that someone has witnessed a miracle); or each Ai could
be interpreted as the denial of a claim that a particular individual has exhibited psychic
powers (in which case A would be the claim that some individual has exhibited psychic
powers); or, finally, each Ai might be taken as asserting that a particular
claimant is not Anastasia, daughter of the Czar of Russia (in which case A would be the
assertion that a particular claimant is Anastasia). Each Ai, let us suppose,
will be strongly confirmed. But surely no one would be tempted to justify belief in A by
indicating as evidence a large set of Ai's.
What general conclusions can be drawn here? First, what has emerged is
that if the epistemic probability argument is successful in establishing the possibility
of rational inconsistent belief, then it will also be the case that strong confirmation
for the thesis that there is a natural regimen which cures AIDS (or that a miracle has
occurred) is possible, in principle, on the basis of a series of entirely negative
findings, or discredited positive claims. The consequent of this conditional is, in my
view, quite incredible. I conclude, therefore, that the antecedent must be denied, that
is, that the epistemic probability argument for inconsistency fails.
Second, the diagnosis: the argument for inconsistency fails because
epistemic probability, degree of support, does not conform to the axioms of the classical
probability calculus. In the examples cited, the degree of confirmation of each Bi
can not be combined, or added up, to produce a substantially higher degree of support for
the disjunction. Nor does the degree of confirmation of the conjunction of the Ai's
appear to follow the multiplication rule. (I leave open the possibility that these rules
may be adequate for some cases.) This is not, of course, an entirely novel conclusion.
Others have challenged the fit between the principles of the calculus and degree of
support or confirmation. (5) Still, the issue has been
controversial and far from clear cut. Relatively few philosophers have been prepared to
endorse the view that epistemic probability diverges from the calculus. The argument
advanced here , I believe, is more compelling than previous discussions; the implications
of the conjunction and disjunction rules which have been revealed here are simply
unacceptable. It is one thing to grant, as the epistemic probability argument purports to
show, that there is error in my present justified beliefs. But it is quite another to
suppose that a series of entirely negative findings may provide adequate evidence for a
substantive thesis — for instance, that some individual has exhibited psychic powers.
Is there any possibility of salvaging the disjunction and conjunction
rules of the calculus? Two objections to the analogous arguments would have this effect,
and should be considered here. First, it might be suggested that we erred in the
probability values we ascribed to the Ai's in the examples above. That is, each
Ai should be assigned a probability of 1 on the grounds that we are confident
that each Ai is true. (Similarly, each Bi should be assigned a value
of 0.) Such an assignment would ensure that the probability of the appropriate
conjunctions does not decrease. This result would be achieved, of course, only at the cost
of abandoning a standard convention of probability theory, namely, that 0 and 1 be
reserved for logically impossible and necessary statements, respectively. One unfortunate
consequence of giving up this convention is that we will no longer have any way to
distinguish the degree of confirmation of contingent statements such as the Ai's
from that of logical truths. More significantly, it is difficult to see how one could use
this option to rule out every analogous argument without assigning a value of 1 to any
statement we are clearly justified in believing (and thus a value of 0 to its negation).
But the effect of such an assignment would be the complete collapse of the argument for
the rationality of inconsistent beliefs.
Second: Another suggestion is that the rule of negation must be
rejected. More specifically, it might be proposed that the probability of a statement and
its negation need not equal 1; (6) and that the negation of
any statement we are justified in believing will have a probability of 0. This proposal
would also clearly stop the analogous arguments. The probability of the disjunction of the
Bi's would not increase given that each Bi has a probability of 0.
In the version which makes use of the conjunction rule, we could not infer a high
probability for ~(A1 & ... & An) from the low probability of
the conjunction. Of course, for the very same reasons, both versions of the argument for
the rationality of inconsistency would also fail.
Each of these proposals, it appears, blocks the analogous arguments
only to the extent that it also defeats the argument for inconsistency. So there is no way
to accept the latter without also granting the former. Further, each objection involves
rejecting some aspect of the traditional calculus. What seems clear, then, is that
epistemic probability cannot conform to the calculus as a whole. It remains a plausible
diagnosis, although not the only one possible, that the rules for conjunction and
disjunction are, at least in part, the source of the problem. (7)
Let me, in closing, briefly pursue some consequences of this diagnosis.
If the epistemic probability of conjunctions and disjunctions does not conform to the
calculus, then a natural alternate model would be that the probability of a disjunction is
equal to that of its most probable disjunct, and the probability of a conjunction is equal
to that of its least probable conjunct. (8) Certainly, this
model seems plausible in the cases discussed above. Unfortunately, it does not appear to
hold in every instance. Let me illustrate just with regard to the disjunction rule
(analogous points can be made concerning conjunction). The most obvious exception is found
in the category of logical truths. Clearly, the probability of P v ~P, for example, will
be higher than that of either of its components if neither P nor ~P has a probability of
1. But the problem also occurs with contingently true disjunctions. Suppose there are
exactly three candidates in an election, Adams, Brown and Cowley. No new names can now be
added to the list of candidates, and no candidate has a clear lead. Let A be the statement
that Adams will win; and so on. The disjunction A v B v C will have a very high degree of
confirmation, since one of the three must win; and it will be higher than any of its
disjuncts, since no candidate is highly likely to be the winner. So again the alternate
model fails, in this case for a contingent statement. It is interesting to note that what
unites these two examples is that the probability of the disjunction, in either case, is
not affected by the assumption of the negation of any disjunct. In the second case, for
instance, Cr(A v B v C) = Cr(A v B v C / ~A).
These are two clear-cut counterexamples to the new model; there may
well be others. However, it would be rash to suppose that epistemic probability is never
correctly calculated by the alternate rules. Nor, for that matter, should we infer that
the traditional rules are never accurate. What seems clear is that neither model holds
universally.
To conclude: I take the analogous arguments to show that degree of
confirmation, or epistemic probability, does not conform to the classical probability
calculus. This conclusion should have significant implications for epistemology generally.
One immediate consequence is that the epistemic probability argument for inconsistency
must fail. Thus, one threat to the traditional view of consistency is now removed.
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