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# Rationality And Inconsistent Beliefs Essay

3308 words - 13 pages

Many believe that there is something inherently irrational about accepting each element
of an inconsistent set of propositions. However, arguments for this doctrine seem lacking
other than those that appeal to the principle that the set of propositions that one rationally
accepts is (or should be) closed under logical consequences, or those that note that error
is made inevitable when one accepts an inconsistent set. After explaining why the
preceding sorts of arguments do not succeed, I consider a novel attempt by Keith Lehrer
to undermine the chief argument in favor of the claim that it can sometimes be rational to
accept inconsistent sets. For reasons that will be described, Lehrer’s argument fails.
I. Inconsistency and Deductive Closure
One cannot accept both that it is rational to accept inconsistent sets, and that the set of
propositions that one rationally accepts is closed under logical consequences. Together
these two propositions imply that it is rational to knowingly accept a logically
contradictory statement. But clearly it is not rational to knowingly accept a contradiction.
Thus, we must give up the principle that our rational acceptances are closed under logical
consequences, or else deny that it is ever rational to accept an inconsistent set. This
dilemma is sometimes appealed to as a premise in an argument for the claim that it is
irrational to accept each element of an inconsistent set. According to this argument, since
our rational acceptances are closed under logical consequences, it must be irrational to
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Rationality and Inconsistent Beliefs
2
accept inconsistent sets. Versions of this argument have recently been offered by Ryan
(1996) and Evnine (1999).
The preceding sort of argument is unacceptable because our rational acceptances
are not closed under logical consequences. The conjunction of a set of propositions that
are each individually well confirmed (and thereby rational to accept) need not be well
confirmed. So assuming that it is rational to accept propositions whose probability is less
than one, it follows that our rational acceptances are not closed under logical
consequences, or that it is rational to accept propositions that are not well confirmed.
Clearly, it is not rational to accept propositions that are not well confirmed. It follows
that our rational acceptances are not closed under logical consequences.
Deductive closure has also been appealed to in making the argument that in cases
where each element of an inconsistent set is probable, the elements of the set interact and
defeat one another. For example, John Pollock appeals to the claim that our set of
rational acceptances are closed under logical consequences in order to defend what he
calls “the principle of collective defeat”.(62) Pollock calls a set minimal inconsistent
provided that it is inconsistent, but has no inconsistent proper subsets. The principle of
collective defeat states that the least well...

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