ANALYSIS 63.4, October 2003, pp. 000–000. © Peter Milne
The simplest Lewisstyle triviality proof yet?
P
ETER
M
ILNE
In his celebrated ‘Probabilities of conditionals and conditional probabili
ties’ David Lewis showed that the identification of the probability of con
ditionals
a
⇒
c
with the conditional probability of consequent given ante
cedent when that antecedent has nonzero probability,
i.e. for all
a
and
c
,
P
(
a
⇒
c
) =
P
(
c

a
), when
P
(
a
) > 0,
trivializes the probability distribution in question. Lewis presented three
triviality results:
LEWIS 1
If
P
(
a
c
) > 0 and
P
(
a
~
c
) > 0 then
P
(
c

a
) =
P
(
c
);
LEWIS 2
P
assigns nonzero probabilities to at most two of any set
of pairwise inconsistent propositions;
LEWIS 3
P
takes at most four values.
The premisses for Lewis’s results are
(i)
the probability expansion rules,
P
(
a
⇒
c
) =
P
((
a
⇒
c
)
b
) +
P
((
a
⇒
c
)
~
b
)
=
P
(
a
⇒
c

b
)
P
(
b
) +
P
(
a
⇒
c
~
b
)
P
(~
b
),
the first holding generally, the second when 0 <
P
(
b
) < 1;
(ii) the family of probability distributions satisfying the identification
of conditional probability and probability of conditional is closed
under conditionalization, so that
P
(
a
⇒
c

b
) =
P
(
c

a
b
) when
P
(
a
b
) > 0.
Lewis used the second expansion rule, expanding the probability of the
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 Fall '06
 FITELSON
 Conditional Probability, Probability, The Lottery, Probability space, David Lewis

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