Inductively Strong Arguments
We have already seen that an argument is valid just in case it's impossible
for the premise(s) to be true and the conclusion false. By contrast, we
noted that an argument is inductively strong just in case the conclusion
is likely to be true given that the premises are true. Armed with the notion
of conditional probability, we can now try to be more precise:
An argument is inductively strong just in case the probability that
the conclusion is true, given the premise(s), is greater than that of its negation,
given the premise(s).
That is, if P1,....Pn are the premises and C is the
conclusion, then the argument is inductively strong if and only if Pr(C|P1,...Pn)>Pr(-C|P1,...Pn),
or, which is the same, Pr(C|P1,...Pn)>1/2.
Notice that while being deductively valid does not admit of degrees, being
inductively strong does.
So, it turns out that by doing some probability we have done some inductive
logic. For example, in the case of the dreaded disease and the test
for it, we showed that the probability of not having the very rare
disease given that one tested positive for it was much greater than the
probability of having the very rare disease given than one tested positive
for it (1001/1002 is much greater than 1/1002). Or, in the Monty
Hall case, we found out that the probability that the car is behind door
C (which we didn't originally choose) given that behind another
door B there's a goat is greater than the probability that the car is behind
door A (which we originally chose) given that behind B there's a goat (2/3
is greater than 1/3).
As you already know, conditional probability is influenced by base (prior) probabilities.
That is, Pr(A|B) is influenced by [Pr(A)/Pr(B)]. So, it's necessary to be
careful.