To confirm or infirm
(support or undermine)
assumptions by showing their conclusions do (not)
conform to the observable facts.
is a related sense of "induction", namely generalization or hypothesis, but
for various reasons this is better named
The definition given above is not the standard one, for
which the reader is referred to Problems of Induction and Mill's Methods of
1. The sense of "induction" used in this
lemma rests much on probability
theory, and is related to the deductive fallacy of affirming the conclusion,
which it avoids.
First, to explain this fallacy. One basic deductively
rule of inference is modus ponens: From (A) and (A-->B), it follows that
(B). The converse of this, from (B) and (A-->B), it follows that (A), is the
fallacy of affirming the conclusion. It is easy to see this is a fallacy -
take (A)=(This is a cat) and (B)=(This is an animal) - yet obvious that this
fallacy is quite common (especially in political argumentation), and also
obvious that it conforms to an intuition that may be stated thus: From (B) and
(A-->B), it follows that (A) is more probable than it was before
learning (B). For to use the same example: Learning that (This is an animal)
at least excludes the cases that (This is a plant) etc. and thus makes the
hypothesis that (This is a cat) somewhat more probable, if not much.
Second, to show how this
intuition is supported by probability
theory. In probability theory, there is an elementary theorem that p(A&B) <=
p(A). Indeed, this follows from the theorem that p(A)=p(A&B)+p(A&~B).
by definition p(A&B)=p(B|A).p(A), where p(B|A) is the conditional probability
that B given that A. In these terms, and still using the above example, what
we are interested in is p(A|B) which is equal to p(B|A).p(A) : p(B).
supposing that indeed B follows deductively from A, p(B|A)=1 and so
p(A|B)=p(A):p(B). Since p(B)<=1, it follows on the same supposition that
p(A|B)>=p(A) - and thus we know that if we learn that (B) and know that
(A-->B) we can infer that p(A|B) is at least as great as p(A) before learning
that (B) and is greater if p(B) itself is less than 1, which is the normal
case, since we need not to explain (B) if we know it is certain to start with.
This explains in principle how we can
confirm or infirm assumptions using probability theory, and how this avoids
the deductive fallacy of confirmation, namely essentially through the
probabilistic analysis of conjunction, that is more subtle than the deductive
2. The account just given also explains in principle how one can learn from
by framing a hypothesis that accounts for the facts one wants to explain, and
then by deriving consequences from it that can be tested in experience, that
are not certainly true, and
then proceeding as above.
The reason to insert the qualification "in principle"
is that the above account is not complete, especially in the following two
(I) The account does not give an explanation or rules
to attribute probabilities to either theories or empirical predictions deduced
from them, and
(II) Probability theory by itself does not provide any means to settle
the probability of any statement that is not 0 nor 1.
For more, see: Problem of Induction