A Proposed Proof of the Validity of Induction
Mark Andrews
Abstract: The validity of induction may be shown by first
assuming its invalidity, then by showing that this
assumption reduces to absurdity. The assumption that
inductive reasoning is invalid requires a conclusion that
violates the law of identity, because the assumption leads
to the conclusion that something is not what it is.
Induction permits the rejection of predictions that
are contrary to events in the past. The assumption that the
future will resemble the past is unneeded. The only
necessary premise is the law of identity.
A. Introduction. I have not found a discussion of
induction that attempts a proof by denying its validity and
then looking for an absurd result.
This brief essay is my
offer.
B.1. Proposed proof. Begin by denying the predictive
value of inductive reasoning: occurrence P does not
necessarily produce result Q in every case.
1. Assume that (P  Q) is valid randomly. At any given time
there exists some possibility, greater than zero, that P
will occur but Q will not. That event is (P  not-Q).
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Call this occurrence Event Y at time Tn, where N is any
random number.
2. At time Tx, P occurs and Q results. Call this Event X.
3. Event X recurs.
4. Time Tx approaches Tn. Eventually the distance (Δt)
between Tx and Tn becomes infinitesimally small. Thus Δt
reduces to zero, because (N minus (N times 0.999…)) is
equivalent to (N minus N), which is equal to zero.
5. When Δt is equal to zero, times Tx and Tn are identical.
Events X and Y occur at the same time. Thus the
following is true: (P  Q) and (P  not-Q).
6. Thus: (not-P or Q) and (not-P or not-Q). Equivalence.
7. Thus: (not-P or (Q and not-Q)). Distribution.
8. Thus: (P  (Q and not-Q)). Equivalence.
9. Thus: (not-P). Reduction to absurdity.
10. Thus: it is not true that occurrence P will not
produce result Q in every case.
B.2. Proposed proof, restated. Begin by denying the
predictive value of inductive reasoning: although (P  Q)
might be true at the first (N minus 1) observations, the
contrary result, (P  not-Q), might occur at observation
N. This contrary result can never be predicted in advance.
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Thus inductive reasoning produces no conclusion. However,
this argument also reduces to absurdity.
1. Begin as follows: (P  Q) might be true on the first (N
minus 1) observations, but (P  not-Q) might still be
true on the Nth observation.
2. As N extends into infinity, the ratio ((N-1)/N)
approaches 1. When ((N-1)/N) = 0.999…, this ratio is
equal to 1. When the ratio equals 1, then the event at
time N and the event at time N-1 occur at the same time.
Thus: the following is true: (P  Q) and (P  not-Q).
3. Thus: (not-P or Q) and (not-P or not-Q). Equivalence.
4. Thus: (not-P or (Q and not-Q)). Distribution.
5. Thus: (P  (Q and not-Q)). Equivalence.
6. Thus: (not-P). Reduction to absurdity.
7. Thus: it is not true that occurrence P will not produce
result Q on the Nth observation.
C. Discussion.
Induction does not permit affirmative proof in the
manner of deductive reasoning. Rather, inductive reasoning
rules out the possibility that future behavior might
contradict behavior observed in the past. This difference
between induction and deduction eliminates the need for an
assumption that inductive reasoning must assume that future
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behavior will resemble past behavior. The only premise
necessary is the law of identity, that something cannot be
what it is not.
David Hume argues that it is impossible to make
predictions based an event never observed before, and about
which nothing is known. He gives the example of a game of
billiards, and the attempt to predict the motion of the
object ball given the motion of the cue ball. Hume asks
what could be predicted “[w]ere any object presented to us,
and were we required to pronounce concerning the effect,
which will result from it, without consulting past
observation…?”.1
All these suppositions [about the expected result] are
consistent and conceivable. Why then should we give the
preference to one, which is no more consistent or
conceivable than the rest?2
In such a case, where there is an absolute absence of
information, the prediction is truly random in relation to
the future event.
But things change immediately. Once the cue ball
strikes the object ball for the first time, there is one
1
David Hume, An Enquiry Concerning Human Understanding.
Section IV, Sceptical Doubts Concerning the Operations of the
Understanding, Part I, §25 (Gutenberg Project,
https://www.gutenberg.org/files/9662/9662-h/9662-h.htm#section4).
2 Id.
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observation of behavior. Whatever the second shot might be,
predictions about it cannot contradict the observed
behavior of the first. The one observation circumscribes
the range of valid predictions. As the number of
observations increases, the range of valid predictions
narrows.
This range narrows because the process of elimination
is itself valid. If one particular shot of the cue ball
caused behavior inconsistent with earlier observations,
that result would mean that something that is a part of the
new event is different from what it was in the past. That
something might be the cue ball, the object ball, the cue,
or the table. But something must explain the contrary
result, or else the new observation requires the conclusion
that something in the process is not what it is.
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