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Sunday, October 26, 2014

A Deductive Model of Inductive Reasoning

There is a widespread belief that inductive reasoning has the following two characteristics:

  1. It entails making a general or predictive claim based on past observations.
  2. The conclusion does not follow as a matter of logical necessity from the premises.

It is also commonly supposed that the second of these characteristics follows necessarily from the first.  It is believed that any time we make general and/or predictive claims based on past experiences, we are drawing a conclusion which does not follow as a matter of necessity from our premises.

I think this is incorrect.  I will take the first characteristic as a defining feature of inductive reasoning and show that the second characteristic does not obtain.  In other words, I will provide a deductive model of inductive reasoning such that (1) a general or predictive claim is based on past observations, and (~2) the conclusion follows as a matter of logical necessity from the premises.

I will take as a starting point an example from Hume:

Premise:  The sun has risen in the east every morning until now.
Conclusion: The sun will rise in the east tomorrow.

As stated, this argument is incomplete--or, rather, the full set of premises are hidden.  They can be explicitly formulated in deductive form as follows:

Premise 1:  The sun has risen in the east every morning until now.
Premise 2:  Some X causes the sun to rise in the east in the morning.
Premise 3: Unless some Y prevents X from causing the sun to rise in the east in the morning, the sun will continue to rise in the east in the morning.
Premise 4: If the sun continues to rise in the east in the morning, the sun will rise in the east tomorrow.
Premise 5: There is no Y preventing X from causing the sun to rise in the east in the morning.
CONCLUSION: The sun will rise in the east tomorrow.

This is a deductively valid argument.  Furthermore, I believe it adequately represents how inductive reasoning (of the sort indicated in Hume's example) actually occurs.

Of course, we can question the truth of any or all of the premises.  That, however, is not the point.  The point is that (1) and (~2) obtain.

The model can be generalized as follows:

P1: A has been observed to occur in condition B.
P2: Some X causes A to occur in condition B.
P3: Unless some Y prevents X from causing A to occur in B, A will continue to occur in B.
P4: There is no Y preventing X from causing A to occur in B.
CONCLUSION: A will occur in B.

It may be observed that P2 and P3 imply determinism.  P2 says that the repeat occurence of A in B is the result of a cause--it is determined by X.  P3 states that the same effects will follow from the same causes in the same conditions unless a new condition is introduced which negates the cause.  (This new condition may simply be the absence of the cause, or it may be a counter-cause.)  These premises may be questioned.  We may ask what justifies our acceptance of them.  However, such openness to questioning  does not negate the formal validity of the argument.  To say that an argument is formally valid is only to say that it is coherent and that the conclusion follows necessarily from the premises.  Inductive reasoning is, on my account, formally valid.  It is a case of deductive reasoning.  So we can accept (1) whilst rejecting (2).

The two weaknesses of inductive reasoning seem to be these:  First, we cannot be sure that the same effects will always follow the same causes in the same conditions; second, we cannot be sure that the same causes are working in the same conditions--i.e., we can never be sure that P4 holds.  We might simply be ignorant of some future Y which will prevent X from causing A to occur in B.  Again, however, these weaknesses do not affect the formal validity of the reasoning.  They only affect the soundness of particular instances of inductive reasoning.  Furthermore, one can be justified in believing that which is not certain; so the fact that we cannot be sure does not mean we cannot be justified in accepting the premises.

Update:  In response to a helpful commenter, I have more directly addressed the well-known Humean "problem of induction."  I wrote in the comments section below: "Empiricists like Hume hold that [some of the premises my deductive schema relies on] cannot be justified except inductively, which makes inductive reasoning circular. However, if we take a Quinean perspective and reject Humean empiricism--if we say that all premises, even simple observational statements, are theory-laden--then there is no simple distinction to be drawn between observational statements and the premises required for induction. So the epistemological problem is no longer a problem of induction per se; it is rather a problem of how we justify premises in general. At least, that is the sort of direction I'm leaning in."