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Friday, September 27, 2013

The Structure of Arguments: Deduction, Induction and Abduction

So this is a brief overview of three types of logical arguments.


So most folks who have ever taken a course in logic of seen the classical syllogism. The structure typically involves a Major Premise, a Minor Premise and a Conclusion. One common syllogism is of the following form:
Major Premise: All S are P.
Minor Premise: x is S.
Conclusion: Therefore, x is P

This argument is a valid deductive argument. What that means is that, given true premises, the conclusion is also true.

This does not, however, mean that the argument is sound. An argument is sound provided that (1) it is valid and (2) the premises are true. You can have valid arguments which are not sound. Consider the following argument:
Major Premise: All odd numbers are prime.
Minor Premise: 3 is an odd number.
Conclusion: Therefore, 3 is prime.
This is a valid deductive argument. It even has a true conclusion since 3 is, in fact, prime.  But the major premise is actually false since, for example, 9 is an odd number that is not prime.

Contrast that with the following argument:
Major Premise: All even numbers greater than 2 are not prime.
Minor Premise: 4 is an even number greater than 2.
Conclusion: Therefore, 4 is not prime.
This argument is valid. But it is also sound since the premises are all true.

There are many more types of valid deductive arguments. Deduction is not, however, my main focus here.


So next up we have induction. In terms of the syllogism, we'll be moving the statements around a bit.
Major Premise: x is S.
Minor Premise: x is P.
Conclusion: Therefore, All S are P.
The old "Major premise" becomes the conclusion of the argument. The "Minor Premise" becomes the new Major Premise. And the former "Conclusion" becomes the new minor premise.

There are two questions that ought to be asked here:
  1. Why can't "x is P" be the major premise?
  2. Why can't we (also) conclude, "All P are S"?
Induction involves taking a (random) sample from a population and then characterizing it. The Major premise in this case tells us from which population we sampled from. The Minor Premise tells us what property we are measuring within that sample.

While one could make an argument with only one observation, it's better to think of "x" as a sample, a collection of observations.

This argument is not deductively valid. In other words, even if you have true premises, that doesn't guarantee that the conclusion is true. So these arguments are fallible; the conclusion could still be wrong.

One way to improve the confidence we have in the conclusion would be to take larger samples or to perform the experiment over many times (assumably with different samples).

Ideally, they should be random samples, an idea that goes back to Ronald Fisher. (For a good book on the history of statistics, I highly recommend The Lady Tasting Tea.)

There's a related problem here known as the Raven Paradox, proposed by Carl Hempel. I believe the way that I have characterized induction, one in which we consider from which population we sampled from, addresses these concerns. To be clear, that's not the only way to address it but it is a way (and one I adhere to).


The last type of argument is known as abduction. This form of argument is probably the most ignored in epistemological discussions in spite of the fact that it has great importance in actual practice (its importance may even exceed induction and deduction but that's a debate for another place.)

So we're going to swap some statements around again. Here's the form:
Major Premise: x is P.
Minor Premise: All S are P.
Conclusion: Therefore, x is S.
Now, I just rotated the statements around like I did before. But I'm not sure if there's any reason to privilege the major over the minor premise in this case.

The abductive argument begins by noting that something has a property P. Then we note that all S are P. We then conclude (or my preference, posit) that it must be S. To state it another way, if we posit that it is S, it would explain why it is P.

This type of reasoning is very common in the sciences. But it's also common in diagnostics and troubleshooting. If you're computer isn't working correctly, for example, you begin to collect a variety of hypotheses which, if granted true, explain why you're computer isn't working. The goal is to find the best hypothesis which explains all of the various information.

Previously, I mentioned that abduction is commonly used in diagnostic medicine as its exemplified in Dr. House and Abductive Reasoning.

Like induction, this argument is fallible. The truth of the premises does not guarantee the truth of the conclusion. But we can improve our confidence by collecting more information. For example, we might know that "all S are M". Then we can test whether or not "x is M". If the test fails (e.g. "x is not M"), then we rule out our hypothesis. Otherwise we "confirm" our hypothesis. But this confirmation is still fallible; it could still turn out that our hypothesis is incorrect.

Quick Summary

There is more to science and research than induction and deduction. Abduction plays a very important role in how we engage in our world and it's frequently ignored (perhaps due to ignorance). This is an attempt to increase awareness on this other type of reasoning.

... well, assuming it is a distinctive type of reasoning. There's some debate over that (or whether it's equivalent to some form of Bayesian analysis). But that's outside the scope of this initial post.

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