Intro to Logic: Abductive Reasoning

by Luke Muehlhauser on November 19, 2009 in Intro to Logic

logic

Welcome to my course Intro to Logic (index). Here, we learn the basic skills of good thinking and their benefits in real life.

Earlier, I discussed two very different forms of argument: deductive and inductive. A deductive argument claims that if its premises are true, its conclusion must be true. An inductive argument claims that if its premises are true, its conclusion is probably true.

There is another kind of argument used by philosophers: an argument to the best explanation. Philosophers may argue that the best explanation for common moral talk is that people commonly assume moral values are absolute, or that God is the best explanation for the origins of the universe.1

A piece of Sherlock Holmes’ reasoning can illustrate:

sherlock

The portly client puffed out his chest with an appearance of some little pride and pulled a dirty and wrinkled newspaper from the inside pocket of his greatcoat. As he glanced down the advertisement column, with his head thrust forward and the paper flattened out upon his knee, I took a good look at the man and endeavored, after the fashion of my companion, to read the indications which might be presented by his dress or appearance.

I did not gain very much, however, by my inspection. Our visitor bore every mark of being an average commonplace British tradesman, obese, pompous, and slow. He wore rather baggy gray shepherd’s check trousers, a not over-clean black frock-coat, unbuttoned in the front, and a drab waistcoat with a heavy brassy Albert chain, and a square pierced bit of metal dangling down as an ornament. A frayed top-hat and a faded brown overcoat with a wrinkled velvet collar lay upon a chair beside him. Altogether, look as I would, there was nothing remarkable about the man save his blazing red head, and the expression of extreme chagrin and discontent upon his features.

Sherlock Holmes’s quick eye took in my occupation, and he shook his head with a smile as he noticed my questioning glances. “Beyond the obvious facts that he has at some time done manual labour, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else.” Mr. Jabez Wilson started up in his chair, with his forefinger upon the paper, but his eyes upon my companion.

“How, in the name of good-fortune, did you know all that, Mr. Holmes?” he asked. “How did you know, for example, that I did manual labour. It’s as true as gospel, for I began as a ship’s carpenter.”

“Your hands, my dear sir. Your right hand is quite a size larger than your left. You have worked with it, and the muscles are more developed.”

“Well, the snuff, then, and the Freemasonry?”

“I won’t insult your intelligence by telling you how I read that, especially as, rather against the strict rules of your order, you use an arc-and-compass breastpin.”

“Ah, of course, I forgot that. But the writing?”

“What else can be indicated by that right cuff so very shiny for five inches, and the left one with the smooth patch near the elbow where you rest it upon the desk?”

“Well, but China?”

“The fish that you have tattooed immediately above your right wrist could only have been done in China. I have made a small study of tattoo marks and have even contributed to the literature of the subject. That trick of staining the fishes’ scales of a delicate pink is quite peculiar to China. When, in addition, I see a Chinese coin hanging from your watch-chain, the matter becomes even more simple.”

Mr. Jabez Wilson laughed heavily. “Well, I never!” said he.2

Though impressive, Holmes is wrong to say that he deduced all that about his client. A deduction would look like this:

  1. Everyone wearing an arc-and-compass breastpin is a Freemason.
  2. Mr. Wilson is wearing an arc-and-compass breastpin.
  3. Therefore, Mr. Wilson is a Freemason.

But is this a sound argument? Surely not. It is quite plausible that many non-Freemasons throughout history have, say, purchased an arc-and-compass breastpin at a pawn shop. So premise (1) is probably false.

Holmes can more charitably be interpreted as making an argument to the best explanation. He seems to be arguing that the best explanation of Mr. Wilson’s arc-and-compass breastpin is that Mr. Wilson is a Freemason. Now it could certainly be false that Mr. Wilson is a Freemason, but given the facts we know, it does seem to be the best explanation of his arc-and-compass breastpin.

So the conclusion of an abductive argument, the supposed “best explanation,” is not as secure as the conclusion of a sound deductive argument. It is not necessarily even probable, as is the conclusion of a strong inductive argument. Rather, the conclusion of a good abductive argument is merely the best explanation we know of.

In general, an abductive argument looks like this:

  1. The surprising fact, E, is observed.
  2. But if H were true, E would be a matter of course.
  3. Hence, there is reason to suspect that H is true.

Here, E stands for the explanandum, the fact to be explained, and H stands for the hypothesis, aka the proposed explanans (explanation).3

In the breastpin example, the abductive argument could be seen as follows:

  1. The surprising fact, an arc-and-compass breastpin on Mr. Wilson, is observed.
  2. But if Mr. Wilson is a Freemason, an arc-and-compass breastpin on Mr. Wilson would be a matter of course.
  3. Hence, there is reason to suspect that Mr. Wilson is a Freemason.

Notice that if an abductive argument were taken to be a deductive argument, it would commit the fallacy of affirming the consequent. If we wrote the above abductive argument as a deductive argument instead, we would get:

  1. Mr. Wilson wears an arc-and-compass breastpin.
  2. If Mr. Wilson is a Freemason, then he would wear an arc-and-compass breastpin.
  3. Therefore, Mr. Wilson is a Freemason.

But this takes the form of

  1. P
  2. If Q then P.
  3. Therefore, Q.

And this is obviously fallacious. We might as well argue:

  1. Mr. Wilson wears underwear.
  2. If Mr. Wilson was a woman, then he would wear underwear.
  3. Therefore, Mr. Wilson is a woman.

So when making an abductive argument, we must be careful to construe it as an abductive argument, not as a deductive argument, or else we may be accused of fallacious reasoning.

Abduction is also widely used in the sciences. See Peter Lipton’s Inference to the Best Explanation.

  1. But not all such cosmological arguments for God’s existence are abductive. For example, William Lane Craig’s formulation of the Kalam cosmological argument is deductive. And, though Richard Swinburne claims in The Existence of God to offer an inductive cosmological argument, what he really offers is an abductive argument, as shown by Greg Dawes in Theism and Explanation, starting at page 102. []
  2. Arthur Conan Doyle, The Red-Headed League (1982). []
  3. C.S. Peirce, who proposed abduction, used this outline for abductive reasoning but used different letters, A and C. I have used Greg Dawes’ modification, from Theism and Explanation, page 21. []

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{ 22 comments… read them below or add one }

Alex November 19, 2009 at 6:58 am

There is one issue that you have overlooked. To determine the adequacy of an explanation, you have focused only on the probability of an otherwise surprising (improbable) observation if the proposed hypothesis is true (or P(O|H) in Bayesian-speak). But to determine whether the proposed explanation is better than some other one (i.e. to compare the values of P(H|O) for the hypotheses), you also have to take into account the intrinsic probability of the proposed hypothesis, P(H).

For example, John winning a 1-in-1,000,000 lottery would be quite improbable if the lottery was fair, but would be certain if God, who wanted John to win, intervened. So if you put all proposed explanations on the same level and compare P(O|H), you will always end up with an “explanation” that is so intrinsically improbable that it doesn’t explain anything at all.

Or as Darwin put it in the Origin (when comparing common descent with special creation, in terms of explaining certain patterns in nature), attempted explanations in terms of special creation are typically just a “restatement of the fact.”

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lukeprog November 19, 2009 at 8:06 am

Alex,

Thanks, but there is more than that which I left out! Also, there are competing theories of intrinsic probability. This was only an introduction to the idea of abductive reasoning – I shall return later for a more thorough discussion.

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Rich November 19, 2009 at 8:40 am

Probably need some Occam’s razor in here, also.

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Alex November 19, 2009 at 11:10 am

Luke: Agreed; I myself think the notion of intrinsic probability is the weakest point of Bayesian accounts of explanation. I’m pretty sure intrinsic probability exists, but I’m not holding my breath for a successful formal account that captures our intuitions about it. I’m awaiting further high-quality posts on abduction.

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Taranu November 19, 2009 at 11:40 am

Luke, how many deductive arguments for atheism are you aware of, apart from the logical problem of evil argument?

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lukeprog November 19, 2009 at 12:41 pm

Taranu,

Well, it depends where you want to draw the line on how different two arguments need to be before they are considered separate arguments. Only about a half dozen deductive arguments for atheism have been widely discussed, but there are others. You could make up as many valid deductive arguments for atheism as you want, it’s just that the premises wouldn’t necessarily be very strong.

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Rhys Wilkins November 19, 2009 at 3:57 pm

Hi Luke, champion effort as always.

Since you are on the topic of ‘inference to the best explanation’, are you thinking about doing an article on why God is never the best explanation? (since He can literally be invoked to explain anything, it does not advance our knowledge etc etc) would be great to hear your thoughts on this!

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Scott November 19, 2009 at 5:59 pm

Taranu: Luke, how many deductive arguments for atheism are you aware of, apart from the logical problem of evil argument?  

The Impossibility of God (http://www.amazon.com/o/ASIN/1591021200) deals with deductive arguments. Its companion, The Improbability of God (http://www.amazon.com/o/ASIN/1591023815) is more of the inductive, god-is-unlikely angle. Quite the dynamic duo of books.

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Thomas Reid November 19, 2009 at 6:35 pm

Rhys Wilkins: Hi Luke, champion effort as always.Since you are on the topic of ‘inference to the best explanation’, are you thinking about doing an article on why God is never the best explanation? (since He can literally be invoked to explain anything, it does not advance our knowledge etc etc) would be great to hear your thoughts on this!  

Yes, I’d like to hear that subject discussed as well. I’m not sure what it means. Rhys, I’d appreciate it if you could further develop your thoughts here, or perhaps post a reference.

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Alex November 19, 2009 at 6:50 pm

Rhys: Since he is a fan of Greg Dawes’ book, I don’t think Luke actually thinks that God can *never* be the best explanation. The claim that existing God hypotheses don’t work and that there are some in-principle difficulties that any theistic explanations must overcome is much more plausible than the absolutist view.

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Bill Maher November 19, 2009 at 8:26 pm

Luke, I am sending you a really cool e-mail involving some important information about technology theory. It should be very entertaining for you :-)

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Jeff H November 19, 2009 at 8:28 pm

I was wondering if someone could explain a bit more how abductive reasoning differs from inductive reasoning. I’ve never been that clear on abductive reasoning, since all the philosophy courses I’ve taken (which, admittedly, are not many) have only mentioned deduction and induction. Abduction and induction seem very similar to me, and both seem to go from data to theory. So how does “the best explanation” differ from “the most probable explanation”?

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majinrevan666 November 19, 2009 at 10:58 pm

Very interesting.
I was wondering about a couple of things.

Is an argument whose conclusion is that something is probable an inductive or deductive argument?
E.G,

1.Most men are mortal.

2.I am a man.

3.Therefore, I am probably mortal.

Secondly, are abductive arguments basically the fallacy of affirming the consequent, only it replaces affirmation
with tentative affirmation?

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lukeprog November 20, 2009 at 2:34 am

Rhys,

That problem is actually central to my rejection of theism. I am planning an entire series on the topic. It is also a central topic of my “debates” with Tom Wilson and Vox Day.

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lukeprog November 20, 2009 at 2:35 am

Thomas Reid,

The best work on the subject so far is Gregory Dawes’ Theism and Explanation.

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lukeprog November 20, 2009 at 2:37 am

Jeff H,

More of this will be covered in the future, but…

Abduction is often thought of as a species of inductive argument, if we consider inductive arguments to be those that simply are not deductive. But another way to use the terms is to say that an inductive argument moves from an enumeration of specific cases to a general statement about how things are, whereas an abductive argument is about offering the best explanation – either by way of confirmation theory or “explanationism” or whatever.

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lukeprog November 20, 2009 at 2:41 am

majinrevan666:

An inductive argument would look like this:

1. Most men are mortal.
2. I am a man.
3. Therefore, I am mortal.

(You do not include the “probably” in the conclusion. It is assumed, since this is an inductive argument.)

As for affirming the consequent, I guess you could say it that way. But keep in mind there is a fantastically successful research program – much more successful than philosophy – that uses careful abduction as its core method. That’s called science. So that’s why we take it seriously as a tentative way to affirm things – but only if certain criteria are met.

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Thomas Reid November 20, 2009 at 3:53 am

lukeprog: Thomas Reid,The best work on the subject so far is Gregory Dawes’ Theism and Explanation.  

Thanks, I’ll check this out.

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lukeprog November 20, 2009 at 7:32 am

Also, guys, see my recent post to Tom Gilson in our debate.

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Rich November 23, 2009 at 1:10 pm

lukeprog: 1. Most men are mortal.
2. I am a man.
3. Therefore, I am mortal.

No.

It would be

1. All men I have observed are Mortal

Induction takes from known examples that are a subset.

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lukeprog November 23, 2009 at 6:51 pm

Rich, you’re right; that is truer to the tradition of induction found in Mill, etc.

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Rich November 23, 2009 at 7:52 pm

The “problem of induction” and “uniformity of nature” are my favourites ;-)

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