Intro to Logic: Abductive Reasoning

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.

A piece of Sherlock Holmes’ reasoning can illustrate:

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.

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

Everyone wearing an arc-and-compass breastpin is a Freemason. Mr. Wilson is wearing an arc-and-compass breastpin. 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:

The surprising fact, E, is observed. But if H were true, E would be a matter of course. 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).

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

The surprising fact, an arc-and-compass breastpin on Mr. Wilson, is observed. But if Mr. Wilson is a Freemason, an arc-and-compass breastpin on Mr. Wilson would be a matter of course. 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:

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

But this takes the form of

P If Q then P. Therefore, Q.

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

Mr. Wilson wears underwear. If Mr. Wilson was a woman, then he would wear underwear. 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.