The Problem of Induction

Last time we discussed a few arguments which seem to show that we can have no knowledge of the properties of things which are outside of us. Let’s set those problems to the side, for now, and just grant that perception does, at least some times, give us knowledge of the properties of the things we encounter in our experience.

Of course, not all the knowledge that we take ourselves to have is the kind of direct perceptual knowledge exempliﬁed by me claiming, on the basis of visual experience of my hands, that I have hands. Sometimes we take ourselves to know things which go beyond our own perceptual experience.

Science is a rich source of such knowledge claims. Consider the following claims for which we seem to have good scientiﬁc evidence:

Massive objects attract one another. The earth rotates on its axis about once every 24 hours. Dinosaurs (not counting birds) became extinct about 60 million years ago.

These claims are not, on a natural interpretation, claims which we can know to be true directly on the basis of sense experience: for example, though we can observe some massive objects attracting each other, we certainly have not observed this of all presently existing massive bodies, let alone all massive bodies past and future.

However, experience clearly plays some role in our justiﬁcation for believing that massive bodies attract: we take ourselves to know this because we have observed many massive bodies interacting, and infer from the fact that we have observed this in many cases that it is also true of unobserved cases. Today we will be focusing a simple variety of inductive reasoning, known as enumerative induction, which is exempliﬁed by the following argument.

The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... ______

The sun will come up tomorrow.

In this sort of reasoning, we infer from a list of observed instances of some generalization that some unobserved instance is also true. It is not essential that the conclusion be some claim about the future; it could be about any unobserved state of affairs, including the distant past.

Enumerative induction seems to play a central role in the justiﬁcation of scientiﬁc claims, even where this is not immediately obvious. Consider, for example, the following claim:

Typical adult members of the species tyrannosaurus rex were about 13 feet tall.

How would you defend a claim of this sort? Would enumerative induction play a role?

In the reading for today, Hume critically examines the sort of inductive reasoning exempliﬁed by the argument above. We can begin this examination with a question: Is this argument valid? Today we will be focusing a simple variety of inductive reasoning, known as enumerative induction, which is exempliﬁed by the following argument.

The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... ______

The sun will come up tomorrow.

In the reading for today, Hume critically examines the sort of inductive reasoning exempliﬁed by the argument above. We can begin this examination with a question: Is this argument valid?

One way to pose Hume’s challenge to inductive reasoning is to ask: given that this argument (and other examples of enumerative induction) is not valid, how can induction be a reasonable way of forming beliefs?

A natural response to this question is that we can, of course, make the argument we just discussed valid by adding a premise.

If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. A natural response to this question is that we can, of course, make the argument we just discussed valid by adding a premise.

The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

Does this make the argument valid?

A valid argument only gives provides good reason for believing its conclusion if we can know that the premises of that argument are true. So what we have to ask is: can we know that the if-then statement above is true and, if so, how?

Hume raises a dilemma for any attempt to explain how we can know this extra premise. This dilemma is based on a distinction between two kinds of knowledge. The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

Hume raises a dilemma for any attempt to explain how we can know this extra premise. This dilemma is based on a distinction between two kinds of knowledge.

“All the objects of human reason or inquiry may naturally be divided into two kinds, to wit, relations of ideas, and matters of fact. Of the first kind are the sciences of geometry, algebra, and arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. ...Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. ... Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; ...The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise tomorrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. ...”

Here Hume not only distinguishes between knowledge of relations of ideas and knowledge of matters of fact; he also gives an argument that the extra premise in our argument, is not just a matter of the relations of ideas, and therefore cannot be known by “the mere operation of thought.” How would you state Hume’s argument for this point? The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

Hume raises a dilemma for any attempt to explain how we can know this extra premise. This dilemma is based on a distinction between two kinds of knowledge.

Knowledge that some claim p is the case

p is a matter of the relations of ideas p is a matter of fact

p can be known by mere thought p can only be known by experience

The opposite of p is a contradiction, or unintelligible The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

Knowledge that some claim p is the case

p is a matter of the relations of ideas p is a matter of fact

p can be known by mere thought p can only be known by experience

The opposite of p is a contradiction, or unintelligible

Given that the opposite -- the negation -- of the extra premise is not unintelligible, the extra premise cannot be known by mere thought, and is not simply a matter of the relations of ideas.

So if we can know that this extra premise is true, we must know this by experience. But can we know this by experience? The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

Knowledge that some claim p is the case

p is a matter of the relations of ideas p is a matter of fact

p can be known by mere thought p can only be known by experience

The opposite of p is a contradiction, or unintelligible

So we can’t know the truth of the extra premise by thought alone, or by experience. But if we cannot know that it is true, it is hard to see how the reasoning exempliﬁed by our argument above could be rational. The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

One plausible idea is that we know this extra premise because it follows from something else that we do know: that nature is uniform, in the sense that the future is, in general, like the past. Our extra premise seems just to be an instance of this general claim of the uniformity of nature; hence if we can know the latter, we can know the former.

Hume considers this idea, and raises the question: how do we know that nature is uniform? This cannot simply be a matter of the relations of ideas, since the opposite of the claim of the uniformity of nature is not contradictory, or unintelligible.

It seems more promising to say that we know of the uniformity of nature by experience. After all, aren’t we constantly observing the uniformity of the laws governing the world around us? The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

It seems more promising to say that we know of the uniformity of nature by experience. After all, aren’t we constantly observing the uniformity of the laws governing the world around us?

Let’s be a bit more precise about this. What we want to know is:: that is what would give us good reason to believe the extra premise in our argument.

The future, relative to today, 3/26/09, will be like the past, relative to today

Knowledge of this would give us good reason to believe the extra premise in our argument.

What we have observed is that, in the past, the future has always been like the past. That is, we know by experience claims of the following sort:

The future, relative to 3/25/09, was like the past, relative to 3/25/09. The future, relative to 3/24/09, was like the past, relative to 3/24/09.

Do claims like this provide a good reason for believing that the future, relative to today, will be like the past, relative to today? The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

Do claims like this provide a good reason for believing that the future, relative to today, will be like the past, relative to today?

One way to approach this question is by asking whether the following argument is valid.

The future, relative to 3/25/09, was like the past, relative to 3/25/09. The future, relative to 3/24/09, was like the past, relative to 3/24/09...... ______The future, relative to today, 3/26/09, will be like the past, relative to today

Is this argument valid?

While this argument is not valid, it seems that it could be made valid by the addition of an extra premise.

If the future, relative to 3/25/09, was like the past, relative to 3/25/09, and the future, relative to 3/24/09, was like the past, relative to 3/24/09, and ...... , then the future, relative to today, 3/26/09, will be like the past, relative to today. The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

The future, relative to 3/25/09, was like the past, relative to 3/25/09. The future, relative to 3/24/09, was like the past, relative to 3/24/09...... If the future, relative to 3/25/09, was like the past, relative to 3/25/09, and the future, relative to 3/24/09, was like the past, relative to 3/24/09, and ...... , then the future, relative to today, 3/26/09, will be like the past, relative to today. ______The future, relative to today, 3/26/09, will be like the past, relative to today

This argument is valid; however, it does not seem to provide a convincing defense of the uniformity of nature.

This is because the extra premise in our argument for the claim that the future (relative to today) will be like the past (relative to today) says that if the future relative to every past time was like the past relative to every past time, then the future relative to this time will be like the past relative to this time. But this just states the thesis of the uniformity of nature. The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... If the sun came up on 3/26/09 and the sun came up on 3/25/09 and ... then the sun will come up tomorrow. ______

The sun will come up tomorrow.

To sum up: we can defend our initial argument for the claim that the sun will come up tomorrow by defending the claim of the uniformity of nature; but the uniformity of nature is not knowable by mere thought, not knowable directly by experience, and can be established by argument only where the argument includes the uniformity of nature itself as a premise. To sum up: we can defend our initial argument for the claim that the sun will come up tomorrow by defending the claim of the uniformity of nature; but the uniformity of nature is not knowable by mere thought, not knowable directly by experience, and can be established by argument only where the argument includes the uniformity of nature itself as a premise.

How might we respond to Hume’s argument?

One might grant the conclusion that we cannot know things on the basis of inductive reasoning. But this seems implausible. After all, inductive reasoning is used widely in science; and doesn’t it seem as though science, at least sometimes, gives us knowledge of the world?

However, one might soften the blow of this conclusion by suggesting that we fall back on the slightly weaker claims that inductive reasoning, even if it cannot provide knowledge, provides us with good reasons, or justiﬁcation, for beliefs. So perhaps we do not know that the sun will come up tomorrow, and perhaps we do not know the truth of scientiﬁc theories; but inductive reasoning does give us good reason to believe that the sun is likely to come up tomorrow.

The problem with this “fallback position” is that Hume’s argument seems to show that we don’t even have good reason for believing in the results of inductive reasoning. The only justiﬁcation we can give for that sort of reasoning relies on that very sort of reasoning -- and this means that inductive reasoning seems to be in the same position as a belief supported only by an argument whose only premise expresses that very belief.

This might suggest that we will have more luck with another “fallback position.” Perhaps our mistake is that we are trying to see how we could have good reason to believe that the sun will come up tomorrow. But perhaps we don’t have any good reason for believing this; perhaps we only have good reason for believing claims about the probability, or likelihood, that the sun will come up tomorrow. This might suggest that we will have more luck with another “fallback position.” Perhaps our mistake is that we are trying to see how we could have good reason to believe that the sun will come up tomorrow. But perhaps we don’t have any good reason for believing this; perhaps we only have good reason for believing claims about the probability, or likelihood, that the sun will come up tomorrow.

Then we might think that the following argument will fare better than our original argument:

The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... ______

There is a 90% chance that the sun will come up tomorrow.

Is this argument valid? The answer to this question depends on what one means by “90% chance.” If this is just a claim about past frequencies of events of this sort, then the argument is of course valid, but irrelevant to our question: we already knew that we could know facts about observed past events on the basis of observation.

An alternative is that we are talking about subjective probabilities: perhaps we are saying that it would be rational for someone who knows the premises to believe that there is a 90% chance that the sun will come up tomorrow. But this is just the conclusion we were trying to establish: we were trying to ﬁgure out how, and why, it could be rational to have a belief of this sort. This might suggest that we will have more luck with another “fallback position.” Perhaps our mistake is that we are trying to see how we could have good reason to believe that the sun will come up tomorrow. But perhaps we don’t have any good reason for believing this; perhaps we only have good reason for believing claims about the probability, or likelihood, that the sun will come up tomorrow.

Then we might think that the following argument will fare better than our original argument:

The sun came up on 3/26/09. The sun came up on 3/25/09. The sun came up on 3/24/09...... ______

There is a 90% chance that the sun will come up tomorrow.

Is this argument valid?

A ﬁnal alternative is that we are talking about some sort of objective probability: the real chance, given the way the world is, that the sun will come up tomorrow. But then it is far from obvious that the argument above is valid: isn’t the past history of the sun coming up consistent with there being quite a good chance that the sun will not come up tomorrow?

So it is not obvious that, even if we restrict ourselves to probabilistic claims, we can solve the problem. So it is not obvious that, even if we restrict ourselves to probabilistic claims, we can solve the problem.

It is worth being clear about what that problem is. We have not given a direct argument that the use of enumerative induction is irrational; rather, we have shown that it seems very difﬁcult to give a justiﬁcation of enumerative induction which is not circular, in the sense that it presupposes the legitimacy of inductive reasoning.

It is interesting to compare induction in this respect with deduction.

Suppose that I believe some claim q, and you ask me to provide my justiﬁcation for believing q; I might respond with an argument of the following form:

(A) p. ______(B) If p, then q. (Z) q.

Suppose now that you respond like this: “Yes, I concede that p is true, and I also concede that if p is true, then q is true. But I still don’t see why I should accept q!”

How should I reply?

It seems that I should say something like this: “Look, if p is true, and it is also true that if p, then q, q has to be true.” (A) p. ______(B) If p, then q. (Z) q.

Suppose now that you respond like this: “Yes, I concede that p is true, and I also concede that if p is true, then q is true. But I still don’t see why I should accept q!”

How should I reply?

It seems that I should say something like this: “Look, if p is true, and it is also true that if p, then q, q has to be true.”

Now suppose you respond like this:

Yes, I see your point. What you are saying is that if (A) and (B) are true, then (Z) is true. I grant this. So let’s add this premise to our argument:

______(C) If (A) and (B) are true, then (Z) is true. (Z) q.

Suppose now that you respond like this: “Yes, I concede that p is true, and I also concede that if p is true, then q is true. But I still don’t see why I should accept q!”

How should I reply?

It seems that I should say something like this: “Look, if p is true, and it is also true that if p, then q, q has to be true.”

Now suppose you respond like this:

Yes, I see your point. What you are saying is that if (A) and (B) are true, then (Z) is true. I grant this. So let’s add this premise to our argument. But I still don’t see why the conclusion, q, should be true. (A) p. (B) If p, then q.

______(C) If (A) and (B) are true, then (Z) is true. (Z) q.

Now suppose you respond like this:

This line of thought comes from a short piece by Charles Dodgson (also known as Lewis Carroll) entitled “What the Tortoise said to Achilles.” (In the story, Tortoise is playing your role, and Achilles is playing mine.) NOTES. 279

"No doubt such a reader might exist. He might say 'I accept as true the Hypothetical Proposition that, {f A and B be true, Z must be true; but, I don't accept A and B as true.' Such a reader would do wisely in abandoning Euclid, and taking to football." "And might there not also be some reader who would say 'I accept A and B as true, but I don't accept the Hypothetical'?" "Certainly there might. He, also, had better take to football." "And neither of these readem," the Tortoise continued, "is as yet under any logical necessity to accept Z as true ? " "Quite so," Achilles assented. "Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true." "A tortoise playing football would be--" Achilles was beginning ''-an anomaly, of course," the Tortoise hastily interrupted. "Don't wander from the point. Let's have Z first, and football afterwards !" "I'm to force you to accept Z, am I?" Achilles said musingly. "And your present position is that you accept A and B, but you don't accept the Hypothetical-" "Let's call it C," said the Tortoise. "-but you don't accept (C) If A and B are true, Z must be true." " That is my present position," said the Tortoise. "Then I must ask you to accept C." "1'11 do so," said the Tortoise, "as soon as you've entered it in that note-book of yours. What else have you got in it 2" '' Only a few (A)memoranda," p. said Achilles, nervously fluttering the leaves: ''a few memoranda(B) If p, then q.of-of the battles in which I have distin- guished myself !" '' Plenty of blank______(C) leaves,If (A) and I (B)see are !" true,the Tortoisethen (Z) is cheerilytrue. remarked. "We shall need them all(Z) !" q. (Achilles shuddered.) "Now write as I dictate :- (A) Things that are equal to the same are equal to each other. (B) The two sides of this Triangle are things that are equal to the Here’ssame. the continuation of the story from this point, picking up with how I (Achilles) might respond to you granting the truth of premise (C), but still refusing to accept the (C) If A and B are true, Z must be true. conclusion.(2) The two sides of this Triangle are equal to each other." "You should call it D, not 2," said Achilles. "It comes next to the other three. If you ac;ept A and B and C, you must accept 2." "And why must I ? "Because it follows logically from them. If A and B and C are true, Z must be true. You don't dispute that, I imagine ? " '' If A and B and C are true, Z must be true," the Tortoise thought- fully repeated. "That's anotl~erHypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I ? " "You might," the candid hero admitted; "though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical." "Very good. I'm quite willing to grant it, as soon as you've written it down. We will call it (D) If A and B and C are true, Z must be true. Have you entered that in your note-book?" "I have !" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at last we've got to the end of this ideal race-course ! Now that you accept A and B and C and D, of coulae you accept S" "Do I ? " said the Tortoise innocently. "Let's make that quite clear, I accept A and B and C and D. Suppose I still refused to accept Z?" (A) p. (B) If p, then q.

______(C) If (A) and (B) are true, then (Z) is true. (Z) q.

280 NOTES.

"Then Logic would take you by the throat, and force yon to do it !" Achilles triumphantly replied. ''Logic would tell you 'You ca'n't help yourself. Now that you've accepted A and B and C and D,you naust accept Z!' So you've no choice, you see." "Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your book, please. We will call it (E) If A and Band Cand D are true, Zmust be true. Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, gou see?" "I see," said Achilles ; and there was ti touch of sadness in his tone. Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his note-book, which appeared to be nearly full. The Tortoise was saying " Have you got that last step written down P Unless I've lost count, that makes a thousand and one. There are several millions more to come. And would you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century-would you Whatmind does adopting this story a pun tell usthat about my the cousin justiﬁcation the Mock-Turtle of deductive reasoning?will then make,Can we and think of the Tortoise as someoneallowing who yourself refuses to to be accept re-named the legitimacy Taught- of Us deductive 2" reasoning unless given a noncircular justiﬁcation for it? Could"As Achilles you please have given!" replied a better the answer weary to thewarrior, Tortoise in than the he hollowdid? tones of despair, as he buried his face in his hands. "Provided that you, for part, will adopt a pull the Mock-Turtle never made, and allow yoursel:: be re-named A Kill-Ease !"

Cnn~b~idge: Printed at the Ug~iversityPress. What does this story tell us about the justiﬁcation of deductive reasoning? Can we think of the Tortoise as someone who refuses to accept the legitimacy of deductive reasoning unless given a noncircular justiﬁcation for it? Could Achilles have given a better answer to the Tortoise than he did?

One thing this example suggests that is that deductive reasoning can only be justiﬁed by deductive reasoning; if so, perhaps induction is not in such bad shape, even if we can give no answer to Hume’s problem.

On the other hand, this point still leaves us with a problem. It seems clear that inductive and deductive reasoning are better ways of forming beliefs than, for example, astrology. But what would this difference consist in, if we could give an argument, using premises from astrology, for the reliability of the astrological method of belief formation?