induction hilary 07

Induction

General Philosophy

January 25, 2007

Antony [email protected]

http://users.ox.ac.uk/~sfop0118/

General Philosophy, January 25, 2007 Induction

Reasoning

We often gain knowledge by reasoning.

If that reasoning is deductively valid,the conclusion must be true if the initialpremises are true.

The flipside: the content of theconclusion adds nothing to what is alreadycontained in the premises. Deductivereasoning is conclusive because, in a sense,if you are committed to the premises youare already committed to the conclusion.

Antony Eagle 2


Reasoning in Science

Not all reasoning is deductive. In sciencewe reason from evidence to accepting ahypothesis that goes beyond the content ofthat evidence.

It is compatible with our evidence, forexample, that invisible fairies move all thefundamental particles around, rather thanatomic forces. So our belief in these forces,while reasonable, goes beyond our evidence.

This is an example of inductive reasoning:reasoning where the premises (our evidence)provide good, though not conclusive,reasons to believe the conclusion (thehypothesis of forces, in this case).

Antony Eagle 3


Induction and Support

Evidence: E1, . . . , En

Supports: +

Hypothesis: H

What is this ‘support’ relation?

One idea: evidence supports a hypothesisexactly in case that evidence makes thehypothesis more likely to be true.

If we toss a coin 10 times, and itlands heads on each toss, that makes thehypothesis that the coin will land headsnext toss more likely, and so supports thathypothesis.

Yet since even a fair coin might landheads 10 times in a row, this is notconclusive evidence for that hypothesis.

Antony Eagle 4


Inductive Inference

Take the coin toss example again. Thereare many hypotheses that involve bias: e.g.

H1 The coin has probability 1 of heads.

H0.9 The coin has probability 0.9 of heads.

Both H1 and H0.9 are supported by theevidence of 10 heads in a row. But they areincompatible: they can’t both be true.

InductionBelieve the claim that is most

likely given one’s evidence.

Given this evidence, we should favour H1.

Antony Eagle 5


Hume on Induction

Let the course of things be allowedhitherto ever so regular; that alone. . . proves not, that, for the future,it will continue so. (Enquiry §4.2)

Hume talks about the past and future,but the point generalises:

Question

Will our judgements oflikelihood — grounded in theevidence we have accumulated— continue to be dependablein areas we have not yet gainedany evidence about?

Antony Eagle 6


Hume’s Central Claim

If there be any suspicion, thatthe course of nature may change,. . . all experience becomesuseless. . . (Enquiry §4.2)

Two questions arise:

1. Is there such a suspicion that the courseof nature may change?

2. Why does experience become useless if itcan?

Antony Eagle 7


Question 1: Reasons forsuspicion

That there are no demonstrativearguments in this case seemsevident; since it implies nocontradiction that the course ofnature may change. (Enquiry §4.2)

Hume means that any body ofpropositions purely concerning our evidenceis consistent with any body of propositionspurely concerning di↵erent topics (like thefuture, or the unobserved) that are not partof our evidence.

We might suspect, then, that since it isconsistent that the course of nature doeschange, that it may.

Antony Eagle 8


Question 1: Circular appeal toevidence

To rule this out, we would need to appealto evidence; but any such argument fromevidence must presuppose that the pastevidence will continue to be reliable:

It is impossible that any argumentsfrom experience can prove thisresemblance of the past to thefuture; since all these argumentsare founded on the supposition ofthat resemblance. (Enquiry §4.2)

Antony Eagle 9


Question 2: The uselessness ofexperience

Experience isn’t useless just because itdoesn’t determine what will happen in thefuture. Rather, it is useless because itdoesn’t even tell us what is likely in thefuture.

This is because what we judge to be likely

depends on what our experience has been.

We judge the future’s resembling thepast to be likely because ‘past futures’have always resembled ‘past pasts’. Butpossibly ‘future pasts’ will not resemble‘future futures’ at all: and in that caseour judgements of what is likely, based oncurrent evidence, are irrelevant to what is infact true.

Antony Eagle 10


Illustration: coin tosses

Suppose we tossed 10 heads. Nowconsider the next 10 tosses. There are210 = 1024 possible sequences: from 10heads, all the way to 10 tails, and everythingin between.

The past evidence is compatible with allof these; but only the ‘all heads’ sequenceis compatible with H1, and 1023 sequencesare incompatible with this hypothesis. If thefuture may change (and we don’t assumethat it is like the past) surely all of thesesequences are equally possible continuationsof the past sequence. So, while it is mostlikely on current evidence, most of the waysthat the world could go on make H1 false.

Antony Eagle 11


Hume’s Argument in Summary

1. For all we know, the course of nature maychange.

2. If for all we know, the course ofnature may change, then current evidenceis useless both for forming conclusivebeliefs about the future, and for makingjudgements of future likelihood.

C. Therefore, current evidence is useless forforming beliefs about the future.

This is valid; so what justifies ourcontinued reliance on inductive inference?

Antony Eagle 12


The Practical Response Defused

to ask whether it is reasonableto place reliance on inductiveprocedures is like asking whetherit is reasonable to proportion[one’s] convictions to . . . theevidence. Doing this is what ‘beingreasonable’ means. . . (Strawson, ch.9)

Hume agrees:

As an agent, I am quite satisfied inthe point; but as a philosopher . . .I want to learn the foundation ofthis inference (Enquiry, §4.2)

Antony Eagle 13


Russell

Thus we must either accept theinductive principle on the groundof its intrinsic evidence, or forgoall justification of our expectationsabout the future. (Russell, p. 68)

Russell’s response: reject premise 1 ofour reconstructed argument, and claims thatthe inductive principle is knowable a priori

(without recourse to experience).

To defend this, Russell must suggest thatthere is a ‘contradiction’ in the idea thatthe course of nature may change. And thisseems implausible at best.

Antony Eagle 14


Hume’s Own Response: Customor Habit

Hume responded by explaining inductivepractices in the absence of justification. Heclaims that what grounds our inferencesfrom the past to the future is custom orhabit:

Custom, then, is the great guideof human life. It is that principlealone, which renders our experienceuseful to us, and makes us expect,for the future, a similar trainof events with those which haveappeared in the past. (Enquiry, §5.1)

But such habit is no more based onreasons than the instincts of beasts.

Antony Eagle 15


Externalism

However: there is more than one wayfor reasons to justify actions. Imagine that Iam o↵ered some medicine by my duplicitousdoctor, who says that it is poison.

1. I have no reason to take it: I trust thedoctor and she has told me that it ispoison;

2. But of course I have a reason to take it,because in fact it will cure me.

Hume’s argument tells us that we lackreasons in the first sense: no belief we couldform could justify our inductive inferenceswithout circularity. But perhaps we havereasons in the second sense.

Antony Eagle 16


Externalism and Reliabilism

Reliabilism

A procedure P is reliable ina situation S if it actuallydoes eventually converge to thetruth about S (whether or notwe know that it does).

If we follow a reliable procedure, evenwithout knowing that it is reliable, we willeventually reach the truth.

So we have a reason, in the externalistsense, to follow reliable procedures; suchprocedures are justified.

Antony Eagle 17


Induction and Reliabilism

So if, in fact, induction is a reliableprocedure, it is justified.

[T]he world is so constituted thatinductive arguments lead on thewhole to true opinions. . . [W]ejudge mental habits by whetherthey work. . . Induction is such auseful habit, and so to adopt it isreasonable. (Ramsey, 1990, 93–4)

And if we do adopt it, we can give aninductive argument for induction!

Antony Eagle 18


Can we assume that induction isreliable?

Indeed, whether or not it is reliable, weshould arguably keep using induction — solong as we have no evidence against it beingreliable.

What could such evidence consist in?

One piece of evidence would be: twoinductive arguments from the same evidencethat have contrary conclusions. That wouldshow that at least one of these argumentsis defective, and thus that induction isunreliable sometimes.

Is there any such evidence?

Antony Eagle 19


Goodman’s Grue

Grue

Something is grue exactly ifit is green and has beenobserved, or blue and hasn’tbeen observed.

So emeralds that have been dug up aregrue; so are sapphires that haven’t yet beendug up (though they won’t be grue oncethey are dug up, since they’ll remain blue).

Two simple facts about emeralds:

• All emeralds so far observed have beengreen.

• All emeralds so far observed have beengrue.

Antony Eagle 20


Induction on Green and Grue

Now consider the two hypotheses:

Hgreen All emeralds are green.

Hgrue All emeralds are grue.

The ‘simple facts’ are our evidence. Bothhypotheses entail the evidence.

If we don’t presuppose that either ‘grue’or ‘green’ is better, it seems that we shouldthink both hypotheses are equally maximallylikely given the evidence.

And so we should come to believe bothhypotheses.

Antony Eagle 21


The ‘new riddle’

The problem, as Goodman pointed out,is that one cannot believe both hypotheses.For if Hgreen is true, all unobserved emeraldsare green. But if Hgrue is true, all unobservedemeralds are grue — which is to say, allunobserved emeralds are blue.

But it can’t be the case that anything, letalone an unobserved emerald, is both blueand green.

So at least one of these inductiveinferences must be defective — which isevidence that induction is not reliable.

Antony Eagle 22


The dilemma

The obvious solution is to distinguishbetween ‘green’ and ‘grue’: one of themis more natural than the other.

But Hume’s argument shows that wecannot come to possess reasons to believethat one predicate is more natural thananother (for that would involve havingevidence that the future will be a certainway).

So we cannot solve Goodman’s riddlewithout solving Hume’s problem; and wecannot solve Hume’s problem by reliabilistmeans without solving Goodman’s riddle:an uncomfortable position indeed!

Antony Eagle 23


Further Reading

Goodman, Nelson (1954), Fact, Fiction andForecast. Cambridge, MA: Harvard UniversityPress.

Howson, Colin (2000), Hume’s Problem: Inductionand the Justification of Belief . Oxford: OxfordUniversity Press.

Hume, David (1999), An Enquiry Concerning HumanUnderstanding . Oxford: Oxford University Press.

Ramsey, F. P. (1990), “Truth and Probability”. InPhilosophical Papers, Cambridge, UK: CambridgeUniversity Press, pp. 52–94.

Russell, Bertrand (1997), The Problems ofPhilosophy . Oxford: Oxford University Press.

Strawson, P. F. (1952), An Introduction to LogicalTheory . Oxford: Oxford University Press.

Antony Eagle 24

induction hilary 07

Documents