The Problem of Induction

PHLA10 12The Problem of Induction

PHLA10 12The Problem of Induction●●Knowledge versus mere justified belief– Knowledge implies truth– Justified belief does not imply truth– Knowledge implies the impossibility of error– Justified belief does not imply impossibility of errorJustified belief comes in grades of more or less– You are more justified in believing you will lose the 649lottery than in believing this coin will come up heads– We often express this ‘gradation’ in terms ofprobability– The concept of evidence can be expressed in terms ofprobability too● P is evidence in favour of Q = P raises theprobability of Q– Learning you rolled an even number is (some)evidence in favour of you having rolled a six

PHLA10 12The Problem of Induction●●Ordinary skepticism attacks knowledge– Claims that we have no (or almost no) knowledge– Does not deny that some beliefs are more reasonablethan others ...– Does not deny that some beliefs are evidence forothers (e.g. raises their probability)Justified belief skepticism attacks rationality– Claims that we have no reason to think that any beliefis either more or less probable than any other– Denies we have any good reason to think that anybelief is evidence in favour of (or against) any otherpossible belief– (A priori beliefs/probabilities may be an exception)

PHLA10 12The Problem of Induction●Review: what is induction– A method of ‘amplifying’ or adding knowledge (or atleast adding to our stock of beliefs)– Unlike in a valid deductive argument, the conclusion ofan inductive argument is not guaranteed to be true,even if the premises are true● example: (1) Most dogs are pets (2) Fido is a dog (3)therefore, Fido is a pet– Recall what makes a good inductive argument● good sample size● good sample distribution (sample must berepresentative of total)● These requirements assume there are better orworse evidential relations

PHLA10 12The Problem of Induction●●●●●●Two (closely related) forms of induction(1) Generalization (GEN)– example: All observed mammals have hair; thereforeall mammals have hair.(2) Prediction (PRED)– example: All observed reptiles are cold blooded;therefore the next reptile to be observed will be coldblooded.Obviously (GEN) and (PRED) are not deductively validargument forms.But it seems intuitively obvious that the premises give usa good reason to believe the conclusionHume argues that this intuition is unsupportable andwrong

PHLA10 12The Problem of Induction●Hume’s version– Hume believed that all inductivearguments involved one crucialassumption: the Principle of theUniformity of Nature (PUN).– PUN = nature will continue tobehave in the future as it has inthe past / nature will generally besimilar to the way it is around hereDavid Hume (1711-1776)

PHLA10 12The Problem of Induction●How does PUN fit into inductive arguments?– Instead of● All thus far observed mammals have hair, so the nextmammal we meet will have hair– We have● All thus far observed mammals have hair and PUN, sothe next mammal we meet will have hair– Does PUN turn an inductive argument into a deductiveargument?– Perhaps it is meant to.– But what kind of proposition is PUN?● A priori (can be deductively proven)?● A posteriori (can only be inductively proven)?

PHLA10 12The Problem of Induction●●●Is PUN a priori?– Can we give a deductive proof of PUN?– Is it possible that nature should not be uniform?– It seems possible– Therefore, PUN is not a prioriTherefore, PUN is a posteriori– So it must be proven either by observation or induction– We cannot observe PUN (because it is about the future)– So we must give an inductive argument for PUNWhatever this argument might look like it will be aninductive argument.– Therefore, the argument will contain an assumption– The assumption – according to Hume – will be PUN– This is circular reasoning and cannot show PUN

PHLA10 12The Problem of Induction●●●●Example argument:– In the past, PUN has always been true– Therefore (inductively) PUN is trueHume notes that this argument depends on the assumptionthat nature will continue to obey PUNSo the argument ought to be:– In the past, PUN has always been true– PUN– Therefore, PUN is trueThis argument fails because it blatantly assumes what itwants to prove!

PHLA10 12The Problem of Induction●Hume’s attitude towards induction– Hume thought we should reason inductively even thoughwe have no rational reason to do so– He thought we (and many other animals) are naturallystructured to believe in and use induction● Example: Pavlov’s dogs– Hume sometimes called this ‘habit’– He also noticed instincts – which are ‘built in’ by natureand carry information about how organisms ‘expect’ theworld to work– Hume wondered how instincts arose and camesomewhat close to a concept of evolution– But rationality cannot support the beliefs derived byinstinct or by the habit of inductive inference

PHLA10 12The Problem of Induction●●●But is PUN needed for inductivearguments or the attack oninduction?What, exactly, is the content ofPUN?Is nature always ‘uniform’?– Do the seasons of the yearshow uniformity or diversity?– Is the death of animals afeature of natural uniformity ora sudden dis-uniformity in ananimal’s life– It seems impossible to state PUN in any non-trivial way– But PUN is not needed to create the problem ofinduction

PHLA10 12The Problem of Induction●Induction and reliability– We want our inductive knowledge to be secure– Let’s say that a reliable method of inference is one thatusually leads to the truth● ‘usually’ can be thought of as a scale, from the notvery reliable to the highly reliable– examples: prediction of solar eclipses (highlyreliable) to weather prediction (not highly rel.)● This scale can be expressed in terms of probability– The probability of an eclipse given what we knowabout Sun, Earth and Moon is virtually 1– The probability of rain next week given ourcurrent knowledge is slightly more than ½– Sober’s version of the problem of induction● How do we know that induction in general is areliable way to get knowledge?

PHLA10 12The Problem of Induction●Sober’s new version of the problem of induction– How do we know that induction in general is a reliableway to get knowledge?– Now we replay Hume’s point– Either we can deductively prove that induction is areliable way to get knowledge, or– We have to inductively prove it is reliable– There is no way to prove deductively that induction isreliable (because we can consistently imagineinduction failing)– But to prove that induction is reliable inductively is toargue in a circle– PUN plays no part in this argument

PHLA10 12The Problem of Induction●Sober’s version of the problem of induction– Think about what this means– We have zero reason to think that induction is reliable– This implies that we have no reason to believe what isinductively reasonable versus the opposite– Example: we have zero reason to believe that the Sunwill rise tomorrow – it is exactly as reasonable tobelieve it will not rise as that it will ?!– How can that be right?– Can we save induction?

PHLA10 12The Problem of Induction●Strawson’s attempt to save induction– Maybe it is an analytic truth that inductionis a rational way to amplify knowledge– (Recall what an analytic truth is)– Strawson seems to be claiming that theidea that induction is a good way to reasonis part of the concept of rationality– Suppose that is true– Would this prove that induction is reliable?– It would seem notSir Peter Strawson(1919-2006)

PHLA10 12The Problem of Induction●Black’s attempt to save induction– Recall the argument in favour of induction:● Induction has been successful in thepast, so it will be successful in the future– Is the argument in favour of induction reallycircular?– Note the difference between a premise ofan argument and a rule of inference– Black argues that an argument is circularjust in case the conclusion appears (maybeonly implicitly) among the premises– On that understanding, the inductiveargument in favour of induction is notcircular– it just uses the inductive rule of inference Max Black(1909-88)

PHLA10 12The Problem of Induction●Achilles and the Tortoise (logical version)– A has an argument: P, P implies Q, thus Q– T says: well that might be OK, but I’m notsure about the final step– T says: are you sure you don’t need an extrapremise, like: if P and (P implies Q) then Q– A’s new argument: P, P implies Q, If P and (Pimplies Q) then Q, thus Q– T says: well that might be OK, but I’m notsure about the final step– T says: are you sure you don’t need an extrapremise, like: if P and (P implies Q) and [if Pand (P implies Q) then Q], then Q– ….

PHLA10 12The Problem of Induction●Black’s attempt to save induction– Is Black’s notion of circularity right?● Or is there something wrong with an argument thatdefends a form of argument which you can accept only ifyou already accept that form of argument?– We could also ask Black:● even if we could give this inductive proof of induction,would that show that induction is reliable?● No, because counter-induction (CI) is equally supportedby a counter-inductive argument● CI = if X has happened in the past, expect not-X– example: gambler’s fallacy● The CI argument in favour of CI– CI has failed in the past, so expect it to succeed in thefuture● This is a good CI argument!

PHLA10 12The Problem of Induction●●●Sober’s Trip Beyond Foundationalism– Note how Sober divides knowledge claims into 3 levels1) Indubitable beliefs (a priori / introspectible)2) Present and past observations3) Predictions and generalizations– Descartes had problems getting from 1 to 2– Hume adds problems getting from 2 to 3Sober thinks there is no way to– Move deductively from a level to a higher level– Even use lower level stuff as evidence as higher level● That is, IF one is restricted to the lower levelThis is because something is evidence only relative toadditional ‘background beliefs’– Example: phantom limb pain ...

PHLA10 12The Problem of Induction●●●The relativity of evidence– Suppose we have this evidence: we have examined10,000 emeralds and they are all green– Is this evidence for: all emeralds are green– Not if we also believe X: There are many emeralds andthey are 99% green OR all emeralds are green butthere are very few emeralds in the worldNotice that X is not a level 2 statementSober’s thesis: no strictly level n statements justify anylevel n+1 statements– Why? Because of the relativity of evidence– What if we had no level n+1 beliefs?– Then we could say nothing about level n+1 based onlyon level n evidence (except for trivial or a priori truths)– Is that true?

PHLA10 12The Problem of Induction●Anti-foundationalism about justification– Is ‘rational justification’ strictly about inter-leveljustification?● If so, Sober thinks it’s impossible to achieve– Or is there a sense of ‘rational justification’ that takesinto account our current epistemic position?– That is, could we say something like● Given our current epistemic situation (what webelieve now) evidence E would justify belief P– This assumes an idea of ‘shared epistemic situation’– Belief conservatism: if it ain’t broke don’t fix it● Maybe all justification is relative to an epistemicsituation = our current beliefs / world view– So, has induction been justified ???