The Problem of Induction

Aristotle’s Inductions Aristotle’s structure of knowledge consisted of explanations such as: Aristotle’s structure of knowledge consisted of explanations such as: Statue is Bronze Bronze is Brown Statue is Brown Eventually fundamental truths need to be known through sensation or induction Eventually fundamental truths need to be known through sensation or induction

Aristotle’s Inductions ‘Induction proceeds through an enumeration of particular cases’: ‘Induction proceeds through an enumeration of particular cases’: Bronze thing 1 is brown Bronze thing 2 is brown … Bronze is Brown Recall: in a good induction, if the premises are true they make the conclusion probably true Recall: in a good induction, if the premises are true they make the conclusion probably true

Inductions and Science Modern science also uses inductions to discover regularities in nature Modern science also uses inductions to discover regularities in nature Sometimes these are formalised as Laws of Nature Sometimes these are formalised as Laws of Nature Boyle’s Law Boyle’s Law p 1 V 1 = p 2 V 2

Inductions and Science Modern science uses these laws in a similar way to Aristotle Modern science uses these laws in a similar way to Aristotle p 1 = 10kPa V 1 = 1l V 2 = ½l p 1 V 1 = p 2 V P 2 = 20kPa

Hume’s Problem What faith can we have in ‘laws’ derived by induction? What faith can we have in ‘laws’ derived by induction? Hume says: not much. Hume says: not much. There are only two known ways to justify induction, and neither of them will work There are only two known ways to justify induction, and neither of them will work Induction! Induction! Deduction Deduction

Hume’s Problem Induction: We should trust induction because it’s worked in the past Induction: We should trust induction because it’s worked in the past Induction was successful in case 1 Induction was successful in case 2 … Induction is a generally successful method But this depends upon trusting induction! But this depends upon trusting induction!

Hume’s Problem Deduction: We can demonstrate deductively that induction works Deduction: We can demonstrate deductively that induction works Suppose that were true, then there’s some sound argument that goes Suppose that were true, then there’s some sound argument that goes Premiss 1 Premiss 2 … Induction works

Hume’s Problem Deduction: We can demonstrate deductively that induction works Deduction: We can demonstrate deductively that induction works Then when we have a normal inductive argument we can slip in the conclusion of that argument Then when we have a normal inductive argument we can slip in the conclusion of that argument Bronze thing 1 is brown Bronze thing 2 is brown Induction works Bronze is brown But it just isn’t logically true that bronze is brown

Hume’s Problem Probability: Restate principle of induction as a probabilistic rule. Probability: Restate principle of induction as a probabilistic rule. Bronze thing 1 is brown Bronze thing 2 is brown … Bronze thing n is brown Bronze is probably brown But this still isn’t good enough But this still isn’t good enough What if we now find n+1 green bronze things? What if we now find n+1 green bronze things?

Popper’s Solution Reject the idea of induction being fundamental to successful science Reject the idea of induction being fundamental to successful science We think science is successful because induction generates true laws from observations We think science is successful because induction generates true laws from observations If induction doesn’t work, science doesn’t work But science does work So induction must work

Popper’s Solution We’ve already seen how science works We’ve already seen how science works Elenchus (Aristotle, from Socrates) Elenchus (Aristotle, from Socrates) Modus Tollens Modus Tollens Hypothetico-Deductive Method Hypothetico-Deductive Method Hypothesis 1:There were land bridges Consequence:If there were land bridges there would be traces of them Observation:There are no traces of them Conclusion:There were no land bridges Hypothesis 2:…

Popper’s Solution Science doesn’t work by generating reliably true theories by induction Science doesn’t work by generating reliably true theories by induction Science works by eliminating demonstrably false theories by deduction Science works by eliminating demonstrably false theories by deduction This is the Falsificationist view of Science This is the Falsificationist view of Science

Popper’s Solution If we can’t observationally disprove X, does that prove X? If we can’t observationally disprove X, does that prove X? If Socrates couldn’t disprove Euthyphro’s claim about piety, would that show E was right? If Socrates couldn’t disprove Euthyphro’s claim about piety, would that show E was right? No, the definition is still provisional No, the definition is still provisional When observations are consistent with a theory, and don’t disprove it When observations are consistent with a theory, and don’t disprove it They are said to confirm it. They are said to confirm it. They can’t prove it. They can’t prove it.

Popper’s Solution What would it look like to observationally ‘prove’ a theory? What would it look like to observationally ‘prove’ a theory? If the theory T is true then we should observe X We do observe X Theory T is true This is the fallacy of affirming the consequent This is the fallacy of affirming the consequent

Popper’s Solution What would it look like to observationally ‘prove’ a theory? What would it look like to observationally ‘prove’ a theory? If my battery is flat then my car won’t start My car won’t start My battery is flat This is the fallacy of affirming the consequent This is the fallacy of affirming the consequent

Popper’s Solution Theories are always ‘provisional’ Theories are always ‘provisional’ Better or worse confirmed Better or worse confirmed Science isn’t a structure of necessary truths Science isn’t a structure of necessary truths It’s a system of hypotheses, constantly being improved It’s a system of hypotheses, constantly being improved

Objections to Popper’s Solution Popper’s view explains the process Popper’s view explains the process Ptolemy  Copernicus  Kepler  But, surely, we really do know things about the world, and these things are known by induction But, surely, we really do know things about the world, and these things are known by induction What happens if I drop this pen? What happens if I drop this pen? It falls It falls I turn into a duck I turn into a duck

Goodman’s New Problem Induction works on properties like ‘brown’ Induction works on properties like ‘brown’ Bronze thing number 1 is brown Bronze thing number 2 is brown … Bronze thing number n is brown Bronze is brown

Goodman’s New Problem We can define a property ‘brue’ as: We can define a property ‘brue’ as: Something is ‘brue’ if it is first observed before [tomorrow’s date] and is brown, or is not first examined before [tomorrow’s date] and is blue. B ronze thing number 1 is brue (it was seen before tomorrow and was brown) Bronze thing number 2 is brue (ditto) … Bronze thing number n is brue (ditto) Bronze is brue

Goodman’s New Problem Something wrong here Something wrong here Two different conclusions from two good inductions on exactly the same observations Two different conclusions from two good inductions on exactly the same observations The problem seems to be that ‘brue’ isn’t the right sort of property (it’s not ‘projectible’) The problem seems to be that ‘brue’ isn’t the right sort of property (it’s not ‘projectible’) What makes a predicate projectible? What makes a predicate projectible? If it’s the sort of predicate we’re accustomed to using in inductions If it’s the sort of predicate we’re accustomed to using in inductions Not a very informative answer Not a very informative answer