The Problem of Induction

1. The Problem of Induction Reading: ‘The Problem of Induction’ by W. Salmon

2. Induction and science • We’ve seen that Popper denies that there is a genuine problem about the mode of inference and reasoning used in science. • He claims that the only kind of inference used in science is deductive inference: reasoning which is such that if the premises are true, then the conclusion must be true.

3. Induction and science • In fact, Popper believes that the only logical principle at work in scientific reasoning is the deductive principle known as modus tollens: • From [If p, then q] • And, [not-q] • Infer: [not-p] • Modus tollens shows up in attempts at falsification: A theory entails p, but p turns out false, so the theory is rejected.

4. Induction and science • Popper’s view is not very popular. It is generally held, by philosophers of science and scientists themselves, that scientific reasoning involves non-deductive inferences: inferences that don’t “necessitate” their conclusions, but nevertheless justify belief in them. • Consider the process by which the law that water boils at 100 degrees is established. • A host of instances--then the postulation of the law.

5. Induction and science • Notice that, as far as what is logically possible, the evidence that is amassed—concerning the boiling point of observed samples of water—is consistent with the falsity of the general law. • So the reasoning involved is non-deductive. • That science uses non-deductive reasoning isn’t surprising. It is, it would seem, the only kind of reasoning that can justify conclusions about the unobserved or the future.

6. Induction and science • And we all believe that we have some knowledge of the unobserved, not just the knowledge that science delivers, but everyday, commonsense knowledge as well. • “This sandwich won’t poison me.” • “The ground will support my next steps.” • “The sun will rise again tomorrow.”

7. Induction and science • But there is a problem with the consensus view that science—and common sense—uses non-deductive reasoning. There is an argument, due to David Hume, that concludes that no conclusions based on non-deductive reasoning can ever be justified.

8. Clarifications and distinctions • How we come to have the beliefs about the unobserved that we do have vs. Whether any such beliefs are justified. (The “descriptive problem” vs. the “normative problem”.) • The interesting problem is the normative one. • Knowledge of, or justified belief in, unobserved matters of fact. • Matters of fact vs. relations of ideas • “All bachelors are unmarried.”— relation of ideas

9. Clarifications and distinctions • “(All) water boils at 100 degrees.”—matter of fact. • P is a matter of fact iff its denial is conceivable, possible, consistent. • P is a relation of ideas iff its denial is inconceivable, impossible, inconsistent. • The problem: Are any of our claims about unobserved matters of fact justified (Do any count as knowledge)?

10. Clarifications and distinctions • Non-deductive inference vs. inductive inference. • Non-deductive inferences include various fallacies and mis-reasonings. • Intuitively however, there are kinds of non-deductive inferences that justify the beliefs based on them—these are inferences that conform to certain acceptable-seeming rules. • E.g. the rule that a “representative sample” of instances confirms a generalization.

11. Clarifications and distinctions • These rule-based non-deductive inferences = inductive inferences. • The kind of inferences we use to generate conclusions about the unobserved are inductive, not simply non-deductive. • But Hume’s argument applies to any non-deductive inference and so applies to inductive inferences as well.

12. Clarifications and distinctions • Ampliative inference vs. non-ampliative inference. • Why are we justified in believing the conclusions of our deductive inferences? (How do such inferences manage to necessitate their conclusions?) • Salmon’s answer: such inferences are non-ampliative: There is no information in the conclusion not already contained in the premises.

13. Clarifications and distinctions • All men are mortal. • Socrates is a man. • Therefore, Socrates is mortal. • Inductive inferences are ampliative: Their conclusions say more than their premises do. • a is G • b is G • All Fs are G.

14. The precise problem • Are any of our inductive (i.e. non-deductive, rule-based) inferences concerning unobserved matters of fact justified, given that they can’t be justified in the way in which our deductive inferences are justified (by being non-ampliative)? • Hume’s answer: No.

15. The skeptical argument • A typical induction • Swan 1 is white • Swan 2 is white. • ... • Swan 2,340 is white • Therefore, (All) swans are white. • What justifies the conclusion?

16. The skeptical argument • Conformity to the rule? • What justifies the rule? • Perhaps inductions are deductions in disguise, ones that appeal to a very general principle, the Principle of the Uniformity of Nature (PUN). • (PUN): If a generalization (All Fs are G) held in the past, or of observed instances, then it will hold in the future, or of unobserved instances.

17. The skeptical argument • The observed swans are white. • (PUN): If a generalization holds of the observed instances, then it holds of the unobserved instances as well. • Therefore: (All) swans are white. • Now the argument is deductive. • The problem: What justifies (PUN)?

18. The skeptical argument • The only thing that is even a candidate for justifying (PUN) is an inductive inference. • (PUN) does not express a relation of ideas and it is, in part, about the unobserved/the future. Hence (PUN) is not knowable except by experience, and is not knowable by perception or memory either. • The big problem: (PUN) is our candidate justifier for every inductive argument. To then give it an inductive justification would be to reason in a circle.

19. The skeptical argument • (PUN) Slogan Form: The future will resemble the past. • What might the argument look like? • In the past, the future has resembled the past. (Past pasts have resembled past futures). • Therefore: In the future, the future will resemble the past. (Future futures will resemble future pasts.) • The obvious question: What justifies this induction? (PUN) itself? Circular.

20. Hume’s dilemma • Horn 1: There can be no inductive justification of induction (e.g. via (PUN)) because such justifications presuppose that some inductions are justified. • Horn 2: There can be no deductive justification of induction because deductive inferences are non-ampliative, but inductive inferences are ampliative.

21. Solutions? • Solution 1—the “success of science” solution: The reason we are justified in using inductive methods is that they work. Science, which utilizes such methods, has been massively successful in predicting the course of the future and providing us with an understanding of the natural world. To doubt its methodology is ridiculously irrational. • The problem with solution 1: It appears to be another instance of the inductive justification of induction, which, as we have seen, is circular.

22. Solutions? • Premise: Observed cases of the application of the scientific method have yielded successful predictions. • Conclusion: Unobserved cases of the application of the scientific method will yield successful predictions. • The argument is clearly another inductive, ampliative inference: precisely the sort whose justification is in question.