Inductive Reasoning

1. Inductive Reasoning

2. The Nature of Inductive Reasoning • What is an inductive argument? • Any argument which is not deductive! • I.e., any argument which does not provide a guarantee of the truth of the conclusion if the premises are true. • Inductive arguments are probabalistic.

3. The Nature of Inductive Reasoning • What else? • Inductive arguments are ampliative, while deductive arguments are non-ampliative. • An argument is ampliative (defn) iff there is information contained in the conclusion that is not already contained in the premises. • That’s the trade-off! You lose the guarantee of the truth of the conclusion for amplification.

4. The Nature of Inductive Reasoning • The logical strength of inductive arguments is not dependent on the form of the argument, but rather the content of the premises. (It’s the opposite for deductive arguments.)

5. The Nature of Inductive Reasoning • However, there are 4 forms of inductive arguments that usually regarded as logically strong so long as certain conditions are met. • Inductive generalization • Statistical syllogism • Induction by confirmation • Analogical reasoning

6. Inductive Generalization • Has the following form: Z percent of observed F’s are G It is probable, therefore, that Z percent of all F’s are G.

7. Inductive Generalization • E.g. • 60% of students at STFX who were questioned believe in God. It is probable, therefore, that 60% of students at STFX believe in God.

8. Inductive Generalization • When assessing these arguments, ask: • Is the sample representative? • Is the sample large enough?

9. Statistical Syllogism • Has the following form: Z percent of all F’s are G x is an F Is it probable to the degree 0.Z that x is G

10. Statistical Syllogism • What’s the difference between an inductive generalization and a statistical syllogism? • Inductive generalizations reason fromparticular observations to a general claim about a class. • Statistical syllogisms reason from a general claim about a class to a claim about a particular individual.

11. Statistical Syllogism • E.g., 60% of students at STFX believe in God. Bob is a student at STFX. Therefore, there is a .6 degree of probability that Bob believes in God.

12. Statistical Syllogism • When assessing these arguments, ask: • Is there any additional information about x that has not been included in the premises? • E.g. Bob is President of Catholic League of Students (prob that he believes in God increases). • E.g., Bob is President of the Atheists for the Environment Society (Prob that he believes in God decreases).

13. Induction by Confirmation • Induction can be used to support a hypothesis or theory by providing confirming instances of that hypothesis or theory. • When we propose a theory or hypothesis, there are certain things that ought to be observed if it is actually true (or probable). • These are called observation statements. If we do observe what the theory predicts, then we have confirmed the theory.

14. Induction by Confirmation • Induction by Confirmation then has the following form: If h then o o It is probable that h NB: similar to the formal fallacy of “affirming the consequent”

15. Induction by Confirmation • E.g. If the theory of general relativity is true, then it follows that light rays passing near the sun will bend. During the solar eclipse of 1919 it was observed that light rays passing near the sun did bend. It is probable therefore that the theory of general relativity is true.

16. Induction by Confirmation • When assessing these arguments, ask: • Is the number of confirming instances relatively high? • In general, the more confirming instances the better the theory. • Are there any disconfirming instances? • Any disconfirming instance refutes the theory.

17. Induction by Confirmation • Disconfirming instances are regarded as refutations of a theory because such a refutation takes this form: If h then o Not-o Therefore not-h • That is, a disconfirming instances refutes a theory because we are dealing with a deductively valid argument form: Modus Tollens (denying the consequent).

18. Analogical Reasoning • Analogical reasoning works by comparing things which are similar (analogous) and concluding that properties or relations that one thing has must also be present in the other.

19. Analogical Reasoning • E.g., Last year I put some fertilizer on my strawberries and in the fall got about 20 per cent more strawberries. You should do the same with your strawberries, since you got the same kind of soil. You’ll probably get more strawberries too.

20. Analogical Reasoning • Analogies compare two cases: the subject case, and the analogue case. • The subject case is the case about which we are trying to derive a conclusion (fertilizer on your soil) • The analogue case is the case about which we are more familiar (fertilizer on my soil).

21. Analogical Reasoning • The conclusion in an analogy makes a claim about the subject case, and in particular states that the subject case will (probably) have the target feature. • The target feature (increase in strawberry production) is the feature that is present in the analogue case, and it is being concluded that it (probably) is in the subject case.

22. Analogical Reasoning • There are two kinds of analogical arguments: • Analogical Argument by Properties • Analogical Argument by Relations

23. Analogical Argument by Properties • Analogical Argument by Properties has the following form: x has A, B, C. [analogue case] y has A, B. [subject case] It is probable therefore that y has C [target feature]

24. Analogical Argument by Properties • E.g., Canada geese are water birds that nest in Canada in the early spring and migrate south to warmer climates for the winter months. Ducks are also water birds that nest in Canada in early spring. Therefore, ducks probably migrate south for the winter, too.

25. Analogical Argument by Properties P1 [analogue case]: Canada geese are water birds that nest in Canada in the early spring and migrate south to warmer climates for the winter months. P2 [subject case]: Ducks are also water birds that nest in Canada in early spring. Conclusion: Therefore, ducks probably migrate south for the winter [target feature], too.

26. Analogical Argument by Relations • Analogical Argument by Relations has the following form: x is to y [analogue case] as a is to b [subject case]. x is R to y. It is probable therefore that a is R to b [target feature]

27. Analogical Argument by Relations • E.g., The proposal to give clean needles to prison inmates to stop the spread of AIDS from the use of dirty needles is ridiculous. It is like giving bank robbers normal bullets to stop them from using dum-dum bullets, which are much more damaging to the victim.

28. Analogical Argument by Relations P1: Dum-dum bullets are to normal bullets (as used by bank robbers) [analogue case] as dirty needles are to clean (as used by prison inmates) [subject case]. P2: Although dum-dum bullets are much more damaging to the victim, normal bullets still kill their victims. Further, the role of police officers is to stop bank robbers, not prevent the harms they cause. Conclusion: Although dirty needles are more damaging to the victim (addicts are likely to get HIV, etc.), clean needles can be just as damaging (e.g., overdoses). Further, the role of prison officials is to stop drug use, not prevent the harms caused by it.

29. Analogical Reasoning • When assessing these arguments, ask: • How many entities are we comparing? • What is the variety of dissimilarity? • In how many respects are the entities similar? • Are the respects in which the compared entities are similar relevant to the conclusion? • In what ways are the entities under consideration dissimilar? • How bold or modest is the conclusion?