8. Inductive Arguments

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Geraldine Cannon
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1 8. Inductive Arguments 1 Inductive Reasoning In general, inductive reasoning is reasoning in which we extrapolate from observed experience (e.g., past experience) to some conclusion (e.g., about present or future experience). We can see immediately that inductive reasoning depends on an assumption: It assumes that observed cases can provide information about unobserved cases, that the future will resemble the past. 2 1

We can see immediately that inductive reasoning depends on an assumption: It assumes that observed cases

2 Inductive Arguments Govier (293) lists four characteristics of inductive arguments: 1. Premises and conclusion are all empirical propositions. 2. Conclusion is not deductively entailed by premises. 3. Reasoning used to infer the conclusion is based on the assumption that the regularities described in the premises will persist. 4. Inference is either that unexamined cases will resemble examined ones or that evidence makes an explanatory hypothesis probable. 3 The Problem of Induction Influentially set out by the Scottish philosopher David Hume ( ). 1. Every day I can remember, the sun has risen 2. The sun will rise tomorrow. This inference is not deductively valid. It is logically possible that the sun will not rise tomorrow. Inductive inference cannot provide the kind of certainty that deductive inference (sometimes) can, yet we cannot learn, cannot get by in life, without reasoning inductively. 4 2

Inference is either that unexamined cases will resemble examined ones or that evidence makes an explanatory hypothesis probable.

3 As Govier points out, we can transform inductive inferences into deductively valid arguments by supplying an additional premise: 1. Every day I can remember, the sun has risen 3. The future will resemble the past (assumption) 2.The sun will rise tomorrow. The question then arises, however, whether 3. is an acceptable premise. The reconstructed argument above begs the question because 3. already implicitly assumes the conclusion. 5 Similarly Hume s argument can represented: 1. Only deductively valid arguments demonstrate their conclusion. 2. Inductive arguments are not deductively valid. 3. Inductive arguments do not demonstrate their conclusion. A valid argument; but while premise 2. is true, we can question the truth (the acceptability) of 1. Following Hume, the question of whether or not 1. is acceptable (and what that might mean) is a big question in philosophy. For present purposes, Govier stipulates that it is not acceptable. 6 3

The reconstructed argument above begs the question because 3. already implicitly assumes the conclusion. 5 Similarly Hume s argument can represented: 1.

4 Inductive Generalizations One of the most common forms of inductive inference. Can be represented as an argument in which the premises describe a number of observed objects as having some property and the conclusion asserts, on the basis of these observations, that all or most objects of the same type will have this property 7 Inductive Generalization: Example 1. In 2004, 20% of FNUC students traveled to campus by public transit. Therefore, probably 2. In 2005, about 20% of FNUC students will travel to campus by public transit. The inference is an extrapolation from observed experience this year to a prediction about next year. It rests on an assumption that nothing relevant to the inference will change (e.g., a prolonged transit strike.) Notice: probably the premise does not guarantee the truth of the conclusion, it only asserts that the conclusion is probable. 8 4

objects of the same type will have this property 7 Inductive Generalization: Example 1. In 2004, 20% of FNUC students traveled to campus by public transit. Therefore, probably 2.

5 Samples and Populations It is often impractical to observe every member of the target population (the group under consideration that we wish to generalize about). But, provided that we observe a sample that is sufficiently large and sufficiently representative, we need not observe every member of the target population in order to make a well-supported inductive inference 9 Sampling: Examples Just one observation of the effect of cold metal on a human tongue is enough for most kids to form a good generalization. Similarly, we need not observe the case history of every smoker who has ever lived in order to conclude that smoking is a health hazard. On the other hand, someone who concludes that all the good ones are taken on the basis of two bad dates might sensibly be advised to keep looking. 10 5

inductive inference 9 Sampling: Examples Just one observation of the effect of cold metal on a human tongue is enough for most kids to form a good generalization.

6 Some Nomenclature Most Albertans approve of the Klein government s performance to date. Across the province 56 percent of 1,004 people interviewed approved of the government s performance, compared to 40 percent who disapproved. Sample: 1,004 Albertans Target population: Most Albertans (i.e., more than 50% of about 3,183,312 people) 11 Some More Nomenclature Prediction: Generalization about will probably happen in the future based on what has happened in the past. Retrodiction: Generalization about what probably happened in the past based on what happens in the observable present. (E.g., ash patterns from Mt. St. Helens might be used to explain archaeological evidence from Mt. Vesuvius.) 12 6

7 Good Samples Is a sample of 1,004 Albertans truly representative of most Albertans? Maybe, maybe not. The study cited in Govier deals with voting behaviour, so we can infer that the sample consisted of adults (children where excluded). Moreover, we are told that the study was conducted by telephone (people who do not own telephones likely including homeless and very poor people were excluded). 13 Sample Representativeness A (truly) random sample: Every member of the target population has an equal chance of being chosen. (e.g., the names of 3 million Albertans placed in container and 1,004 names are drawn at random) Strictly speaking, the whole mathematical apparatus of statistics is inapplicable when the sample is not random (297). Yet for obvious practical reasons most statistical samples are not truly random. 14 7

Moreover, we are told that the study was conducted by telephone (people who do not own telephones likely including homeless and very poor people were excluded).

8 Improving Sampling Size matters Your neighbour owns a pit bull and it is quite friendly. Can you rightly conclude from this that all pit bulls are friendly? to a point. As we ve already seen, however, even more important than sample size, is sample representativeness We conduct a survey of attendees at a Mendel Art Gallery fundraiser and find that 70% of them own expensive imported cars. Can we rightly conclude from this that 70% of all people own expensive imported cars? 15 To improve representativeness we can, e.g., randomize the distribution of a questionnaire, and try to ensure that no relevant sub-group in the target population has been excluded. A sample is perfectly representative iff it resembles the population in all respects relevant to the topic being explored. (299) A paradox: In order to choose a perfectly representative sample we would already need to everything relevant to the population we are studying. 16 8

expensive imported cars. Can we rightly conclude from this that 70% of all people own expensive imported cars? 15 To improve representativeness we can, e.g., randomize the distribution of a questionnaire, and try to ensure that no relevant sub-group in the target population has been excluded.

9 Stratified Sampling Another technique to improve sampling. If we know that the target population of some distinct sub-groups A, B, and C, then a good stratified sample will be made of As, Bs and Cs in the same proportion in which they occur in the target population. E.g., Gallup polls, Nielsen ratings 17 Variability Size matters, but not always If a population does not vary at all with respect to the property that we are interested in then a sample of one would be suffice to draw a good generalization. For cases where there is variability, the greater the variability the larger the sample required in order to ensure representativeness 18 9

10 But again the paradox: If already knew the exact degree of variability in the population, there would be no need for the sample. Nonetheless, it is important to try have at least as much variety in the sample as know to exist in the population. Many studies fail in this regard: e.g., medical studies that include only men in order to simplify testing. Opposite case: Zelnorm. 19 Biased Samples A sample that demonstrably misrepresents the target population. E.g., a study about use distributed only via a self-selected sample Notorious case: Carol Gilligan s criticism of Kohlberg s research on moral development

rd: e.g., medical studies that include only men in order to simplify testing. Opposite case: Zelnorm.

11 Five Guidelines for Evaluating Inductive Generalizations 1. Try to determine what the sample is and what the population is. If it is not stated what the population is, make an inference as to what population is intended, relying on the context for cues. 2. Note the size of the sample. If the sample is [smaller] than 50, then, unless the population is extremely uniform or itself very small, the argument is weak. 3. Reflect on the variability of the population with regard to the trait or the property x that the argument is about. If the population is not known to be uniform with regard to x, the sample should be large enough to reflect the variety of the population Reflect on how the sample has been selected. Is there any likely source of bias in the selection process? If so, the argument is inductively weak. 5. Taking the previous considerations into account, try to evaluate the representativeness of the sample. If you can give good reasons to believe that it is representative of the population, the argument is inductively strong. Otherwise, it is weak

If the sample is [smaller] than 50, then, unless the population is extremely uniform or itself very small, the argument is weak. 3.

A Few Basics of Probability

A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study