The Structure of Statements: Translating If and And Statements

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Module 4 Readings and Assignments

Complete the following readings early in the module:

Module 4 online lectures

From your textbook, Schaum’s outline of logic, 2nd , read the following:

From the section, Induction, read:

Statement Strength

Induction by Analogy

Mill's Methods

Scientific Theories

From the section, Propositional Logic, read the following sections:

Argument Forms to Formalization

SCHAUM'S OUTLINE OF: THEORY AND PROBLEMS OF LOGIC

Second Edition

JOHN NOLT, Ph.D.

Associate Professor of Philosophy

University of Tennessee

DENNIS ROHATYN, Ph.D.

Professor of Philosophy

University of San Diego

ACHILLE VARZI, Ph.D.

Assistant Professor of Philosophy

Columbia University

069-7786412

JOHN NOLT is Associate Professor of Philosophy at the University of Tennessee, Knoxville, where he has taught since receiving his doctorate from Ohio State University in 1978. He is the author of Informal Logic: Possible Worlds and Imagination and numerous articles on logic, metaphysics, and the philosophy of mathematics.

DENNIS ROHATYN is Professor of Philosophy at the University of San Diego, where he has taught since 1977. He is the author of Two Dogmas of Philosophy, The Reluctant Naturalist, and many other works. He is a regular symposiast on critical thinking at national and regional conferences. In 1987 he founded the Society for Orwellian Studies.

ACHILLE VARZI is Assistant Professor of Philosophy at Columbia University, New York. His works include Holes and Other Superficialities and Fifty Years of Events: An Annotated Bibliography (both with Roberto Casati) and numerous articles on logic, formal semantics, and analytic metaphysics.

Schaum's Outline of Theory and Problems of LOGIC

Copyright © 1998, 1988 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any forms or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

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ISBN 0-07-046649-1

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Library of Congress Cataloging-in-Publication Data

Front Matter

Chapter 1: Argument Structure

Chapter 2: Argument Evaluation

Chapter 3: Propositional Logic

Chapter 4: The Propositional Calculus

Chapter 5: The Logic of Categorical Statements

Chapter 6: Predicate Logic

Chapter 7: The Predicate Calculus

Chapter 8: Fallacies

Chapter 9: Induction

Chapter 10: The Probability Calculus

Chapter 11: Further Developments in Formal Logic

Back Matter

9.1: STATEMENT STRENGTH

This chapter concerns some common kinds of inductive reasoning. In an inductive argument, the conclusion need not follow from the premises as a matter of logical necessity. Rather, in inductive reasoning we are concerned with the probability of the conclusion, given the premises, i.e., the inductive probability of the argument (see Section 2.3). Since inductive probability depends in turn on the relative strengths of the premises and conclusion, we begin with a discussion of statement strength.

The strength of a statement is determined by what the statement says. The more it says, the stronger it is, regardless of whether what it says is true. A strong statement is true only under specific circumstances; the world must be just so in order for it to be true. A weak statement is true only under a wide variety of possible circumstances; it says nothing very specific, and it demands little of the world for its truth.

SOLVED PROBLEM

9.1

Some of the following statements are quite strong; the rest are quite weak. Distinguish the strong statements from the weak ones.

(a) There are exactly 200 cities with populations over 100,000 in the United States.

(b) Something is happening somewhere.

(d) Hobbits are humanoid creatures, rarely over a meter tall, with ruddy faces and woolly toes, that inhabit burrows on hillsides in a land called The Shire.

(e) If there are any crows, then some of them are male.

(f) Jim's house is the third house on the left as you turn from Concord Street onto Main.

(g) Every organism now alive has obtained its genetic material from once-living organisms.

(h) Some people are sort of weird.

(i) Every vertebrate has a heart.

(j) It is not true that Knoxville, Tennessee, has exactly 181,379 inhabitants this very second.

Solution

(a) Strong

(b) Weak

(d) Strong

(e) Weak

(f) Strong

(g) Strong

(h) Weak

(i) Strong

(j) Weak

It may seem surprising that statement (j) is weak, since this statement is very specific with respect to time, place, and number. But note that all it says is that there were not exactly this many people at this place and time; it therefore says very little, since it allows the population of Knoxville to be anything but 181,379 (and is true even if Knoxville doesn't exist). If we omit the phrase ‘It is not true that’, this statement becomes very strong, but when this phrase is included it is very weak.

Indeed, this holds in all cases; the negation of a weak statement is strong and the negation of a strong statement is weak. Consider, for example, the negation of the weak statement ‘Something exists’. This is ‘Nothing at all exists’, which says a great deal, for it asserts that the universe is utterly devoid of anything. Even the slightest little flicker of existence would make it false. Thus it is quite informative and, fortunately, quite false.

The strength of a statement is approximately inversely related to what is called its a priori probability, that is, probability prior to or in the absence of evidence. (We say “approximately” because there are various conceptions of a priori probability which differ in detail, and there is no agreement as to which, if any, is the exact inverse of statement strength.) The stronger a statement is, the less inherently likely it is to be true; the weaker it is, the more inherently probable it is.

The strongest possible statements are those which say so much that they cannot be true, i.e., self-contradictions. The statement ‘Waldo both is and is not a cat’, for example, is maximally strong and thus has an a priori probability of zero. The strength of this statement is perhaps best appreciated by the observation that since it is self-contradictory it logically implies every statement.

The weakest possible statements are those which are logically necessary. Because logically necessary statements are true under all possible circumstances, in a sense they say nothing at all. The tautology ‘If grass is green, then grass is green’, for example, is logically necessary and therefore maximally weak. Its a priori probability is 1.

Comparisons of strength are not always possible. Among the five strong statements in Problem 9.1, for instance, it is impossible to say which is the strongest. All are stronger than any of the five weak statements, but no method is known for making precise comparisons among such diverse statements.

It is possible, however, to rank some sets of statements with respect to relative strength. This is achieved by the following rules:

Rule 1: If statement A deductively implies statement B but B does not deductively imply A, then A is stronger than B.

Rule 2: If statement A is logically equivalent to statement B (i.e., if A and B deductively imply one another), then A and B are equal in strength.

The justification of these two rules rests on the fact that if statement A logically implies statement B, then there are no possible circumstances in which A is true and B is false. Thus the possible circumstances in which A is true are a subset of the possible circumstances in which B is true. Now if (as in rule 1) B does not imply A, then there are possible circumstances in which B is true and A is false. Hence A is true in fewer possible circumstances than B is, which is just to say that A is stronger than B. But if (as in rule 2), B does imply A, then A and B are true under precisely the same sets of circumstances, and hence they are equal in strength.

SOLVED PROBLEMS

9.2

Rank the following statements in order from strongest to weakest:

(a) Either some cows are horned or some buffalo are horned.

(b) There are cows and buffalo, and all cows and buffalo are horned.

(d) Some cows are horned.

(e) Either some cows are horned, or it is not the case that some cows are horned.

(f) Some cows are both horned and not horned.

Solution

(f), (b), (c), (d), (a), (e). This solution is obtained by applying rule 1 and the predicate calculus. One can see by using, for example, the truth tree method of Chapter 6 that each item in the solution list deductively implies all succeeding items, but that no item in the list deductively implies an earlier item. Note that statement (f) is self-contradictory, while (e) is logically necessary.

9.3

Compare the strengths of the following statements:

(a) Adair admires Adler.

(b) It is not the case that Adair doesn't admire Adler.

(d) Adair admires something which is identical with Adler.

(e) Adair admires Adler, and if it rains then it rains.

Solution

These statements are logically equivalent, as the reader can check again by the techniques of Chapter 6; hence, by rule 2 they are all of equal strength. Notice that statement (e) is just the conjunction of statement (a) with the tautology ‘if it rains, then it rains’. This tautology in effect says nothing, so that conjoining it with statement (a) produces a statement equivalent to (a).

Rules 1 and 2, however, are not applicable in every case. None of the strong statements of Problem 9.1, for example, deductively implies any of the others. Moreover, the differences in strength among them (if any) are too small to be intuitively apparent. Thus we have no way of ordering these strong statements with respect to strength. The same applies to the weak statements of Problem 9.1.

The logical importance of statement strength lies in its relation to inductive probability. The general rule is that inductive probability tends to vary directly with the strength of the premises and inversely with the strength of the conclusion.

SOLVED PROBLEMS

9.4

What is the effect on the following argument form of increasing the number n?

We have observed at least n daisies, and they have all had yellow centers.

∴ If we observe another daisy, it will have a yellow center.

Solution

The premise gets stronger as the number n gets larger. As a result, each increase in n also increases the argument's inductive probability. This argument form is an instance of simple induction, which will be discussed more thoroughly in Section 9.4.

9.5

Suppose that the mean height for an adult male in the United States is 5 feet 10 inches. We wish to use this fact as a premise to draw a conclusion about the height of X, an American man whom we have not yet met. Below are three different conclusions which we could draw. Which conclusion produces the strongest argument?

(a) X is exactly 5 feet 10 inches tall.

(b) X is within a foot of 5 feet 10 inches tall.

Solution

The inductive probability (and hence the strength) of the argument will be higher the weaker we make the conclusion. Conclusion (a) is the strongest of the three conclusions; (b) is the weakest. Thus (b) produces the strongest argument and (a) the weakest.

Strengthening the premises or weakening the conclusion of an argument may not increase the argument's inductive probability, however, if we alter their content in a way that disrupts relevance.

SOLVED PROBLEM

9.6

Evaluate the inductive probability and degree of relevance of the following argument:

Ninety-five percent of American families have indoor plumbing, a telephone, a television, and an automobile.

The Joneses are an American family.

∴ The Joneses have indoor plumbing, a telephone, a television, and an automobile.

What happens to inductive probability and relevance if we replace the conclusion with the following weaker but less relevant statement?

(a) The Chun family of Beijing owns at least one automobile.

What happens if we replace the original conclusion with this weaker but still highly relevant statement?

(b) The Joneses have indoor plumbing.

Solution

The original argument has both high relevance and a fairly high inductive probability, and its conclusion is strong. Conclusion (a) is weaker, but when we substitute it for the original conclusion both the relevance and inductive probability of the argument decrease significantly. If we replace the conclusion by (b), on the other hand, we have weakened the conclusion while preserving relevance and thereby created a stronger argument. The inductive probability of this new argument is at least as high as that of the original, and we may reasonably suppose that it is somewhat higher.

Since in practice we are seldom concerned with modifications to premises or conclusions unless they preserve relevance, we seldom encounter exceptions to the rule that strengthening premises or weakening the conclusion increases inductive probability.

9.2: STATISTICAL SYLLOGISM

Inductive arguments are divisible into two types, according to whether or not they presuppose that the universe or some aspect of it is or is likely to be uniform or lawlike. Those which do not require this presupposition may be called statistical arguments; the premises of a statistical argument support its conclusion for purely statistical or mathematical reasons. Those which do require it we shall call Humean arguments, after the Scottish philosopher David Hume, who was the first to study them thoroughly and to question this presupposition.

SOLVED PROBLEMS

9.7

What type of inductive argument is the following?

98 percent of college freshmen can read beyond the sixth-grade level.

Dave is a college freshman.

∴ Dave can read beyond the sixth-grade level.

Solution

This argument is statistical. The conclusion is quite likely, given the premises, on statistical grounds alone. (Of course, if there were some evidence that Dave's reading skills were deficient, then the conclusion would not be so likely, but that is not a special feature of this argument; all inductive arguments are vulnerable to contrary evidence. See Section 2.5.)

9.8

What type of inductive argument is the following?

Each of the 100 college freshmen surveyed knew how to spell ‘logic’.

∴ If we ask another college freshman, he or she will also know how to spell ‘logic’.

Solution

This is a Humean argument. It is clearly nondeductive, since it is possible for its premise to be true while its conclusion is false. And it is not statistical, since its inductive probability depends on how likely it is that future observations of college freshmen resemble past ones. Any estimate of the inductive probability of this argument presupposes an appraisal of the degree of uniformity or lawlikeness of certain events—in this case, events involving the spelling abilities of college freshmen. That is the hallmark of a Humean argument.

In the remainder of this section and in Section 8.3, we focus on inductive arguments that are statistical. We shall come back to various forms of Humean inference in the second part of this chapter.

The most obvious value for the inductive probability of a statistical argument is simply the percentage figure divided by 100. Thus, the inductive probability of the argument of Problem 9.7 is .98. At least, this is true according to the so-called logical interpretation of inductive probability. Many theorists prefer some version of the so-called subjective interpretation, according to which inductive probability is a measure of a particular rational person's degree of belief in the conclusion, given the premises. According to the subjective view, the inductive probability of the argument of Problem 9.7 may deviate from .98, depending on the knowledge and circumstances of the person whose degree of belief is being measured. It is not possible to explain the details of these two kinds of interpretations here (though some further remarks are made in Chapter 10). Instead, we will simply presuppose a logical interpretation.

The form of the argument of Problem 9.7 is called statistical syllogism and can be represented as follows:

n percent of F are G.

x is F.

∴ x is G.

Here ‘F’ and ‘G’ are to be replaced by predicates, ‘x’ by a name, and ‘n’ by a number from 0 to 100. The inductive probability of a statistical syllogism is (by our logical interpretation) simply n/100. Note that in the case in which n = 100, the argument becomes deductive and its inductive probability is 1. For n < 50, it is more natural for the argument to take the form:

n percent of F are G.

x is F.

∴ x is not G.

We will regard this form too as a version of statistical syllogism. Its inductive probability is 1 − n/100, and it becomes deductive whenever n = 0.

In some cases, the statistics used to draw the conclusion of a statistical syllogism are not numerically precise. This is illustrated by the arguments in the next problem.

SOLVED PROBLEM

9.9

Evaluate the inductive probability of the following statistical syllogisms:

(a) Madame Plodsky's diagnoses are almost always right.

Madame Plodsky says that Susan is suffering from a kidney stone.

∴ Susan is suffering from a kidney stone.

(b) Most of what Dana says about his past is false.

Dana says that he lived in Tahiti and had two wives there.

∴ Dana did not live in Tahiti and have two wives there.

I will take a commercial airliner to Chicago.

∴ My Chicago flight will not end in a crash.

Solution

Each of these arguments has an inductive probability greater than. 5, though to none of them can we assign a precise inductive probability. The terms ‘almost always’ and ‘only a tiny fraction’ in arguments (a) and (c) indicate very small and very large percentages, respectively. These arguments have reasonably high inductive probabilities. ‘Most’ in argument (b) simply means more than half; the inductive probability of this argument is therefore only slightly better than. 5.

Reasonably high inductive probability, of course, is only one of the criteria an argument must meet in order to demonstrate the probable truth of its conclusion. It must also have true and relevant premises, and insofar as possible it must satisfy the requirement of total evidence (Section 2.5). The premises of a statistical syllogism are automatically relevant in virtue of its form. But they may not be true, and they may not be all that is known with respect to the conclusion.

Argument (a) of Problem 9.9, for example, may well have a false first premise, especially if Madame Plodsky is some sort of fortune-teller. It is an argument from authority, whose strength depends on Madame Plodsky's reliability. Although we depend on arguments from authority for much of our knowledge (i.e., we depend on the fact that much of what others tell us is true), such dependence easily becomes fallacious if the authority's reliability is in doubt or contrary evidence is supposed. If the first premise, the one asserting the authority's reliability, is omitted, then the argument is no longer a statistical syllogism. Its inductive probability drops significantly, and the remaining premise now lacks relevance to the conclusion, since in the absence of evidence that the authority is reliable, his or her pronouncements are not clearly relevant to the facts. The result is a fallacy of appeal to authority, a mistake discussed in Section 8.2.

Argument (b), in contrast to argument (a), reasons from the unreliability of a person's pronouncements. This is a form of ad hominem argument (argument against the person). If the premises are true and there is no suppressed evidence, argument (b) is a reasonably good argument. However, many ad hominem arguments, instead of addressing the veracity of the person in question, attack the person's character, reputation, or circumstances. If, for example, we replaced the first premise of argument (b) by the statement ‘Dana is a terrorist’, then the inductive probability of the argument would drop, the premises would lack relevance, the argument would commit an ad hominem fallacy (see Section 8.2). Without some premise connecting Dana's alleged terrorist activities to the reliability of his pronouncements about his past, this new argument would provide little, if any, support for its conclusion.

Argument (c) would be a very strong argument if its premises were known to be true and no evidence were suppressed. However, there is a subtle problem here. The conclusion of argument (c) concerns a future flight. Does its first premise also concern future flights, or just flights that have already occurred? That is, are we to read the first premise as

(1) Only a tiny fraction of all commercial airline flights, past, present, and future, end in crashes. or as

(2) Only a tiny fraction of all past commercial airline flights have ended in crashes.

If we interpret the first premise of argument (c) as in alternative 1, then how can we know that it is true? Perhaps it is, at least if we take ‘future’ to mean the near or foreseeable future, but there may be some doubt. Interpreted in this way, however, the argument is clearly a statistical syllogism with relevant premises and a high inductive probability. If, on the other hand, we interpret the first premise of argument (c) as in alternative 2, then this premise is clearly true, but the argument is, strictly speaking, no longer a statistical syllogism; the flight referred to in the conclusion is not among the flights mentioned in the first premise, since it is a future flight. Under this interpretation, the first premise is weaker and (in conjunction with the second) less relevant to the conclusion than under interpretation 1, so that the argument's inductive probability is considerably lower. Indeed, since the argument now moves from premises about the past to a conclusion about the future, its reliability depends on how closely we can expect the future to resemble the past—specifically, on how consistent or uniform the pattern of airline crashes will be. This is the Humean presupposition, and hence under interpretation 2 the argument is Humean.

This is not to say that argument (c) is not a good argument. It may be a good argument under either interpretation, provided that we know the premises to be true and that there is no suppressed evidence. But under interpretation 1 we may not know the first premise to be true, and under interpretation 2, the argument's inductive probability is less than it is under interpretation 1 (how much less depends on the strength of our presupposition of uniformity).

SOLVED PROBLEM

9.10

Arrange the following arguments in order of decreasing inductive probability.

(a) 85 percent of the Snooze missiles fired so far have missed their targets.

A Snooze missile was fired on July 4, 1997.

∴ This Snooze missed its target.

(b) A Snooze missile will be fired tomorrow.

∴ This Snooze will miss its target.

A Snooze will be fired tomorrow.

∴ This Snooze will miss its target.

(d) No Snooze missiles have ever missed their targets.

A Snooze was fired on July 4, 1997.

∴ This Snooze missed its target.

(e) 95 percent of Snooze missiles fired so far have missed their targets.

A Snooze was fired on July 4, 1997.

∴ This Snooze missed its target.

(f) No Snooze missiles have ever missed their targets.

A Snooze will be fired tomorrow.

∴ This Snooze will miss its target.

Solution

(e), (a), (c), (b), (f), (d). Argument (e) is stronger than argument (a) because its first premise is stronger. Argument (a) is stronger than argument (c) because (c) extrapolates from past to future and hence presupposes the uniformity of nature, while (a) does not. Argument (c) in turn is stronger than argument (b), because (c) is based on stronger premises; (b) offers no real evidence in support of its conclusion. The conclusions of arguments (f) and (d) are unlikely, given their premises; the inductive probability in each case is less than .5. In fact, (d) deductively implies the negation of its conclusion, so that the inductive probability of argument (d) is actually 0.

9.3: STATISTICAL GENERALIZATION

Statistical syllogism is an inference from statistics concerning a set of individuals to a (probable) conclusion about some member of that set. Statistical generalization, by contrast, moves from statistics concerning a randomly selected subset of a set of individuals to a (probable) conclusion about the composition of the set as a whole. It is the sort of reasoning used to draw general conclusions from public opinion polls and other types of random surveys.

SOLVED PROBLEM

9.11

Evaluate the inductive probability of the following argument:

Fewer than 1 percent of 1000 ball bearings randomly selected for testing from the 1997 production run of the Saginaw plant failed to meet specifications.

∴ Only a small percentage of all the ball bearings produced during the 1997 production run at the Saginaw plant fails to meet specifications.

Solution

The inductive probability of this argument is quite high. The size of the sample and the randomness of the selection strongly justify the generalization expressed by the conclusion. Moreover, the generalization itself involves a certain approximation, as indicated by the phrase ‘only a small percentage’. This weakens the conclusion, adding strength to the argument.

The general form of statistical generalization is as follows:

n percent of s randomly selected F are G.

∴ About n percent of all F are G.

The number s indicates the size of the sample. F is a property which defines the population about which we are generalizing (in the case of Problem 9.11, bearings produced during the 1997 production run at the Saginaw plant). And G is the property studied by the survey (in this case, the property of failing to meet specifications).

To say that the sample was randomly selected is to say that it was selected by a method which guarantees that each of the F's had an equal chance of being sampled. This in turn implies that each s-membered subset of the F's has an equal chance of being chosen. Now it is a mathematical fact (whose proof is beyond the scope of this discussion) that if s is sufficiently large, most s-membered subsets of a given population are approximately representative of that population. In particular, for most s-membered subsets of the set of F's, the proportion of G's is about the same as it is among the F's generally. Hence if a fairly large sample of F's is randomly selected, it is likely, though not certain, that the proportion of G's which it contains will approximate the proportion of G's among all the F's.

The inductive probability of a statistical generalization is determined by purely mathematical principles. There is no need to presuppose any sort of natural uniformity. Consequently, statistical generalization is a statistical form of inference, not a Humean form.

The success of statistical generalization depends crucially on the randomness of the sampling technique. If the sample is not randomly chosen, then the sampling technique may favor samples having either an unusually high or an unusually low number of G's. In such cases the sample is said to be biased. Attempts to apply statistical generalization with a nonrandom sampling technique commit the fallacy of biased sample, which is one form of the fallacy of hasty generalization discussed in Section 8.5. The resulting arguments are not true statistical generalizations, since true statistical generalization requires randomness. Their inductive probabilities are typically quite low.

SOLVED PROBLEM

9.12

Evaluate the inductive probability of the following argument:

I spoke to my three friends. They took that course and they all got an A.

∴ Virtually everybody who took that course got an A.

Solution

This is obviously a weak argument. Here ‘F’ designates the students who took the course and ‘G’ those who got an A. Assuming a normal class size (number of F's), a sample size of three is not large enough to justify the generalization expressed by the conclusion. Moreover, the sample is biased: the friends of the arguer do not constitute a random group of F's. The argument commits the fallacy of hasty generalization.

The inductive probability of a genuine statistical generalization is primarily a function of two quantities: the sample size s and the strength of the conclusion. Increasing s strengthens the premise in a way relevant to the conclusion and thus enhances the argument's inductive probability. But to determine the argument's inductive probability we also need to take account of the strength of its conclusion. Notice that in the form given above, the conclusion is ‘About n percent of F are G’. If it said ‘Exactly n percent of F are G’, it would be much too strong, and the argument's inductive probability would in almost all cases be close to zero. It is very unlikely that a random sample would contain exactly the same proportion of G's as the population from which it was selected. Therefore, if we want our conclusion to be reliable, we must allow it a certain margin of error, and this is what the term ‘about’ (or a similar expression) signifies.

If we delineate this margin of error precisely, then there are mathematical methods for determining the argument's inductive probability numerically. The details of these methods are beyond the scope of this book, but some examples will illustrate the point.1 Suppose we take ‘about n percent’ to mean n%± 3%. Then if s = 1000, the inductive probability of the argument turns out to be quite high, about .95 or perhaps a little higher. If we decrease s to 100 while keeping the conclusion the same, the inductive probability drops to something on the order of. 5. For samples much smaller than 100, it becomes unlikely that the proportion of G's in the sample is within 3 percent of the proportion of G's in the population. In other words, the argument's inductive probability drops below .5.

If we interpret ‘about’ less strictly, we weaken the conclusion and hence raise the argument's inductive probability. Suppose that s = 100, but instead of concluding that n%± 3% of F are G, we conclude that n% ± 10% of F are G. Now once again we have a strong argument, with an inductive probability of. 95 or slightly more. If we allow an even greater margin of error in the conclusion, the inductive probability gets still closer to 1. If we are willing to accept a very wide margin of error (say, ±30%), we can get an inductive probability of. 95 with a sample as small as 20 or so. These figures remain relatively constant, regardless of the population size (number of F's), provided that this number is fairly large.

Inductive probability is thus enhanced both by increasing s (thereby strengthening the premise) and by increasing the margin of error in the conclusion (which weakens the conclusion). If the conclusion is too strong to be supported with reasonable inductive probability by the premise, the argument is said to commit the fallacy of small sample, which is another version of the fallacy of hasty generalization (Section 7.4).

Though the inductive probability of a statistical generalization varies mainly with s and the margin of error of the conclusion, n also has a small effect. If n is very large or very small (near 0 or 100), the argument's inductive probability is slightly higher (other things being equal) than it is if n is close to 50.

SOLVED PROBLEM

9.13

Arrange the following arguments in order of decreasing inductive probability.

(a) 50 percent of 100 randomly selected Americans said that they favored the president's handling of the economy.

∴ Exactly 50 percent of all Americans would say (if asked under the survey conditions) that they favor the president's handling of the economy.

(b) 50 percent of 1000 randomly selected Americans said that they favored the president's handling of the economy.

∴ 50% ± 10% of all Americans would say (if asked under the survey conditions) that they favor the president's handling of the economy.

∴ Exactly 50 percent of all Americans favor the president's handling of the economy.

(d) 50 percent of 100 randomly selected Americans said that they favored the president's handling of the economy.

∴ 50% ± 1% of all Americans would say (if asked under the survey conditions) that they favor the president's handling of the economy.

(e) 50 percent of 100 randomly selected Americans said that they favored the president's handling of the economy.

∴ Exactly 50 percent of all Americans favor the president's handling of the economy.

(f) 50 percent of 100 randomly selected Americans said that they favored the president's handling of the economy.

∴ 50% ± 10% of all Americans would say (if asked under the survey conditions) that they favor the president's handling of the economy.

Solution

(b), (f), (d), (a), (e), (c). Argument (b) is like argument (f), except that it employs a larger sample; so (b) is stronger than (f). Argument (f) is stronger than argument (d) because argument (d) has a stronger conclusion. For the same reason, (d) is stronger than (a). Argument (e) is still weaker than argument (a) because the conclusion of argument (e) is less relevant, being an assertion about what people actually believe, as opposed to what they would say. Argument (c), finally, has the smallest inductive probability, since it is like argument (e), except that its first premise is weaker, since it fails to indicate that the sample was random.

Inductive probabilities of statistical generalizations are usually suppressed in reports of the findings of surveys and public opinion polls. A report might say, for example, “Sixty-two percent of the voters approved the president's handling of the economy, subject to a sampling error of ±3%.” What this means is that the sample was large enough (in this case about 1000) to ensure a. 95 probability that the interval 62% ± 3% contains the actual proportion of voters who approve of the president's handling of the economy. Statisticians customarily take a probability of. 95 as practical certainty and hence do not mention that a probability is involved here. But their conclusion that 62% ± 3% of the voters approve of the president's handling of the economy is in fact derived by statistical generalization from the premise that 62 percent of their sample approved the president's handling of the economy, a statistical generalization whose inductive probability is only .95. Thus it should be kept in mind that there is still a. 05 probability that the proportion of all voters who support the president's handling of the economy lies outside the range 62% ± 3%.

Several precautions must be observed in evaluating statistical generalizations. For one thing, as noted earlier, the sample must be randomly selected. This does not mean that the sample must be known to contain the same proportion of G's as the population at large. If this were known, there would be no need for statistical generalization; we could simply deduce the proportion of G's in the population from the proportion of G's in the sample. It simply means that the sampling technique must ensure that the proportion of G's in the sample is likely to be close to the proportion of G's in the population at large.

Of course, if in making a statistical generalization someone claims that a certain sample is random but in fact it is not, then the premise of the statistical generalization is false, and the argument must be rejected. This may occur, for example, when what seems to be a random sampling method (e.g., picking names from a phone book) actually is not. Choosing names from a phone book will not, for example, provide a random sample of all homeowners, since those without phones have no chance of being chosen.

Like all inductive arguments, statistical generalization is vulnerable to suppressed evidence. If two or more random surveys get distinctly different results from true premises, then none of them alone constitutes a good argument. The requirement of total evidence demands that all of them be weighed in assessing the probability of the conclusion on which they bear.

Even slight deviations from the form of statistical generalization can seriously weaken the argument. This is illustrated by the next example.

SOLVED PROBLEM

9.14

Evaluate the following argument:

Only 10 percent of 1000 randomly selected Americans answered “yes” to the question “Have you ever committed a felony?”

∴ About 10 percent of all Americans have committed felonies.

Solution

This argument deviates from the form of statistical generalization, because the premise concerns what the sampled Americans (F's) said, while the conclusion concerns what Americans did. That is, the property assigned to the variable G in the premise (answering “yes” to the question) is not the same as the property assigned to the variable G in the conclusion (actually having committed a felony). But for the argument to be a legitimate instance of statistical generalization, the same property must be assigned to both occurrences of the variable. The only conclusion we can legitimately draw by statistical generalization from the premise is:

About 10 percent of all Americans would answer “yes” to the question “Have you ever committed a felony?” (if asked under the survey conditions).

With this new conclusion, the argument's inductive probability is fairly high, though we can't say exactly how high, because of the vagueness of the term ‘about’. The original argument lacked clear relevance, and its inductive probability was much lower. For it is quite possible (indeed likely, given the sensitive subject matter of the survey) that some of the answers received were less than honest.

Problem 9.14 illustrates a general difficulty with polling human beings: How can we be sure that the respondents are telling the truth? In many cases (as, for example, when voters are asked which candidates they prefer) there is little motive for dishonesty, and it seems fairly safe to assume that their responses generally reflect actual opinions. But the assumption of truthfulness should not be made uncritically.

A related problem concerns human response to the way in which a question is asked. Suppose we wish to survey public opinion on a new piece of legislation by Senator S. The way we phrase our question may drastically affect the responses. If we ask, “Do you favor Senator S's governmentbloating socialist bill?” we are likely to generate many more negative responses than if we asked the question more neutrally: “Do you favor Senator S's bill on government aid to the poor?” And this in turn will probably generate more negative responses than the positively worded “Do you favor Senator S's popular new bill to bring much-needed aid to the victims of poverty in America?” The final form of the argument, however, may conceal the way in which the question was asked:

SOLVED PROBLEM

9.15

Evaluate the following argument:

51 percent of 100 randomly selected registered voters said that they favored Senator S's bill.

∴ About 51 percent of all voters favor Senator S's bill.

Solution

Like the argument of Problem 9.14, this is not a statistical generalization, since it moves from what the voters said to what they actually think. But this argument has an additional source of weakness in that we are not given the exact phrasing of the question. The question itself, for all we know, may have substantially biased the responses either for or against the bill.

Biased questions are frequently a problem in polls which are poorly conducted or conducted by those with a vested interest in the outcome.

9.4: INDUCTIVE GENERALIZATION AND SIMPLE INDUCTION

Statistical generalization allows us to arrive at a conclusion concerning an entire population from a premise concerning a random sample of that population. The randomness of the sample ensures the probability of the conclusion on purely mathematical grounds. But often it is not possible to obtain a random sample. This is true, for example, if the relevant population concerns future objects or events. Since these objects or events do not yet exist at the time the sample is taken, they have no chance of being included in the sample. Therefore, since randomness requires that each member of the population have an equal chance of being selected, no sample which we take now can be random for a population which includes future objects or events.

SOLVED PROBLEM

9.16

Evaluate the randomness of the sample in the following generalization:

The Bats have won 10 of the 20 games they have played so far this season.

∴ The Bats will finish the season having won about half of their games.

Solution

The conclusion of the argument concerns a population (all Bats games this season) which includes future games. The sample only concerns games played so far. It is therefore not a random sample with respect to the relevant population.

The general form of the argument of Problem 9.16 may be represented as follows:

n percent of s thus-far-observed F are G.

∴ About n percent of all F are G.

In the example given, n is 50, s is 20, F is ‘Bats games this season’, and G is ‘are (or will be) won by the Bats’. We shall call this form inductive generalization.

Inductive generalization differs from statistical generalization in that its premise does not claim that the sample is random. Without a claim of randomness, the reasoning cannot be justified by mathematical principles alone. No mathematical principle can guarantee, for example, that the Bats will not suddenly improve dramatically and win all their remaining games—or finish with a long losing streak. Nor does any principle of mathematics ensure that such radical changes are not probable. The inference of Problem 9.16 therefore presupposes something substantial, namely, that the course of events (in this case, ball games) exhibits or is likely to exhibit a certain uniformity over time; that is, that future instances of wins are likely to occur with about the same frequency as past instances of wins. Inductive generalizations are therefore Humean inferences.

Inductive generalizations are weaker arguments than statistical generalizations, for the kind of uniformity they presuppose is always to some degree uncertain. But since there is no universally accepted way of calculating the inductive probabilities of Humean arguments, we cannot say exactly how much weaker. In other respects, however, evaluation of inductive generalizations employs the same principles as evaluation of statistical generalizations. Thus, for example, in both kinds of generalization, inductive probability increases as s gets larger.

One of the most notable forms of inductive generalization occurs when n = 100, so that we have:

All the s thus-far-observed F are G.

∴ All F are G.

This form has been widely regarded as the means by which scientific laws (which can often be expressed in the form ‘All F are G’) are justified. Thus, for example, our knowledge that water freezes at +32 degrees Fahrenheit is said to be based on the fact that all the (very many) samples of pure water observed thus far have had a freezing point of +32 degrees Fahrenheit. Yet inductive generalization is a relatively weak form of reasoning. Some theorists reject it as too weak to establish universal laws. They argue that if s is small relative to the population of F's, the inductive probability of the inference is near zero, and that for infinite populations and finite s it is strictly zero. Some have questioned whether inductive generalization really is the way in which we justify scientific laws—and, indeed, whether such laws can be justified at all. But others have disputed these contentions, and no consensus has been reached.

Despite this widespread disagreement, there are certain comparative principles on which most logicians agree. If we assume that the inductive probability of a given inductive generalization is not strictly zero, then clearly it may be increased by increasing s. (This is an instance of the general rule that strengthening the premise strengthens the argument.) Likewise, since reasoning is strengthened by weakening the conclusion, the smaller the population of F's, the greater the inductive probability of the argument.

The most extreme way to weaken the conclusion of such an inference is to reduce the population it mentions to one individual. This yields the following form, which is called simple induction, induction by enumeration, or the simple predictive inference:

n percent of the s thus-far-observed F are G.

∴ If one more F is observed, it will be G.

In general, simple inductions are much stronger than inductive generalizations from the same premises.

SOLVED PROBLEM

9.17

Evaluate the relative strength of the following arguments:

(a) All the (very many) objects observed thus far exert gravitational force in proportion to their mass.

∴ All objects exert gravitational force in proportion to their mass.

(b) All the (very many) objects observed thus far exert gravitational force in proportion to their mass.

∴ The next observed object will exert gravitational force in proportion to its mass.

Solution

Argument (a) is a typical example of an inductive generalization; its conclusion has the form of a scientific law. Argument (b) is a simple induction whose conclusion is much weaker than the conclusion of (a). Since both arguments are based on the same premise, this means that (b) is itself considerably stronger than (a).

Like all inductive generalizations (of which they are a special case), simple inductions get stronger as s increases, so long as n > 50. And like statistical syllogisms, simple inductions are highly sensitive to the value of n. They are strongest when n = 100 and weakest when n = 0. If n <50, a simple induction will provide more support for the negation of its conclusion than for the conclusion itself.

Yet unlike statistical syllogisms, simple inductions do not become deductive when n = 100. This is because they are Humean inferences, whose strength depends on an uncertain presupposition of the uniformity of nature. As with all Humean arguments, there is no generally accepted method for calculating the inductive probability of a simple induction.

SOLVED PROBLEM

9.18

Arrange the following arguments in order of decreasing inductive probability.

(a) Exactly 99 percent of 500 observed meteorites contained iron.

∴ If another meteorite is observed, it will contain iron.

(b) Exactly 99 percent of 500 observed meteorites contained iron.

∴ All meteorites contain iron.

∴ If a meteorite is observed, it will contain iron.

(d) All 500 meteorites observed thus far contain iron.

∴ If another meteorite is observed, it will contain iron.

(e) All 500 meteorites observed thus far contain iron.

∴ All meteorites contain iron.

(f) All 1000 meteorites observed thus far contain iron.

∴ If another meteorite is observed, it will contain iron.

(g) All 500 meteorites observed thus far contain iron.

∴ All meteorites we will ever observe contain iron.

Solution

(c), (f), (d), (a), (g), (e), (b). Argument (c) is a deductive argument. Its inductive probability is thus higher than that of the others, which are not deductive. Argument (f) is stronger than argument (d) because of its stronger first premise; it employs a larger sample. Argument (a) is similar, but lower in inductive probability, since according to its first premise the percentage of iron-bearing meteorites is smaller. The first premise of argument (g) describes a sample like that of argument (d), but its inductive probability is still much lower than that of either (a) or (d), since its conclusion is exceedingly strong—much stronger than the conclusion of (a) or (d). The conclusion of argument (e) is stronger still; hence (e) is even weaker. The inductive probability of argument (b) is strictly zero, since according to its first premise five meteorites which do not contain iron have already been observed.

9.5: INDUCTION BY ANALOGY

Another important kind of Humean argument is argument by analogy. In an argument by analogy we observe that an object x has many properties, F1, F2,. . ., Fn, in common with some other object y. We observe also that y has some further property G. Hence we consider it likely (since x and y are analogous in so many other respects) that x has G as well. The general form of the argument may be represented as follows:

F1x & F2x & . . . & Fnx

F1y & F2y & . . . & Fny

∴ Gx

SOLVED PROBLEM

9.19

Evaluate the following argument:

Specimen x is a single-stemmed plant with lanceolate leaves and five-petaled blue flowers, about 0.4 meter tall, found growing on a sunny roadside.

Specimen y is a single-stemmed plant with lanceolate leaves and five-petaled blue flowers, about 0.4 meter tall, found growing on a sunny roadside.

Specimen y is a member of the gentian family.

∴ Specimen x is a member of the gentian family.

Solution

This is a reasonably strong argument by analogy. The argument is Humean because no logical or mathematical principle can guarantee that similarities in external appearance, size, and shape correspond to taxonomic sameness. The argument thus presupposes a more or less orderly correspondence between the characteristics mentioned and taxonomic type. Its strength is in part a function of the strength of this presupposition.

Like inductive arguments generally, analogical arguments may be strengthened by strengthening their premises or by weakening their conclusions. We raise the inductive probability of the argument of Problem 9.19, for example, if we weaken the conclusion to:

Specimen x is a member of the gentian family or some closely related family.

We can also raise its inductive probability by noting more properties that x and y have in common, thus strengthening each of the first two premises. We might, for example, observe that x and y also produce similar kinds of seeds.

However, a simple count of the properties constituting the analogy is only a rough way to gauge premise strength. Some properties count more than others. We may note, for example, that both x and y have the property of being composed of matter. But this property provides only a weak and very general analogy between the two in comparison with more specific properties, like having lanceolate leaves or having five-petaled blue flowers. Thus the strength of the premises depends not only on the number of properties x and y are claimed to have in common, but also on the specificity of these properties. The more specific the resemblances are, the stronger the argument.

Another consideration in analogical reasoning is the relevance of the properties F1, F2,. . ., Fn to the property G (see Section 8.5). Problem 9.19 is relatively strong in part because all the properties mentioned in the first two premises are likely to be relevant to taxonomic classification (i.e., to the property G, the property of being a member of the gentian family). But where relevance is lacking and the conclusion is strong enough to be of much interest, the argument's inductive probability will be quite low.

SOLVED PROBLEM

9.20

Estimate the inductive probability of the following argument by analogy:

Person x was born on a Monday, has dark hair, is 5 feet 8 inches tall, and speaks Finnish.

Person y was born on a Monday, has dark hair, is 5 feet 8 inches tall, and speaks Finnish.

Person y likes brussels sprouts.

∴ Person x likes brussels sprouts.

Solution

The inductive probability is low, because the properties F1, F2,. . ., Fn mentioned in the first two premises are almost surely irrelevant to the property G (the property of liking brussels sprouts).

Still, in advance of careful investigation, it is not always clear what is relevant and what is not. It might turn out that dark-haired people have a gene which predisposes them to have a taste for brussels sprouts, so that having dark hair, for example, is relevant after all! This, of course, is unlikely. But it does sometimes happen that previously unsuspected but genuine connections are suggested by analogical reasoning which appears at first to lack relevance.

Relevance is often difficult to determine, but in analogical arguments its role is especially problematic. Perhaps the best advice that can be given is simply that in evaluating analogical reasoning, common sense ought to prevail.

Analogical considerations can be combined with induction by enumeration to yield hybrid argument forms. For example, instead of comparing x with just one object y, we may compare it with many different objects, all of which have the properties F1, F2,. . ., Fn, and G. This strengthens the argument by showing that G is associated with F1, F2,. . ., Fn in many instances, not just in one.

SOLVED PROBLEM

9.21

Arrange the following analogical arguments in order of decreasing inductive probability:

(a) A common housefly x, 8 millimeters long, is being placed in a tightly closed jar.

A common housefly y, 8 millimeters long, was placed in a tightly closed jar.

y died within a day.

∴ x will die within a day.

(b) A common housefly x, 8 millimeters long and 14 days old, is being placed in a tightly closed jar.

A common housefly y, 8 millimeters long and 14 days old, was placed in a tightly closed jar.

y died within a day.

∴ x will die within a day.

A common housefly y, 8 millimeters long, was placed in a tightly closed jar.

y died within a day.

∴ x will die within 12 hours.

(d) A common housefly x, 8 millimeters long and 14 days old, is being placed in a tightly closed jar.

Common houseflies y, z, and w, each 8 millimeters long and 14 days old, were placed in tightly closed jars.

y, z, and w died within a day.

∴ x will die within a day.

(e) A common housefly x, 8 millimeters long and 14 days old, is being placed in a tightly closed jar.

Common houseflies y, z, and w, each 8 millimeters long and 14 days old, were placed in tightly closed jars.

y, z, and w died within a day.

∴ x will die eventually.

(f) A common housefly x, 8 millimeters long, is being placed in a tightly closed jar in Wisconsin.

A common housefly y, 8 millimeters long, was placed in a tightly closed jar in Wisconsin.

y died within a day.

∴ x will die within a day.

(g) A common housefly x is being placed in a tightly closed jar.

A common housefly y was placed in a tighly closed jar.

y died within a day.

∴ x will die within 12 hours.

Solution

(e), (d), (b), (f), (a), (c), (g). Argument (e) is stronger than argument (d) because its conclusion is weaker. Argument (d) is stronger than argument (b) because of its stronger second premise; the analogy in (b) is based on a sample of just one fly, instead of three. Argument (b) is stronger than argument (f) because the state in which the experiment is performed is surely less relevant to the conclusion than the age of the fly. Yet (f) is still marginally stronger than (a), which is like (f) except that it does not mention the state in which the experiment was performed (and hence has very slightly weaker premises). Argument (c) is like argument (a), except that it has a stronger conclusion; so (c) is weaker than (a). Finally, argument (g) is slightly weaker still, since it does not mention the size of the flies and hence has weaker premises.

Analogical arguments, like all inductive arguments, are vulnerable to contrary evidence. If any evidence bearing negatively on the analogy is suppressed, then the argument violates the requirement of total evidence and should be rejected. (The conclusion should then be reconsidered in the light of the total available evidence.) Contrary evidence to analogical arguments often takes the form of a relevant disanalogy. (For more on faulty analogies, see Problem 8.30.)

SOLVED PROBLEM

9.22

Evaluate the following argument by analogy:

Jim Jones was the leader of a religious movement which advocated peace, brotherhood, and a simple agrarian way of life.

Mahatma Gandhi was the leader of a religious movement which advocated peace, brotherhood, and a simple agrarian way of life.

Mahatma Gandhi was a saintly man.

∴ Jim Jones was a saintly man.

Solution

The argument has true premises, a fairly high inductive probability, and a reasonable degree of relevance. But it suppresses crucial contrary evidence: Jim Jones was the leader of a fanatical cult which he incited to acts of murder and mass suicide. Since the argument ignores this crucially relevant disanalogy between Jones and Gandhi, it should be rejected.

9.6: MILL'S METHODS

Often we wish to determine the cause of an observed effect. Logically, this is a two-step procedure. The first step is to formulate a list of suspected causes which, to the best of our knowledge, includes the actual cause. The second is to rule out by observation as many of these suspected causes as possible. If we narrow the list down to one item, it is reasonable to conclude that this item is probably the cause.

The justification of the first step (i.e., the evidence that the actual cause is included on our list of suspected causes) is generally inductive. The eliminative reasoning of the second step is deductive. Since both inductive and deductive reasoning are involved, the reasoning as a whole is inductive. (See Section 2.3.)

We arrive at the list of suspected causes by a process of inductive (frequently analogical) reasoning. Suppose, for example, that we wish to find the cause of a newly discovered disease. Now this disease will resemble some familiar diseases more than others. We note the familiar diseases to which it is most closely analogous and then conclude (by analogy) that its cause is probably similar to the causes of the familiar diseases which it most closely resembles. This will give us a range of suspected causes.

Suppose, for example, that the familiar diseases which the new disease most closely resembles are all viral infections. The suspected causes will then be viral. Close observation of the disease victims will establish which viruses are present in their tissues. We will conclude that the actual cause is probably one of these viruses. These viruses, then, form our list of suspected causes.

At this stage, however, our investigation is only half finished. For it is quite likely that we will find several kinds of virus in the tissues of the victims. To determine which of these actully caused the disease, we now employ a deductive process designed to eliminate from our list as many of the suspected causes as possible. The kind of eliminative process we use will depend on the kind of cause we are looking for.

Here we shall discuss four different kinds of causes and, corresponding to each, a different method of elimination. The eliminative methods were named and investigated by the nineteenth-century philosopher John Stuart Mill. Mill actually discussed five such methods, but the fifth (the method of residues) does not correspond to any specific kind of cause and will not be discussed here. Before discussing Mill's methods, however, we need to define the kinds of causes to which they apply.

The first kind of cause is a necessary cause, or causally necessary condition. A necessary cause for an effect E is a condition which is needed to produce E. If C is a necessary cause for E, then E will never occur without C, though perhaps C can occur without E. For example, the tuberculosis bacillus is a necessary cause of the disease tuberculosis. Tuberculosis never occurs without the bacillus, but the bacillus may be present in people who do not have the disease.

A given effect may have several necessary causes. Fire, for example, requires for its production three causally necessary conditions: fuel, oxygen (or some similar substance), and heat.

The second kind of cause is a sufficient cause, or causally sufficient condition. A condition C is a sufficient cause for an effect E if the presence of C invariably produces E. If C is a sufficient cause for E, then C will never occur without E, though there may be cases in which E occurs without C. For example (with respect to higher animal species), decapitation is a sufficient cause for death. Whenever decapitation occurs, death occurs. But the converse does not hold; other causes besides decapitation may result in death.

A given effect may have several sufficient causes. In addition to decapitation, as just noted, there are many sufficient causes for death: boiling in oil, crushing, vaporization, prolonged deprivation of food, water, or oxygen—to name only a few of the unpleasant alternatives.

Some conditions are both necessary and sufficient causes of a given effect. That is, the effect never occurs without the cause nor the cause without the effect. This is the third kind of causal relationship. For example, the presence of a massive body is causally necessary and sufficient for the presence of a gravitational field. Without mass, no gravitational field can exist. With it, there cannot fail to be a gravitational field. (This does not mean, of course, that one must experience the gravitational field. Moving in certain trajectories relative to the field will produce weightlessness, but the field is still there.)

The fourth kind of causal relation we shall discuss is causal dependence of one variable quantity on another. A variable quantity B is causally dependent on a second variable quantity A if a change in A always produces a corresponding change in B. For example, the apparent brightness B of a luminous object varies inversely with the square of the distance from that object, so that B is a variable quantity causally dependent on distance. We can cause an object to appear more or less bright by varying its distance from us.

An effect (such as apparent brightness) may be causally correlated with more than one quantity. If the object whose apparent brightness we are investigating is a gas flame, its apparent brightness will also depend on the amount of fuel and oxygen available to it, and on other factors as well.

SOLVED PROBLEM

9.23

Classify the kind of causality intended by the following statements:

(a) Flipping the wall switch will cause the light to go on.

(b) Closing the electricity supply from the main lines will cause the light to go off.

(d) Pulling the trigger will cause the gun to fire.

(e) Raising the temperature of a gas will cause an increase in its volume.

(f) Raising the temperature of the freezer above +32 degrees Fahrenheit will cause the ice cubes in the freezer to melt.

(g) Killing the President will cause new presidential elections.

(h) Raising the temperature in the environment will cause the death of many plants.

Solution

(a) Necessary (but not sufficient: the light will not go on unless the light bulb is working).

(b) Sufficient (but not necessary: the light will go off also if the wall switch is turned to the “off” position).

(d) Necessary (but not sufficient: the gun won't fire unless it is loaded).

(e) Dependent (the higher the temperature, the higher the volume).

(f) Necessary and sufficient.

(g) Sufficient (but not necessary).

(h) Dependent (the higher the temperature, the greater the number of plants that will die).

Now, to reiterate, Mill's methods aim to narrow down a list of suspected causes (of one of the four kinds just described) in order to find a particular cause for an effect E. Each of the four methods listed below is appropriate to a different kind of cause:

Mill's Method of:

Rules Out Conditions Suspected of Being:

Agreement

Necessary causes of E

Difference

Sufficient causes of E

Agreement and difference

Necessary and sufficient causes of E

Concomitant variation

Quantities on which the magnitude of E is causally dependent

If by using the appropriate method we are able to narrow the list of suspected causes down to one entry, then (presuming that a cause of the type we are seeking is included in the list) this entry is a cause of the kind we are looking for. We now examine each of the four methods in detail.

The Method of Agreement

Mill's method of agreement is a deductive procedure for ruling out suspected causally necessary conditions. Recall that if a circumstance C is a causally necessary condition of an effect E, then E cannot occur without C. So to determine which of a list of suspected causally necessary conditions really is causally necessary for E, we examine a number of different cases of E. If any of the suspected necessary conditions fails to occur in any of these cases, then it can certainly be ruled out as not necessary for E. Our hope is to narrow the list down to one item.

Solution

Only one of the five suspected causes (namely, V1) is present in each of the four patients with the disease. This proves that none of the suspected causes, except possibly V1, really is causally necessary for E.

Once V2 through V5 are eliminated as necessary causes, it follows deductively that

(1) If the list V1 through V5 includes a necessary cause of E, then V1 is that necessary cause.

This is the conclusion which Mill's method of agreement yields. If we wish to advance further to the unconditional conclusion

(2) V1 is a necessary cause of E

then we need the premise

(3) The list V1 through V5 includes a necessary cause of E.

Such a premise cannot in general be proved with certainty, but can only be established by inductive reasoning. Typically, such inductive reasoning will be analogical. In the case in question, it may look something like this:

(4) Disease E has characteristics F1, F2,. . ., Fn.

(5) The known diseases similar to E have characteristics F1, F2,. . ., Fn.

(6) Viruses are necessary causes of the known diseases similar to E.

∴ (7) Some virus is a necessary cause of E.

Here the characteristics F1, F2,. . ., Fn might be such things as infectiousness or the presence of fever. To get from statement 7 to statement 3, we need to add to statement 7 the premise that

(8) The only viruses present in the cases of patients 1 through 4, who had E, were V1 through V5.

Statement 8 together with statement 7 deductively implies 3, since (by definition) any necessary cause for E must occur in every case of E. The entire argument may now be summarized in the following diagram:

The basic premises in statements 4, 5, 6, and 8 are obtained by observation or previous investigation. Statement 1 is the conclusion obtained by using Mill's method of agreement. The reliability of the argument as a whole depends on the adequacy of the analogical inference (the inference from 4, 5, and 6 to 7) and on the truth of the basic premises. The premise in statement 8, for example, could prove to be false if our observations of the patients were not sufficiently thorough. That would undermine the whole argument, since in that case the real cause of E might be a virus that was present but undetected in the cases we studied. The adequacy of the analogical inference, of course, depends on the factors discussed in Section 9.5. We should be especially wary of suppressed evidence. (Does E have any unusual characteristics which suggest a nonviral cause?)

Not every application of the method of agreement works out so neatly. Suppose V1 and V2 both occur in all cases of E that we examine. Does that mean both are necessary for E? No, this does not follow. We may not have examined a large enough sample of patients to rule out one or the other. Our investigation is inconclusive, and we need to seek more data.

It may also happen that the method of agreement rules out all the suspected causes on our list. In that case, statement 3 is false, and so either 7 or 8 must be false as well. That is, either the necessary cause is not viral (as our analogical argument led us to suspect) or we failed to detect some other virus that was present in the patients. If this occurs, we need to recheck everything and probably gather more data before any firm conclusions can be drawn.

The Method of Difference

If we are seeking a sufficient cause rather than a necessary cause, the method to use is the method of difference. Recall that a sufficient cause for an effect E is an event which always produces E. If cause C ever occurs without E, then C is not sufficient for E. Often it is useful, however, to speak of sufficient causes relative to a restricted class of individuals. A small quantity of a toxic chemical, for example, may be sufficient to produce death in small animals and children but not in healthy adults. Hence, relative to the class of children and small animals it is a sufficient cause for death, but relative to a larger class which includes healthy adults it is not. Claims of causal sufficiency are often implicitly to be understood as relative to a particular class of individuals or events.

Solution

Once V2 through V5 are eliminated as necessary causes, it follows deductively that

(1) If the list V1 through V5 includes a necessary cause of E, then V1 is that necessary cause.

This is the conclusion which Mill's method of agreement yields. If we wish to advance further to the unconditional conclusion

(2) V1 is a necessary cause of E

then we need the premise

(3) The list V1 through V5 includes a necessary cause of E.

(4) Disease E has characteristics F1, F2,. . ., Fn.

(5) The known diseases similar to E have characteristics F1, F2,. . ., Fn.

(6) Viruses are necessary causes of the known diseases similar to E.

∴ (7) Some virus is a necessary cause of E.

Here the characteristics F1, F2,. . ., Fn might be such things as infectiousness or the presence of fever. To get from statement 7 to statement 3, we need to add to statement 7 the premise that

(8) The only viruses present in the cases of patients 1 through 4, who had E, were V1 through V5.

The Method of Difference

What is the sufficient cause of P?

Solution

Since P failed to occur in person 2 in the presence of F2 through F5, clearly none of these foods is sufficient for P. On the assumption, then, that a sufficient cause for P occurs among F1 through F5, it follows that the cause is F1.

The weakest part of this reasoning is the assumption that a sufficient cause for P occurs among F1 through F5. As with the analogous premise in our discussion of necessary causes, it generally cannot be proved deductively but must be supported by an inductive argument. In this case, we might argue that in most past cases in which food poisoning has occurred, some toxic substance present in one food has been the culprit and that it has been sufficient to produce the poisoning in anyone who consumed a substantial amount of the food.

Once again, it is important to see how this sort of inductive reasoning could go wrong. Perhaps none of the foods is by itself sufficient for P, but ingestion of F1 and F2 together causes a chemical reaction which results in toxicity. Under these conditions we would still observe the poisoning in person 1 and no effect on person 2, but the assumption that a sufficient cause for P occurs among F1 through F5 would be false.

It may also happen that none of F1 through F5, or any combination of F1 through F5, is sufficient for P. A toxin may be present, say in F1, but consumption of this toxin may produce P only in certain susceptible individuals. That is, F1 may be sufficient for P in certain people but not in every member of the population we are concerned with. If this is so, then once again the assumption that a sufficient cause for P occurs among F1 through F5 is false, and so is the unqualified conclusion that F1 is sufficient for P. These errors can occur even if we make no faulty observations, because of the fallibility of the reasoning needed to establish this assumption. Therefore, caution is needed in applying the method of difference.

In summary, Mill's method of difference is used to narrow down a list of suspected sufficient causes for an effect E. It does this by rejection of any item on the list which occurs without E. We hope to make enough observations to narrow the list down to one item. If we do, then we may conclude deductively that if there is a sufficient cause for E on the list, it is the one remaining. However, to establish that our list contains a sufficient cause, we must rely on induction from past experience.

The Joint Method of Agreement and Difference

Mill's joint method of agreement and difference is a procedure for eliminating items from a list of suspected necessary and sufficient causes. It incorporates nothing new; it merely involves the simultaneous application of the methods of agreement and difference.

If C is a necessary and sufficient cause of E, then C never occurs without E and E never occurs without C. Hence, if we find any case in which C occurs but E does not or E occurs but C does not, C can be ruled out as a necessary and sufficient cause for E (though it may still be a necessary or sufficient cause, as the case may be).

SOLVED PROBLEM

9.26

Suppose that a student in a college dormitory notices a peculiar sort of interference on her television set. She has seen similar kinds of interference before and suspects that its necessary and sufficient cause (provided the television is on) is the nearby operation of some electrical appliance. This leads her to formulate the following list of suspected causes:

S = electric shaver

H = hair dryer

D = clothes dryer

W = washing machine

She now observes which appliances are operating in nearby rooms while her television is on. The results are as follows (I is the interference):

Which of the suspected causes, if any, is the necessary and sufficient cause of the interference?

Solution

The only one of the suspected necessary and sufficient causes which is always present when I is present and always absent when I is absent is D. Hence, if one of the suspected necessary and sufficient causes really is necessary and sufficient for I, it must be D.

If the student goes on to conclude that D is actually necessary and sufficient for I, once again the weakest point of her reasoning will be the assumption that a necessary and sufficient cause was included on her list of suspected causes. As before, this premise can be justified only by induction from past experience with similar situations.

The Method of Concomitant Variation

Mill's method of concomitant variation differs from the other methods in that it is not concerned with the mere presence or absence of cause and effect, but with their relative magnitudes. It is a means of narrowing down a list of variable magnitudes suspected of being responsible for a specific change in the magnitude of an effect E. A variable is rejected as not responsible for a particular change if that variable remains constant throughout the change. If all but one of a list of variables remain constant while the magnitude of an effect changes, then, presuming that the variable responsible for the change appears on the list, it must be the one which has not remained constant.

SOLVED PROBLEM

9.27

A houseplant exhibits a sudden spurt of growth. We suspect that the variables relevant to its growth rate are these:

S = sunlight

W = water

F = fertilizer

T = temperature

But we observe that only one of these variables, namely, the amount of water the plant receives, has been altered recently. This observation may be schematized as follows:

Here G is the growth rate and the plus signs stand for increases of magnitude. No plus sign indicates no change. Which, if any, of the variables on our list is causally relevant to the observed change in the growth rate of the plant? Solution

Since the amount of water the plant receives is the only one of the variables on our list that has changed, only it among these variables could be responsible for the observed change in growth rate.

where the minus signs indicate decreases in magnitude. Then we will be still more confident that the rate of watering is the variable responsible for the observed changes in growth rate.

As with the other three methods, the process of eliminating S, F, and T as possible causes of the observed effect is deductive, but induction from past experience with plants is required to support the premise that one of the four variables on our list caused the changes in G.

The increased confidence provided by case 3 is due to the additional support case 3 lends to this premise. For repetition of instances of the correlation between W and G enhances by simple induction the probability that W and G have varied and will continue to vary together. If we were perfectly confident that the variable responsible for the change was one of the four on our list, this additional confirmation would be superfluous, and cases 1 and 2 alone would suffice to establish that the responsible variable is W.

9.7: SCIENTIFIC THEORIES

The most sophisticated forms of inductive reasoning occur in the justification or confirmation of scientific theories. A scientific theory is an account of some natural phenomenon which in conjunction with further known facts or conjectures (called auxiliary hypotheses) enables us to deduce consequences which can be tested by observation. Often a theory is expressed by a model, a physical or mathematical structure claimed to be analogous in some respect to the phenomenon for which the theory provides an account.

For example, prior to the twentieth century there were two theories of the phenomenon of light, the corpuscular theory and the wave theory. According to the corpuscular theory (whose most notable advocate was Isaac Newton), light consists of minute particles, or corpuscles, expelled in straight trajectories by luminous objects. According to the wave theory (first propounded by the Dutch astronomer Christian Huygens), light consists of spherical waves spreading out from luminous objects like the circular ripples from a stone dropped into a lake. According to the wave theory, light waves are propagated through a fluid substance, the ether, which permeates the universe. Now, both theories were able to account for the phenomenon of color and for many of the reflective and refractive properties of light. But by the end of the nineteenth century the wave theory had temporarily won out, because of its superiority in explaining diffraction effects—patterns of light and dark bands formed when light is passed through a small aperture. Such patterns are predicted by the wave theory but difficult to explain by the corpuscular theory.

Each theory modeled light as a physical structure—moving particles, in one case; waves in a fluid medium, in the other. Both, however, were succeeded in the twentieth century by the quantum theory, in which light is modeled as a mathematical structure that has some features of both waves and particles but is not completely analogous to any familiar physical structure.

This example shows that scientific theories are justified primarily by their success in making true predictions. By ‘prediction’ we mean a statement about the results of certain tests or observations, not necessarily a statement about the future. Even theories about the past make predictions in this sense, since (in conjunction with appropriate auxiliary hypotheses) they imply that certain tests or observations will have certain results. A theory about the evolution of dinosaurs, for example, will have implications concerning the sorts of fossils we should expect to find in certain geological strata. These implications, then, are among its predictions. Since a theory's predictions are deduced from the theory together with its auxiliary hypotheses, if any of them prove false, then either the theory itself or one or more of the auxiliary hypotheses must be false. (One cannot deduce a false conclusion from a set of true premises.) If we are confident of all the auxiliary hypotheses, then we may confidently reject the part of the theory used to derive the prediction. The corpuscular theory of light, together with what seems to be the most reasonable auxiliary hypotheses about the way small particles ought to behave, implies that diffraction ought not to occur. Since it does occur, nineteenth-century physicists, confident of these auxiliary hypotheses, rejected the corpuscular theory.

In this example, a deductive process was used to refute a scientific theory. Often, however, theorists are not completely confident of the truth of the auxiliary hypotheses; hence there may be controversy about the soundness of the deduction used to reject the theory. If one or more of the auxiliary hypotheses are indeed false, then the falsity of a prediction made with the aid of those hypotheses does not entail the falsity of the theory.

Whereas the reasoning by which scientific theories are refuted is deductive, the reasoning by which they are confirmed is inductive. After the demise of the corpuscular theory of light, the wave theory became increasingly confirmed. Unlike the corpuscular theory, the wave theory (in conjunction with plausible auxiliary hypotheses about the orientation and amplitude of the waves) does predict diffraction effects. Hence, when these were observed, confidence in the wave theory increased.

However, confirmation of a prediction (or even many predictions) of a theory does not prove deductively that the theory is true. Theories together with their auxiliary hypotheses always imply many more predictions than can actually be tested. Even if all the predictions tested so far have been verified, some untested prediction may still be false. That would imply the falsity of the theory, provided that the auxiliary hypotheses are true. Hence, from a logical point of view, confidence in any scientific theory should never be absolute.

Nevertheless, it is often held that as more and more of the predictions entailed by a theory are verified, the theory itself becomes more probable. This principle may be formulated more precisely as follows:

(P): If E is some initial body of evidence (including auxiliary hypotheses) and C is the additional verification of some of the theory's predictions, the probability of the theory given E & C is higher than the probability of the theory given E alone.

Principle (P) seems to be the principle underlying the inductions by which scientific theories are confirmed. But it is not self-evidently true, and it is not provable as a law of logic or probability theory. Moreover, some of its instances are evidently false, suggesting that (P) needs further restriction.

To illustrate this point, consider the situation with respect to theories of light at the time when serious attention was first paid to diffraction phenomena in the middle of the nineteenth century. What happened historically was that the corpuscular theory was rejected and the wave theory accepted. But one might have accounted for diffraction by maintaining a corpuscular theory, augmented by the hypothesis that a strange force acts on the corpuscles of light as they pass through small apertures, separating them into distinct sheaths and thus giving rise to the observed effects. Alternatively, one might have argued that the diffraction phenomena are an illusion due to peculiarities of our cameras and eyes. Or one might have rejected both the wave and corpuscular theories and argued that light is something else entirely—say, filaments or strands emitted from luminous objects. This could be made compatible with the known properties of light by adopting sufficiently ingenious auxiliary hypotheses. One could create such alternatives ad infinitum.

Each of these theories, if augmented by appropriate auxiliary hypotheses, predicts diffraction phenomena as well as the other properties of light known in the nineteenth century. Does the observation of diffraction, then, make each more probable, as unrestricted use of (P) suggests? This seems doubtful. In practice, only the wave theory was regarded as having been confirmed or rendered more probable. Theories like those mentioned in the previous paragraph were not seriously considered. The reason is that the auxiliary hypotheses required by these other theories (such as the hypothesis that a strange force affects light corpuscles traveling through small apertures) were themselves unjustified. They were not plausible independent of the theory. Auxiliary hypotheses which have no independent justification and are adopted only to make a theory fit the facts are called ad hoc hypotheses.

In practice, principle (P) is not applied equally to all theories, but preferentially to those theories which do not require ad hoc hypotheses. The wave theory predicted diffraction by means of auxiliary hypotheses which seemed perfectly natural. All competing theories were either extremely complex in themselves or required complex and ad hoc auxiliary hypotheses. So even though other theories could be made to imply the same predictions, only the wave theory was regarded as substantially confirmed by the observation of diffraction. (We might note, incidentally, that the wave theory itself was succeeded by the quantum theory primarily because of the discovery of new phenomena which could not be predicted by the wave theory unless it too were burdened with ad hoc hypotheses.)

Not only is (P) applied preferentially to theories which do not require ad hoc hypotheses; it is also (as suggested by the example above) applied preferentially to theories which are themselves simple. That is, other things being equal, simple theories are regarded as more highly confirmed by verification of their predictions than are complex theories. Various restrictions on (P) have been proposed by various theorists, but they are generally controversial and need not be discussed here.

Supplementary Problems

Arrange each of the following sets of statements in order from strongest to weakest.

(1)

(a) Iron is a metal.

(b) Either iron is a metal or copper is a metal.

(d) It is not true that iron is not a metal.

(e) Iron, zinc, and copper are metals.

(f) Something is a metal.

(g) Some things are both metals and not metals.

(h) Either iron is a metal or it is not a metal.

(i) Iron and zinc are metals.

(2)

(a) Most Americans are employed.

(b) There are Americans, and all of them are employed.

(d) At least 90 percent of Americans are employed.

(e) At least 80 percent of Americans are employed.

(f) Someone is employed.

(3)

(a) About 51 percent of newborn children are boys.

(b) Exactly 51 percent of newborn children are boys.

(d) It is not true that all newborn children are not boys.

(e) Somewhere between one-fourth and three-fourths of all newborn children are boys.

(4)

(a) Leonardo was a great scientist, inventor, and artist who lived during the Renaissance.

(b) Leonardo did not live during the Renaissance.

(d) Leonardo was a Renaissance artist.

(e) Leonardo was not a Renaissance artist.

(f) Leonardo was not a Renaissance artist and scientist.

Arrange the following sets of argument forms in order from greatest to least inductive probability.

(1)

(a) 60 per cent of observed F are G.

x is F.

∴ x is G.

where the minus signs indicate decreases in magnitude. Then we will be still more confident that the rate of watering is the variable responsible for the observed changes in growth rate.

9.7: SCIENTIFIC THEORIES

Supplementary Problems

Arrange each of the following sets of statements in order from strongest to weakest.

(1)

(a) Iron is a metal.

(b) Either iron is a metal or copper is a metal.

(d) It is not true that iron is not a metal.

(e) Iron, zinc, and copper are metals.

(f) Something is a metal.

(g) Some things are both metals and not metals.

(h) Either iron is a metal or it is not a metal.

(i) Iron and zinc are metals.

(2)

(a) Most Americans are employed.

(b) There are Americans, and all of them are employed.

(d) At least 90 percent of Americans are employed.

(e) At least 80 percent of Americans are employed.

(f) Someone is employed.

(3)

(a) About 51 percent of newborn children are boys.

(b) Exactly 51 percent of newborn children are boys.

(d) It is not true that all newborn children are not boys.

(e) Somewhere between one-fourth and three-fourths of all newborn children are boys.

(4)

(a) Leonardo was a great scientist, inventor, and artist who lived during the Renaissance.

(b) Leonardo did not live during the Renaissance.

(d) Leonardo was a Renaissance artist.

(e) Leonardo was not a Renaissance artist.

(f) Leonardo was not a Renaissance artist and scientist.

Arrange the following sets of argument forms in order from greatest to least inductive probability.

(1)

(a) 60 per cent of observed F are G.

x is F.

∴ x is G.

(b) 20 per cent of F are G.

x is F.

∴ x is G.

x is F.

∴ x is G.

(2)

(a) All ten observed F are G.

∴ All F are G.

(b) All ten observed F are G.

∴ If three more F are observed, they will be G.

∴ If two more F are observed, they will be G.

(d) All ten observed F are G.

∴ If two more F are observed, at least one of them will be G.

(e) All F are G.

∴ If an F is observed, it will be G.

(3)

(a) 8 of 10 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(b) 80 of 100 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(d) My doctor prescribes product X.

∴ All doctors prescribe product X.

(e) My doctor prescribes product X.

∴ Some doctor(s) prescribe(s) product X.

(f) All 10 doctors we asked prescribe product X.

∴ All doctors prescribe product X.

(4)

(a) Objects a, b, c, and d all have properties F and G.

Objects a, b, c, and d all have property H.

Object e has properties F and G.

Object e has property H.

(b) Objects a, b, c, and d all have properties F, G, and H.

Objects a, b, c, and d all have property I.

Object e has properties F, G, and H.

∴ Object e has property I.

Object a has property G.

Object b has property F.

∴ Object b has property G.

(d) Object a has property F.

∴ Object b has property F.

(e) Object a has properties F and G.

Object a has property H.

Object b has properties F and G.

∴ Object b has property H.

(f) Object a has property F.

∴ Objects b and c have property F.

(5)

(a) Objects a, b, c, d, and e have property F.

∴ All objects have property F.

(b) Objects a, b, c, d, and e have property F.

∴ Objects f and g have property F.

∴ All objects have property F.

(d) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Objects f and g have property G.

(e) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Object f has property G.

III

Each of the following problems consists of a list of observations. For each, answer the following questions. Are the observations compatible with the assumption that exactly one cause of the type indicated (necessary, sufficient, etc.) is among the suspected causes? If so, do the observations enable us to identify it using Mill's methods? If they do, which of the unsuspected causes is it, and by what method is it identified?

Answers to Selected Supplementary Problems

(1) (g), (e), (i), (a) and (d), (b), (c), (f), (h) ((a) and (d) are of equal strength)

(4) (a), (d), (c), (b), (e), (f)

(2) (e), (d), (c), (b), (a)

(4) (b), (a), (e), (c), (d), (f)

III

(3) None of the suspected causes is necessary for E (method of agreement).

(6) G is the only one of the suspected causes which could be necessary and sufficient for E (joint method of agreement and difference).

(9) H is the only one of the suspected causal variables on which E could be dependent (method of concomitant variation).

9.5: INDUCTION BY ANALOGY

F1x & F2x & . . . & Fnx

F1y & F2y & . . . & Fny

∴ Gx

SOLVED PROBLEM

9.19

Evaluate the following argument:

Specimen x is a single-stemmed plant with lanceolate leaves and five-petaled blue flowers, about 0.4 meter tall, found growing on a sunny roadside.

Specimen y is a single-stemmed plant with lanceolate leaves and five-petaled blue flowers, about 0.4 meter tall, found growing on a sunny roadside.

Specimen y is a member of the gentian family.

∴ Specimen x is a member of the gentian family.

Solution

Specimen x is a member of the gentian family or some closely related family.

SOLVED PROBLEM

9.20

Estimate the inductive probability of the following argument by analogy:

Person x was born on a Monday, has dark hair, is 5 feet 8 inches tall, and speaks Finnish.

Person y was born on a Monday, has dark hair, is 5 feet 8 inches tall, and speaks Finnish.

Person y likes brussels sprouts.

∴ Person x likes brussels sprouts.

Solution

The inductive probability is low, because the properties F1, F2,. . ., Fn mentioned in the first two premises are almost surely irrelevant to the property G (the property of liking brussels sprouts).

SOLVED PROBLEM

9.21

Arrange the following analogical arguments in order of decreasing inductive probability:

(a) A common housefly x, 8 millimeters long, is being placed in a tightly closed jar.

A common housefly y, 8 millimeters long, was placed in a tightly closed jar.

y died within a day.

∴ x will die within a day.

(b) A common housefly x, 8 millimeters long and 14 days old, is being placed in a tightly closed jar.

A common housefly y, 8 millimeters long and 14 days old, was placed in a tightly closed jar.

y died within a day.

∴ x will die within a day.

A common housefly y, 8 millimeters long, was placed in a tightly closed jar.

y died within a day.

∴ x will die within 12 hours.

(d) A common housefly x, 8 millimeters long and 14 days old, is being placed in a tightly closed jar.

Common houseflies y, z, and w, each 8 millimeters long and 14 days old, were placed in tightly closed jars.

y, z, and w died within a day.

∴ x will die within a day.

(e) A common housefly x, 8 millimeters long and 14 days old, is being placed in a tightly closed jar.

Common houseflies y, z, and w, each 8 millimeters long and 14 days old, were placed in tightly closed jars.

y, z, and w died within a day.

∴ x will die eventually.

(f) A common housefly x, 8 millimeters long, is being placed in a tightly closed jar in Wisconsin.

A common housefly y, 8 millimeters long, was placed in a tightly closed jar in Wisconsin.

y died within a day.

∴ x will die within a day.

(g) A common housefly x is being placed in a tightly closed jar.

A common housefly y was placed in a tighly closed jar.

y died within a day.

∴ x will die within 12 hours.

Solution

SOLVED PROBLEM

9.22

Evaluate the following argument by analogy:

Jim Jones was the leader of a religious movement which advocated peace, brotherhood, and a simple agrarian way of life.

Mahatma Gandhi was the leader of a religious movement which advocated peace, brotherhood, and a simple agrarian way of life.

Mahatma Gandhi was a saintly man.

∴ Jim Jones was a saintly man.

Solution

9.6: MILL'S METHODS

A given effect may have several necessary causes. Fire, for example, requires for its production three causally necessary conditions: fuel, oxygen (or some similar substance), and heat.

SOLVED PROBLEM

9.23

Classify the kind of causality intended by the following statements:

(a) Flipping the wall switch will cause the light to go on.

(b) Closing the electricity supply from the main lines will cause the light to go off.

(d) Pulling the trigger will cause the gun to fire.

(e) Raising the temperature of a gas will cause an increase in its volume.

(f) Raising the temperature of the freezer above +32 degrees Fahrenheit will cause the ice cubes in the freezer to melt.

(g) Killing the President will cause new presidential elections.

(h) Raising the temperature in the environment will cause the death of many plants.

Solution

(a) Necessary (but not sufficient: the light will not go on unless the light bulb is working).

(b) Sufficient (but not necessary: the light will go off also if the wall switch is turned to the “off” position).

(d) Necessary (but not sufficient: the gun won't fire unless it is loaded).

(e) Dependent (the higher the temperature, the higher the volume).

(f) Necessary and sufficient.

(g) Sufficient (but not necessary).

(h) Dependent (the higher the temperature, the greater the number of plants that will die).

Mill's Method of:

Rules Out Conditions Suspected of Being:

Agreement

Necessary causes of E

Difference

Sufficient causes of E

Agreement and difference

Necessary and sufficient causes of E

Concomitant variation

Quantities on which the magnitude of E is causally dependent

The Method of Agreement

SOLVED PROBLEM

9.24

Suppose we are looking for the necessary cause of a certain disease E, and we have by using our background knowledge and expertise formulated a list of five viral agents, V1 through V5, which we suspect may cause E. We examine a number of patients with E and check to see which of the suspected causes is present in each case. The results are as follows:

Solution

Once V2 through V5 are eliminated as necessary causes, it follows deductively that

(1) If the list V1 through V5 includes a necessary cause of E, then V1 is that necessary cause.

This is the conclusion which Mill's method of agreement yields. If we wish to advance further to the unconditional conclusion

(2) V1 is a necessary cause of E

then we need the premise

(3) The list V1 through V5 includes a necessary cause of E.

(4) Disease E has characteristics F1, F2,. . ., Fn.

(5) The known diseases similar to E have characteristics F1, F2,. . ., Fn.

(6) Viruses are necessary causes of the known diseases similar to E.

∴ (7) Some virus is a necessary cause of E.

Here the characteristics F1, F2,. . ., Fn might be such things as infectiousness or the presence of fever. To get from statement 7 to statement 3, we need to add to statement 7 the premise that

(8) The only viruses present in the cases of patients 1 through 4, who had E, were V1 through V5.

The Method of Difference

SOLVED PROBLEM

9.25

A number of people have eaten a picnic lunch which included five foods, F1 through F5. Many of them are suffering from food poisoning. It is assumed that among the five foods is one which is sufficient to produce the poisoning among this group of people. Now suppose that we find two individuals, one of whom has eaten all five foods and is suffering from food poisoning and the other of whom has eaten all but F1 and is feeling fine. Thus if P is the effect of poisoning, the situation is this:

What is the sufficient cause of P?

Solution

The Joint Method of Agreement and Difference

SOLVED PROBLEM

9.26

S = electric shaver

H = hair dryer

D = clothes dryer

W = washing machine

She now observes which appliances are operating in nearby rooms while her television is on. The results are as follows (I is the interference):

Which of the suspected causes, if any, is the necessary and sufficient cause of the interference?

Solution

The Method of Concomitant Variation

SOLVED PROBLEM

9.27

A houseplant exhibits a sudden spurt of growth. We suspect that the variables relevant to its growth rate are these:

S = sunlight

W = water

F = fertilizer

T = temperature

But we observe that only one of these variables, namely, the amount of water the plant receives, has been altered recently. This observation may be schematized as follows:

Solution

Since the amount of water the plant receives is the only one of the variables on our list that has changed, only it among these variables could be responsible for the observed change in growth rate.

Notice that this method does not eliminate the possibility that changes in S, F, or T also affect G. What it shows is that these three variables were not responsible for the particular changes observed here. Hence, if some variable on our list was responsible, it must have been W. To verify our conjecture further, we may cut back on the water the plant receives. Suppose that we then find this:

where the minus signs indicate decreases in magnitude. Then we will be still more confident that the rate of watering is the variable responsible for the observed changes in growth rate.

9.7: SCIENTIFIC THEORIES

Supplementary Problems

Arrange each of the following sets of statements in order from strongest to weakest.

(1)

(a) Iron is a metal.

(b) Either iron is a metal or copper is a metal.

(d) It is not true that iron is not a metal.

(e) Iron, zinc, and copper are metals.

(f) Something is a metal.

(g) Some things are both metals and not metals.

(h) Either iron is a metal or it is not a metal.

(i) Iron and zinc are metals.

(2)

(a) Most Americans are employed.

(b) There are Americans, and all of them are employed.

(d) At least 90 percent of Americans are employed.

(e) At least 80 percent of Americans are employed.

(f) Someone is employed.

(3)

(a) About 51 percent of newborn children are boys.

(b) Exactly 51 percent of newborn children are boys.

(d) It is not true that all newborn children are not boys.

(e) Somewhere between one-fourth and three-fourths of all newborn children are boys.

(4)

(a) Leonardo was a great scientist, inventor, and artist who lived during the Renaissance.

(b) Leonardo did not live during the Renaissance.

(d) Leonardo was a Renaissance artist.

(e) Leonardo was not a Renaissance artist.

(f) Leonardo was not a Renaissance artist and scientist.

Arrange the following sets of argument forms in order from greatest to least inductive probability.

(1)

(a) 60 per cent of observed F are G.

x is F.

∴ x is G.

(b) 20 per cent of F are G.

x is F.

∴ x is G.

x is F.

∴ x is G.

(2)

(a) All ten observed F are G.

∴ All F are G.

(b) All ten observed F are G.

∴ If three more F are observed, they will be G.

∴ If two more F are observed, they will be G.

(d) All ten observed F are G.

∴ If two more F are observed, at least one of them will be G.

(e) All F are G.

∴ If an F is observed, it will be G.

(3)

(a) 8 of 10 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(b) 80 of 100 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(d) My doctor prescribes product X.

∴ All doctors prescribe product X.

(e) My doctor prescribes product X.

∴ Some doctor(s) prescribe(s) product X.

(f) All 10 doctors we asked prescribe product X.

∴ All doctors prescribe product X.

(4)

(a) Objects a, b, c, and d all have properties F and G.

Objects a, b, c, and d all have property H.

Object e has properties F and G.

∴ Object e has property H.

(b) Objects a, b, c, and d all have properties F, G, and H.

Objects a, b, c, and d all have property I.

Object e has properties F, G, and H.

∴ Object e has property I.

Object a has property G.

Object b has property F.

∴ Object b has property G.

(d) Object a has property F.

∴ Object b has property F.

(e) Object a has properties F and G.

Object a has property H.

Object b has properties F and G.

∴ Object b has property H.

(f) Object a has property F.

∴ Objects b and c have property F.

(5)

(a) Objects a, b, c, d, and e have property F.

∴ All objects have property F.

(b) Objects a, b, c, d, and e have property F.

∴ Objects f and g have property F.

∴ All objects have property F.

(d) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Objects f and g have property G.

(e) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Object f has property G.

III

Answers to Selected Supplementary Problems

(1) (g), (e), (i), (a) and (d), (b), (c), (f), (h) ((a) and (d) are of equal strength)

(4) (a), (d), (c), (b), (e), (f)

(2) (e), (d), (c), (b), (a)

(4) (b), (a), (e), (c), (d), (f)

III

(3) None of the suspected causes is necessary for E (method of agreement).

(6) G is the only one of the suspected causes which could be necessary and sufficient for E (joint method of agreement and difference).

(9) H is the only one of the suspected causal variables on which E could be dependent (method of concomitant variation).

1 Readers who wish to understand the mathematical details of the relationships among n, s, the margin of error, and the probability of the conclusion, given the premise, of a statistical generalization should consult the material on confidence intervals in any standard work on statistics.

9.6: MILL'S METHODS

A given effect may have several necessary causes. Fire, for example, requires for its production three causally necessary conditions: fuel, oxygen (or some similar substance), and heat.

SOLVED PROBLEM

9.23

Classify the kind of causality intended by the following statements:

(a) Flipping the wall switch will cause the light to go on.

(b) Closing the electricity supply from the main lines will cause the light to go off.

(d) Pulling the trigger will cause the gun to fire.

(e) Raising the temperature of a gas will cause an increase in its volume.

(f) Raising the temperature of the freezer above +32 degrees Fahrenheit will cause the ice cubes in the freezer to melt.

(g) Killing the President will cause new presidential elections.

(h) Raising the temperature in the environment will cause the death of many plants.

Solution

(a) Necessary (but not sufficient: the light will not go on unless the light bulb is working).

(b) Sufficient (but not necessary: the light will go off also if the wall switch is turned to the “off” position).

(d) Necessary (but not sufficient: the gun won't fire unless it is loaded).

(e) Dependent (the higher the temperature, the higher the volume).

(f) Necessary and sufficient.

(g) Sufficient (but not necessary).

(h) Dependent (the higher the temperature, the greater the number of plants that will die).

Mill's Method of:

Rules Out Conditions Suspected of Being:

Agreement

Necessary causes of E

Difference

Sufficient causes of E

Agreement and difference

Necessary and sufficient causes of E

Concomitant variation

Quantities on which the magnitude of E is causally dependent

The Method of Agreement

SOLVED PROBLEM

9.24

Solution

Once V2 through V5 are eliminated as necessary causes, it follows deductively that

(1) If the list V1 through V5 includes a necessary cause of E, then V1 is that necessary cause.

This is the conclusion which Mill's method of agreement yields. If we wish to advance further to the unconditional conclusion

(2) V1 is a necessary cause of E

then we need the premise

(3) The list V1 through V5 includes a necessary cause of E.

(4) Disease E has characteristics F1, F2,. . ., Fn.

(5) The known diseases similar to E have characteristics F1, F2,. . ., Fn.

(6) Viruses are necessary causes of the known diseases similar to E.

∴ (7) Some virus is a necessary cause of E.

Here the characteristics F1, F2,. . ., Fn might be such things as infectiousness or the presence of fever. To get from statement 7 to statement 3, we need to add to statement 7 the premise that

(8) The only viruses present in the cases of patients 1 through 4, who had E, were V1 through V5.

The Method of Difference

SOLVED PROBLEM

9.25

What is the sufficient cause of P?

Solution

The Joint Method of Agreement and Difference

SOLVED PROBLEM

9.26

S = electric shaver

H = hair dryer

D = clothes dryer

W = washing machine

She now observes which appliances are operating in nearby rooms while her television is on. The results are as follows (I is the interference):

Which of the suspected causes, if any, is the necessary and sufficient cause of the interference?

Solution

The Method of Concomitant Variation

SOLVED PROBLEM

9.27

A houseplant exhibits a sudden spurt of growth. We suspect that the variables relevant to its growth rate are these:

S = sunlight

W = water

F = fertilizer

T = temperature

But we observe that only one of these variables, namely, the amount of water the plant receives, has been altered recently. This observation may be schematized as follows:

Solution

Since the amount of water the plant receives is the only one of the variables on our list that has changed, only it among these variables could be responsible for the observed change in growth rate.

where the minus signs indicate decreases in magnitude. Then we will be still more confident that the rate of watering is the variable responsible for the observed changes in growth rate.

9.7: SCIENTIFIC THEORIES

Supplementary Problems

Arrange each of the following sets of statements in order from strongest to weakest.

(1)

(a) Iron is a metal.

(b) Either iron is a metal or copper is a metal.

(d) It is not true that iron is not a metal.

(e) Iron, zinc, and copper are metals.

(f) Something is a metal.

(g) Some things are both metals and not metals.

(h) Either iron is a metal or it is not a metal.

(i) Iron and zinc are metals.

(2)

(a) Most Americans are employed.

(b) There are Americans, and all of them are employed.

(d) At least 90 percent of Americans are employed.

(e) At least 80 percent of Americans are employed.

(f) Someone is employed.

(3)

(a) About 51 percent of newborn children are boys.

(b) Exactly 51 percent of newborn children are boys.

(d) It is not true that all newborn children are not boys.

(e) Somewhere between one-fourth and three-fourths of all newborn children are boys.

(4)

(a) Leonardo was a great scientist, inventor, and artist who lived during the Renaissance.

(b) Leonardo did not live during the Renaissance.

(d) Leonardo was a Renaissance artist.

(e) Leonardo was not a Renaissance artist.

(f) Leonardo was not a Renaissance artist and scientist.

Arrange the following sets of argument forms in order from greatest to least inductive probability.

(1)

(a) 60 per cent of observed F are G.

x is F.

∴ x is G.

(b) 20 per cent of F are G.

x is F.

∴ x is G.

x is F.

∴ x is G.

(2)

(a) All ten observed F are G.

∴ All F are G.

(b) All ten observed F are G.

∴ If three more F are observed, they will be G.

∴ If two more F are observed, they will be G.

(d) All ten observed F are G.

∴ If two more F are observed, at least one of them will be G.

(e) All F are G.

∴ If an F is observed, it will be G.

(3)

(a) 8 of 10 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(b) 80 of 100 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(d) My doctor prescribes product X.

∴ All doctors prescribe product X.

(e) My doctor prescribes product X.

∴ Some doctor(s) prescribe(s) product X.

(f) All 10 doctors we asked prescribe product X.

∴ All doctors prescribe product X.

(4)

(a) Objects a, b, c, and d all have properties F and G.

Objects a, b, c, and d all have property H.

Object e has properties F and G.

∴ Object e has property H.

(b) Objects a, b, c, and d all have properties F, G, and H.

Objects a, b, c, and d all have property I.

Object e has properties F, G, and H.

∴ Object e has property I.

Object a has property G.

Object b has property F.

∴ Object b has property G.

(d) Object a has property F.

∴ Object b has property F.

(e) Object a has properties F and G.

Object a has property H.

Object b has properties F and G.

∴ Object b has property H.

(2)

(a) All ten observed F are G.

∴ All F are G.

(b) All ten observed F are G.

∴ If three more F are observed, they will be G.

∴ If two more F are observed, they will be G.

(d) All ten observed F are G.

∴ If two more F are observed, at least one of them will be G.

(e) All F are G.

∴ If an F is observed, it will be G.

(3)

(a) 8 of 10 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(b) 80 of 100 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(d) My doctor prescribes product X.

∴ All doctors prescribe product X.

(e) My doctor prescribes product X.

∴ Some doctor(s) prescribe(s) product X.

(f) All 10 doctors we asked prescribe product X.

∴ All doctors prescribe product X.

(4)

(a) Objects a, b, c, and d all have properties F and G.

Objects a, b, c, and d all have property H.

Object e has properties F and G.

∴ Object e has property H.

(b) Objects a, b, c, and d all have properties F, G, and H.

Objects a, b, c, and d all have property I.

Object e has properties F, G, and H.

∴ Object e has property I.

Object a has property G.

Object b has property F.

∴ Object b has property G.

(d) Object a has property F.

∴ Object b has property F.

(e) Object a has properties F and G.

Object a has property H.

Object b has properties F and G.

∴ Object b has property H.

(f) Object a has property F.

∴ Objects b and c have property F.

(5)

(a) Objects a, b, c, d, and e have property F.

∴ All objects have property F.

(b) Objects a, b, c, d, and e have property F.

∴ Objects f and g have property F.

∴ All objects have property F.

(d) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Objects f and g have property G.

(e) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Object f has property G.

III

(1)

Answers to Selected Supplementary Problems

(1) (g), (e), (i), (a) and (d), (b), (c), (f), (h) ((a) and (d) are of equal strength)

(4) (a), (d), (c), (b), (e), (f)

(2) (e), (d), (c), (b), (a)

(4) (b), (a), (e), (c), (d), (f)

III

(3) None of the suspected causes is necessary for E (method of agreement).

(6) G is the only one of the suspected causes which could be necessary and sufficient for E (joint method of agreement and difference).

(9) H is the only one of the suspected causal variables on which E could be dependent (method of concomitant variation).

9.7: SCIENTIFIC THEORIES

Supplementary Problems

Arrange each of the following sets of statements in order from strongest to weakest.

(1)

(a) Iron is a metal.

(b) Either iron is a metal or copper is a metal.

(d) It is not true that iron is not a metal.

(e) Iron, zinc, and copper are metals.

(f) Something is a metal.

(g) Some things are both metals and not metals.

(h) Either iron is a metal or it is not a metal.

(i) Iron and zinc are metals.

(2)

(a) Most Americans are employed.

(b) There are Americans, and all of them are employed.

(d) At least 90 percent of Americans are employed.

(e) At least 80 percent of Americans are employed.

(f) Someone is employed.

(3)

(a) About 51 percent of newborn children are boys.

(b) Exactly 51 percent of newborn children are boys.

(d) It is not true that all newborn children are not boys.

(e) Somewhere between one-fourth and three-fourths of all newborn children are boys.

(4)

(a) Leonardo was a great scientist, inventor, and artist who lived during the Renaissance.

(b) Leonardo did not live during the Renaissance.

(d) Leonardo was a Renaissance artist.

(e) Leonardo was not a Renaissance artist.

(f) Leonardo was not a Renaissance artist and scientist.

Arrange the following sets of argument forms in order from greatest to least inductive probability.

(1)

(a) 60 per cent of observed F are G.

x is F.

∴ x is G.

(b) 20 per cent of F are G.

x is F.

∴ x is G.

x is F.

∴ x is G.

(2)

(a) All ten observed F are G.

∴ All F are G.

(b) All ten observed F are G.

∴ If three more F are observed, they will be G.

∴ If two more F are observed, they will be G.

(d) All ten observed F are G.

∴ If two more F are observed, at least one of them will be G.

(e) All F are G.

∴ If an F is observed, it will be G.

(3)

(a) 8 of 10 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(b) 80 of 100 doctors we asked prescribed product X.

∴ About 80 percent of all doctors prescribe product X.

(d) My doctor prescribes product X.

∴ All doctors prescribe product X.

(e) My doctor prescribes product X.

∴ Some doctor(s) prescribe(s) product X.

(f) All 10 doctors we asked prescribe product X.

∴ All doctors prescribe product X.

(4)

(a) Objects a, b, c, and d all have properties F and G.

Objects a, b, c, and d all have property H.

Object e has properties F and G.

∴ Object e has property H.

(b) Objects a, b, c, and d all have properties F, G, and H.

Objects a, b, c, and d all have property I.

Object e has properties F, G, and H.

∴ Object e has property I.

Object a has property G.

Object b has property F.

∴ Object b has property G.

(d) Object a has property F.

∴ Object b has property F.

(e) Object a has properties F and G.

Object a has property H.

Object b has properties F and G.

∴ Object b has property H.

(f) Object a has property F.

∴ Objects b and c have property F.

(5)

(a) Objects a, b, c, d, and e have property F.

∴ All objects have property F.

(b) Objects a, b, c, d, and e have property F.

∴ Objects f and g have property F.

∴ All objects have property F.

(d) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Objects f and g have property G.

(e) Objects a, b, c, d, and e have property F.

Objects a, b, c, d, and e have property G.

Objects f and g have property F.

∴ Object f has property G.

III

Answers to Selected Supplementary Problems

(1) (g), (e), (i), (a) and (d), (b), (c), (f), (h) ((a) and (d) are of equal strength)

(4) (a), (d), (c), (b), (e), (f)

(2) (e), (d), (c), (b), (a)

(4) (b), (a), (e), (c), (d), (f)

III

(3) None of the suspected causes is necessary for E (method of agreement).

(6) G is the only one of the suspected causes which could be necessary and sufficient for E (joint method of agreement and difference).

(9) H is the only one of the suspected causal variables on which E could be dependent (method of concomitant variation).

From the section, Propositional Logic, read the following sections:

Argument Forms to Formalization

In summary: To determine whether an argument form of propositional logic is valid, put the entire form on a truth table, making as many lines as determined by the number of distinct sentence letters occurring in the relevant formulas. If the table displays no counterexample, then the form is valid (and hence so is any instance of it). If the table displays one or more counterexamples, then the form is invalid. Since invalid forms may have valid as well as invalid instances, the truth table test does not establish the invalidity of specific arguments. If we formalize an argument and then show that the resulting form is invalid, we are not thereby entitled to infer that the argument is invalid. But if a truth table shows a form to be invalid, then it shows that none of its instances is valid solely in virtue of having that form. Any valid instances must derive their validity at least in part from some feature of the argument which has been lost in the process of formalization. Argument 5 of Section 3.1, for example, is valid despite being an instance of affirming the consequent, which is invalid by the truth table test; when it is formalized as affirming the consequent (i.e., as P → Q, Q ⊢ P), the information that the conclusion follows from the second premise is lost.

In Section 3.1 we have seen that a particular argument may in fact be an instance of several forms, some of which are valid and some of which are not. But if it is an instance of any valid form, then it is valid. For example, the argument

If she loves me, then she doesn't hate me.

It's not true that she doesn't hate me.

∴ She doesn't love me.

is an instance of each of the following forms, only the first two of which are valid:

L → ∼H, ∼ ∼H ⊢ ∼L

L → D, ∼D ⊢ ∼L

L → D, N ⊢ ∼L

L → D, N ⊢ S

I, N ⊢ S

In the third case, for example, we have formalized ‘She loves me’ as ‘L’, ‘She doesn't hate me’ as ‘D’, and ‘It's not true that she doesn’t hate me’ as ‘N’. And even this list is not complete. The reader can probably discover several more forms of which this argument is an instance. In formalizing an argument, we generally select the form which displays the most logical structure (in this case the first form), since if the argument is valid in virtue of any of its forms, it will be valid in virtue of that one. However, if a form with less structure is valid (as is the second form on this list), then it too is an adequate formalization for demonstrating the validity of the argument.

SOLVED PROBLEMS

3.19

Construct a truth table for the following form, and use the table to determine whether the form is valid:

P → Q, P → ∼Q ⊢ ∼P

Solution

The only possible situations in which the premises are both true are those represented by the third and fourth lines of the table. But in these situations the conclusion is also true; hence the form is valid.

3.20

Construct a truth table for the following form, and use the table to determine whether the form is valid:

P → Q ⊢ ∼(Q → P)

Solution

The table displays two kinds of counterexamples. The first is when ‘P’ and ‘Q’ are both true (first line of the table); the second when they are both false (last line of the table). Hence the form is invalid.

3.21

Construct a truth table for the following form, and use the table to determine whether the form is valid:

P ∨ Q, Q ∨ R ⊢ P ∨ R

Solution:

The form is invalid, because in situations in which ‘P’ and ‘R’ are false and ‘Q’ is true (line 6 of the table), the premises are both true while the conclusion is false.

3.22

Construct a truth table for the following form, and use the table to determine whether the form is valid:

P, ∼P ⊢ Q

Solution

Since the premises are mutually inconsistent, there is no possible situation in which both are true. Hence there are no counterexamples; the form is valid. However, notice that every argument of this form is unacceptably flawed as a means of proving its conclusion: in the terminology of Chapter 2, criterion 1 (truth of premises) is violated, so the argument cannot be sound. In addition, such an argument may well violate criterion 3 (relevance). Compare Problem 2.21.

3.23

Construct a truth table for the following form, and use the table to determine whether the form is valid:

R ⊢ P ↔ (P ∨ (P & Q))

Solution

The conclusion of this argument is a tautology, so there is no situation in which the premise is true and the conclusion false. The argument is therefore valid. As in the previous example, however, arguments of this form lack relevance, since their validity is totally independent of the relationship between premise and conclusion (compare Problem 2.19).

3.7: REFUTATION TREES

Truth tables provide a rigorous and complete test for the validity or invalidity of propositional logic argument forms, as well as a test for tautologousness, truth-functional contingency, and inconsistency of wffs. Indeed, they constitute an algorithm, the sort of precisely specifiable test which can be performed by a computer and which always yields an answer after a finite number of finite operations. When there is an algorithm for determining whether or not the argument forms expressible in a formal system are valid, that system is said to be decidable. Thus the truth tables ensure the decidability of predicate logic. But they are cumbersome and inefficient, especially for problems involving more than two or three sentence letters. Refutation trees, the topic of this section, provide a more efficient algorithm for performing the same tasks.

Given a list of wffs, a refutation tree is an exhaustive search for ways in which all the wffs on the list can be true. To test an argument form for validity using a refutation tree, we construct a list consisting of its premises and the negation of its conclusion. The search is carried out by breaking down the wffs on the list into sentence letters or their negations. If we find any assignment of truth and falsity to sentence letters which makes all the wffs on the list true, then under that assignment the premises of the form are true while its conclusion is false. Thus we have refuted the argument form; it is invalid. If the search turns up no assignment of truth and falsity to sentence letters which makes all the wffs on the list true, then our attempted refutation has failed; the form is valid. To illustrate, we examine some simple examples.

SOLVED PROBLEMS

3.24

Construct a refutation tree to show that the form ‘P & Q ⊢ ∼ ∼P’ is valid.

Solution

We begin by forming the list consisting of the premise and the negation of the conclusion:

P & Q

∼ ∼ ∼P

Now the premise is true if and only if ‘P’ and ‘Q’ are both true. Hence we can without distortion replace ‘P & Q’ by these two sentence letters. We show this by writing ‘P’ and ‘Q’ at the bottom of the list and checking off the formula ‘P & Q’ to indicate that we are finished with it. A checked formula is in effect eliminated from the list:

Moreover ‘∼ ∼ ∼P’ is true if and only if the simpler formula ‘∼P’ is true; hence we can check ‘∼ ∼ ∼P’ and replace it by ‘∼P’:\

We have now broken down the original list of formulas into a list of sentence letters or negations of sentence letters, all of which must be true if all the members of our original list are to be true. But among these sentence letters and their negations are both ‘P’ and ‘∼P’, which cannot both be true. Hence it is impossible for everything on the finished list to be true. We express this by writing ‘X’ at the bottom of the list.

The refutation tree is now complete. Our search for a refutation of the original argument form has failed; hence this form is valid.

3.25

Construct a refutation tree to show that the form ‘P ∨ Q, ∼P ⊢ Q’ is valid.

Solution

As before, we begin with a list consisting of the premises followed by the negation of the conclusion:

P ∨ Q

∼P

∼Q

Since both ‘∼P’ and ‘∼Q’ are negations of sentence letters, they cannot be analyzed further; but ‘P ∨ Q’ is true if and only if either ‘P’ or ‘Q’ is true. To represent the fact that ‘P ∨ Q’ can be true in either of these two ways, we check ‘P ∨ Q’ and “branch” the tree, like this:

The tree now contains two paths, each starting with the checked formula ‘P ∨ Q’. The first branches to the left and ends with ‘P’; the second branches to the right and ends with ‘Q’. The three formulas on the initial list can be true if and only if all the formulas on one or both of these paths can be true. But the first path contains both ‘P’ and ‘∼P’, and the second contains both ‘Q’ and ‘∼Q’. Hence not all the formulas on either path can be true. As in the previous problem, we indicate this by ending each path with an X:

This is the finished tree. Since the attempted refutation fails along both paths, the original argument is valid.

3.26

Construct a refutation tree to determine whether the following form is valid:

P ∨ Q, P ⊢ ∼Q

Solution

Again, we form a list consisting of the premises and the negation of the conclusion:

P ∨ Q

∼ ∼Q

‘∼ ∼Q’ is equivalent to the simpler formula ‘Q’, so we check it and write ‘Q’ at the bottom of the list. Then, as in the previous problem, we check ‘P ∨ Q’ and show its truth possibilities by branching the tree:

or their negations. Moreover, both paths represent possible truth value assignments, since neither contains both a sentence letter and its negation. Since each path contains both ‘P’ and ‘Q’, each represents an assignment on which both these letters are true. Moreover, the tree indicates that under this truth value assignment all three formulas on the original list are true, i.e., that the premises of the form ‘P ∨ Q, P ⊢ ∼Q’ are true while its conclusion is false. Hence this form is invalid.

The reader may wish to confirm the findings of these three problems by constructing truth tables for their respective forms. We now consider more systematically the procedure by which these problems were solved.

A refutation tree is an analysis in which a list of statements is broken down into sentence letters or their negations, which represent ways in which the members of the original list may be true. Since the ways in which a statement may be true depend on the logical operators it contains, formulas containing different logical operators are broken down differently. All wffs containing logical operators fall into one of the following ten categories:

Negation

Negated negation

Conjunction

Negated conjunction

Disjunction

Negated disjunction

Conditional

Negated conditional

Biconditional

Negated biconditional

Corresponding to each category is a rule for extending refutation trees. Problems 3.24 to 3.26 illustrated four of these rules. In order to state them, we need first to define the concept of an open path. An open path is any path of a tree which has not been ended with an ‘X’. Paths which have been ended with an ‘X’ are said to be closed. The four rules may now be stated as follows:

Negation (∼): If an open path contains both a formula and its negation, place an ‘X’ at the bottom of the path.

The idea here is that any path which contains both a formula and its negation is not a path all of whose formulas can be true, which is what we are searching for in constructing a refutation tree. Hence we can close this path as a failed attempt at refutation. The negation rule was used in Problems 3.24 and 3.25.

Negated Negation (∼ ∼): If an open path contains an unchecked wff of the form ∼ ∼ϕ, check it and write ϕ at the bottom of every open path that contains this newly checked wff.

This rule was used in Problems 3.24 and 3.26.

Conjunction (&): If an open path contains an unchecked wff of the form ϕ & Ψ, check it and write ϕ and Ψ at the bottom of every open path that contains this newly checked wff.

This rule was used in Problem 3.24.

Disjunction (∨): If an open path contains an unchecked wff of the form ϕ ∨ Ψ, check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write ϕ and at the end of the second of which write Ψ.

This rule was used in Problems 3.25 and 3.26.

A path is finished if it is closed or if the only unchecked wffs it contains are sentence letters or their negations, so that no more rules apply to its formulas. A tree is finished if all its paths are finished. If all the paths of a finished tree are closed (as in Problems 3.24 and 3.25), then the original formulas from which the tree is constructed cannot be true simultaneously. Thus, if the list is constructed from an argument form by negating its conclusion, that form is valid. On the other hand, if one or more of the paths of a finished tree are open (as in Problem 3.26), then the original formulas from which the tree is constructed can be true simultaneously. If the list is constructed from an argument form by negating its conclusion, this means that the form is invalid.

Indeed, the finished tree displays more than just the validity or invalidity of the argument form. Each open path of the finished tree is a prescription for constructing counterexamples. The only unchecked formulas on a finished open path are sentence letters or their negations. Any situation in which the unnegated sentence letters on the path are true and the negated ones are false is a counterexample. For example, the finished tree of Problem 3.26 shows two open paths, each containing both ‘P’ and ‘Q’; thus, any situation in which ‘P’ and ‘Q’ are both true is a counterexample to the form ‘P ∨ Q, P ⊢ ∼Q’.

It is useful to annotate trees by numbering the lines they contain and indicating which rules and lines have been used to add formulas to the tree. We number lines in a column to their left and indicate the lines from which they are derived and the rules used to derive them to their immediate right. Rules are designated by the signs for the connectives they employ. Thus, for example, the annotated version of the tree of Problem 3.25 is as follows:

We now state and illustrate the remaining six rules for generating refutation trees. Together with the four rules already given, they enable us to construct a tree for any finite set of wffs of propositional logic.

Conditional (→): If an open path contains an unchecked wff of the form ϕ → Ψ, check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write ∼ϕ and at the end of the second of which write Ψ.

This rule is based on the fact that ϕ → Ψ is true if and only if either ϕ is false or Ψ is true (see the truth table for the material conditional).

SOLVED PROBLEM

3.27

Construct a refutation tree to determine whether the following form is valid:

P → Q, Q → R, P ⊢ R

Solution

We begin by writing the premises and then the negation of the conclusion (lines 1 to 4). The conditional rule is then applied to line 1 to obtain line 5. The left branch closes at 6, by the negation rule, but the right branch remains open, and so the conditional rule is applied to 2 to obtain line 7. The negation rule then closes the two remaining paths. Since the finished tree is closed, the attempted refutation has failed and the form is valid.

Biconditional (↔): If an open path contains an unchecked wff of the form ϕ ↔ Ψ, check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write both ϕ and Ψ, and at the end of the second of which write both ∼ϕ and ∼Ψ.

This rule is an expression of the fact that ϕ ↔ Ψ is true if and only if ϕ and Ψ are either both false or both true.

SOLVED PROBLEM

3.28

Construct a refutation tree to determine whether the following form is valid:

P ↔ Q, ∼P ⊢ ∼Q

Solution

Notice that it is not necessary to apply the negated negation rule in line 3 in this problem. The tree closes even without this move, since lines 3 and 5 are one the negation of the other. The form is valid.

Negated Conjunction (∼&): If an open path contains an unchecked wff of the form ∼(ϕ & Ψ), check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write ∼ϕ and at the end of the second of which write ∼Ψ.

This rule depends upon the fact that ∼(ϕ & Ψ) is true if and only if either ϕ or Ψ is false.

SOLVED PROBLEM

3.29

Construct a refutation tree to determine whether the following form is valid:

∼(P & Q) ⊢ ∼P & ∼Q

Solution

Again, we begin with the premise and the negation of the conclusion. We analyze these by two steps of negated conjunction (lines 3 and 4). Two of the four paths then close, but two remain open, even after the applications of negated negation at line 5. Since no more rules apply, the tree is finished. And since there are two open paths, the form is invalid. The open paths indicate that situations in which ‘P’ is false and ‘Q’ true or ‘Q’ false and ‘P’ true are counterexamples.

Negated Disjunction (∼∨): If an open path contains an unchecked wff of the form ∼(ϕ ∨ Ψ), check it and write both ∼ϕ and ∼Ψ at the bottom of every open path that contains this newly checked wff.

This rule is an expression of the fact that ∼(ϕ ∨ Ψ) is true if and only if both ϕ and Ψ are false.

SOLVED PROBLEM

3.30

Construct a refutation tree to determine whether the following form is valid:

P → Q ⊢ P ∨ Q

Solution

The negated disjunction rule is applied to line 2 to yield lines 3 and 4. The open path in the finished tree indicates that the form is invalid, and that any situation in which ‘P’ and ‘Q’ are both false is a counterexample.

Negated Conditional (∼ →): If an open path contains an unchecked wff of the form ∼(ϕ → Ψ), check it and write both ϕ and ∼Ψ at the bottom of every open path that contains this newly checked wff.

A negated conditional ∼(ϕ → Ψ) is true if and only if the conditional ϕ → Ψ is false, hence if and only if ϕ is true and Ψ is false; that is the justification for this rule.

SOLVED PROBLEM

3.31

Construct a refutation tree to determine whether the following form is valid:

∼P → ∼Q ⊢ P → Q

Solution

The tree has two open paths, each indicating that the premise is true and the conclusion false when ‘P’ is true and ‘Q’ false. Hence the form is invalid.

Negated Biconditional (∼ ↔): If an open path contains an unchecked wff of the form ∼(ϕ ↔ Ψ), check it and split the bottom of each open path containing this newly checked wff into two branches, at the end of the first of which write both ϕ and ∼Ψ, and at the end of the second of which write both ∼ϕ and Ψ.

This rule is an expression of the fact that a negated biconditional ∼(ϕ ↔ Ψ) is true if and only if the biconditional is false, hence if and only if ϕ is true and Ψ false or ϕ is false and Ψ true.

SOLVED PROBLEM

3.32

Construct a refutation tree to determine whether the following form is valid:

P, P → Q ⊢ P ↔ Q

Solution

Since the finished tree is closed, the form is valid.

Refutation trees are useful for purposes other than testing the validity of argument forms. A list of wffs is truth-functionally consistent if the tree beginning with those formulas (and including no other formulas except for those obtained by applying the rules) contains at least one finished open path. Thus, a consistent list of wffs is one all members of which can be true simultaneously, since a finished open path represents a way to make all the formulas on the list true. If a finished tree contains no open paths, the list of formulas from which it is constructed is inconsistent.

The list may consist of just one formula. If the finished tree for a single formula contains no open paths, then the formula is truth-functionally inconsistent. If it contains more than one open path, then the formula is either tautologous or truth-functionally contingent.

Indeed, refutation trees can also be used to test specifically for tautologousness. A wff is tautologous if and only if its negation is truth-functionally inconsistent. Therefore for any wff ϕ, ϕ is tautologous if and only if all the paths on the finished tree for ∼ϕ are closed (i.e., if and only if ∼ϕ is truth-functionally inconsistent).

SOLVED PROBLEMS

3.33

Construct a refutation tree to determine whether the following wff is tautologous:

(P → Q) ∨ (P & ∼Q)

Solution

We construct the tree for the negation of the wff in question. Since all paths close, our attempt to find a way to make its negation true has failed; it is therefore tautologous.

3.34

Construct a refutation tree to determine whether the following wff is tautologous:

∼(Q → (P & ∼P))

Solution

The left-branching path remains open, showing that ‘∼ ∼(Q → (P & ∼P))’ is true if ‘Q’ is false. Hence ‘∼(Q → (P & ∼P))’ is not tautologous.

The reader should keep the following points in mind when constructing refutation trees:

(1) The rules for constructing trees apply only to whole formulas, not to mere subformulas. Thus, for example, the use of negated negation in the following tree is impermissible:

Although removing double negations from subformulas does not produce wrong answers, it is unnecessary and makes trees more difficult to read. More serious problems may result from

trying to apply some of the rules for binary operators to subformulas, since we have not defined a consistent procedure for doing so.

(2) The order in which rules are applied makes no difference to the final answer, but it is usually most efficient to apply nonbranching rules first. After branching rules have been applied, further steps may require formulas to be written at the bottoms of several paths, which may necessitate more writing than if nonbranching rules had been applied first. Consider, for example, what happens if we apply the branching rule ‘→’ first in Problem 3.30. Then the tree is:

Here the formulas ‘∼P’ and ‘∼Q’ must each be written twice, whereas if we apply the nonbranching rule negated disjunction first, as in Problem 3.30, we need write them only once.

(3) The open paths of a finished tree for an argument form display all the counterexamples to that form. This is true even if not all the sentence letters of the form occur among the unchecked formulas of some open paths. Consider, for example, the invalid form ‘P → Q ⊢ P’. Its tree is:

Both paths are open. The right-branching path indicates that situations in which ‘P’ is false while ‘Q’ is true are counterexamples; on the other hand, the left-branching path indicates simply that situations in which ‘P’ is false are counterexamples. The letter ‘Q’ does not occur among the unchecked formulas of this path. This shows that the falsity of ‘P’ is by itself sufficient for a counterexample, i.e., that any situation in which ‘P’ is false is a counterexample, regardless of the truth value of ‘Q’. Thus the tree indicates that there are two kinds of counterexamples to the form: situations in which ‘P’ is false and ‘Q’ true, and situations in which both are false.

Table 3-1: Refutation Tree Rules

Supplementary Problems

Formalize the following statements, using the interpretation indicated below:

Sentence Letter P

Interpretation

Pam is going.

Quincy is going.

Richard is going.

Sara is going.

(1) Pam is not going.

(2) Pam is going, but Quincy is not.

(3) If Pam is going, then so is Quincy.

(4) Pam is going if Quincy is.

(5) Pam is going only if Quincy is.

(6) Pam is going if and only if Quincy is.

(7) Neither Pam nor Quincy is going.

(8) Pam and Quincy are not both going.

(9) Either Pam is not going or Quincy is not going.

(10) Pam is not going if Quincy is.

(11) Either Pam is going, or Richard and Quincy are going.

(12) If Pam is going, then both Richard and Quincy are going.

(13) Pam is staying, but Richard and Quincy are going.

(14) If Richard is going, then if Pam is staying, Quincy is going.

(15) If neither Richard nor Quincy is going, then Pam is going.

(16) Richard is going only if Pam and Quincy are staying.

(17) Richard and Quincy are going, although Pam and Sara are staying.

(18) If either Richard or Quincy is going, then Pam is going and Sara is staying.

(19) Richard and Quincy are going if and only if either Pam or Sara is going.

(20) If Sara is going, then either Richard or Pam is going, and if Sara is not going, then both Pam and Quincy are going.

Determine which of the following formulas are wffs and which are not. Explain your answer.

(1) ∼(∼P)

(2) P ∼Q

(3) (P → P)

(4) P → P

(5) ∼ ∼ ∼(P & Q

(6) ((P → Q))

(7) ∼(P & Q) & ∼R

(8) (P ↔ (P ↔(P ↔ P)))

(9) (P → (Q → (R & S))

(10) (P → (Q ∨ R ∨ S))

III

Determine whether the following formulas are tautologous, truth-functionally contingent, or inconsistent, using either truth tables or refutation trees.

(1) P → P

(2) P → ∼P

(3) ∼(P → P)

(4) P → Q

(5) (P ∨ Q) → P

(6) (P & Q) → P

(7) P ↔ ∼(P ∨ Q)

(8) ∼((P & Q) ↔ (P ∨ Q))

(9) (P & Q)& ∼(P ∨ R)

(10) (P → (Q & R)) → (P → R)

Test the following forms for validity, using either truth tables or refutation trees.

(1) ∼P ⊢ P → ∼P

(2) P ∨ Q ⊢ P & Q

(3) P → ∼Q ⊢ ∼(P & Q)

(4) P ⊢ (P → (Q & P)) → (P & Q)

(5) P ∨ Q, ∼P, ∼Q ⊢ R

(6) (Q & R) → P, ∼Q, ∼R ⊢ ∼P

(7) ∼(P ∨ Q), R ↔ P ⊢ ∼R

(8) ∼(P & Q), R ↔ P ⊢ ∼R

(9) P ↔ Q, Q ↔ R ⊢ P ↔ R

(10) P → (R ∨ S), (R & S) → Q ⊢ P → Q

Formalize the following arguments, using the interpretation given below, and test their forms for validity using either truth tables or refutation trees.

Sentence Letter

Interpretation

Argument form F has a counterexample.

The premises of argument form F are inconsistent.

The finished tree for argument form F contains an open path.

Argument form F is valid.

(1) If argument form F is valid, then its finished tree contains no open paths. Hence if its finished tree contains an open path, it is invalid.

(2) If the premises of argument form F are inconsistent, then F is valid. Therefore, if the premises of form F are not inconsistent, then form F is invalid.

(3) If argument form F has a counterexample, then its premises are not inconsistent. For it is not the case both that it has a counterexample and that its premises are consistent.

(4) Either argument form F has a counterexample or it is valid, but not both. Hence form F is valid if and only if it has no counterexample.

(5) The premises of argument form F are inconsistent. If the premises of F are inconsistent, then F is valid. F is valid if and only if its finished tree contains no open path. If its finished tree contains no open path, then it has no counterexample. Form F has a counterexample. Therefore, the premises of form F are not inconsistent.

Answers to Selected Supplementary Problems

(5) P → Q.

(10) Q → ∼P.

(15) ∼(R ∨ Q) → P, or equivalently (∼R & ∼Q) → P.

(20) (S → (R ∨ P)) & (∼S → (P & Q)).

(5) ‘P’ and ‘Q’ are wffs by rule 1 in Section 3.3; so ‘∼P’ is a wff by rule 2. Hence ‘(∼P & Q)’ is a wff by rule 3, whence by three applications of rule 2 it follows that ‘∼ ∼ ∼(∼P & Q)’ is a wff.

(10) Not a wff. No rule allows us to disjoin three statements at once.

III

(2) Truth-functionally contingent

(4) Truth-functionally contingent

(6) Tautologous

(8) Truth-functionally contingent

(10) Tautologous

(2) Invalid

(4) Valid

(6) Invalid

(8) Invalid

(10) Invalid

(2) Invalid form (indeed, the argument itself is invalid)

(4) Valid form

1 From now on we shall omit the qualification ‘declarative’, since we shall only be concerned with sentences that can be used to express premises and conclusions of arguments (and by definition these can only be expressed by declarative sentences; see Section 1.1).

2 Some authors use different symbols. Here are some of the most common alternatives:

Logical Operator

Alternative Symbol(s)

It is not the case that

− or ¬

And

· or ∧

Either . . . or

none

If . . . then

⊃

If and only if

≡

3 Recall that the same sentence can be used to express different statements in different circumstances—even statements that disagree in their being true or false (see Chapter 1, footnote 2). However, where there is no danger of confusion, we shall suppress the distinction and speak freely of the truth value of a sentence (or of a wff).

4 Some important families of nonclassical logics are surveyed in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume 3: Non-Classical Logics, Dordrecht, Reidel, 1986.