PRELIMINARIES: LOGIC AND PHILOSOPHICAL METHOD



ARGUMENTS

Copyright 2008 © by Elmar J. Kremer

Philosophy of Religion contains many famous and complex arguments. Studying these arguments can help students who are interested in religion to think more clearly about the subject. It can also help students in a more general way with the discipline of philosophy, for the arguments have to do with fundamental areas of metaphysics (the philosophical study of what exists) and epistemology (the philosophical study of knowledge). It is worthwhile, therefore, to begin by considering what an argument is and how arguments are evaluated.

I: ARGUMENTS

In ordinary English the word 'argument' is often used to stand for a quarrel, as in, "Jones's thoughts were interrupted by the sound of a loud argument in the apartment below." In other cases, the word stands for an attempt to persuade someone or to change someone's mind by any verbal means whatever: "Charles was a tiresome child; he always gave his parents an argument, no matter what he was told to do." But there is also a less colorful and more precise use of the term, as when we speak of arguments for and against the claim that there is a God. Here is an argument, in this sense of the term, in support of a form of skepticism:

(1) If any source of belief is sometimes misleading, it should never be trusted. Each of our external senses (sight, hearing, etc.) is sometimes misleading. Therefore, none of our external senses should ever be trusted.

The first two propositions in this text are offered in support of the third. The term 'argument' is used in philosophy, and in the rest of this text, to refer to arguments in this sense of the term, and can be defined as follows:

Definition: An argument is a set of propositions in which one or more propositions (called the premises) are offered in support of another proposition (called the conclusion).

In ordinary English, arguments are set forth in a variety of ways. The premises may be given first, and then the conclusion, marked with the word 'therefore,' as in the above example. But in other cases, the conclusion is stated before the premises, or in between several premises. The placing of the conclusion and the order of the premises, do not affect the argument. Thus the above argument about skepticism can be reformulated in many ways, including the two following:

(2) None of our external senses should ever be trusted. For each of our external senses (sight, hearing, etc.) is sometimes misleading, and if any source of belief is sometimes misleading, it should never be trusted.

(3) Each of our external senses (sight, hearing, etc.) is sometimes misleading. So, none of our external senses should ever be trusted. For if any source of belief is sometimes misleading, it should never be trusted.

Both of these reformulations--(2) and (3)-- express exactly the same argument as the original passage (1).

When dealing with arguments, it is important to identify correctly which proposition is the conclusion and which are the premises. Sometimes arguments contain special words, like 'therefore,' whose function is to mark the conclusion or the premises. Other words that sometimes mark a proposition as the conclusion are 'so' (as used in the third example), 'thus,' and 'hence'. Words that sometimes mark propositions as premises are 'for' (as in the second and third examples) and 'since'. But be careful. These expressions do not always function in that way. To see this, consider the following examples:

If God exists, there is no evil. But there is evil. Thus, God does not exist.

and

Some unmarried men are very happy. Thus, it is reported that St. Francis of Assisi was a very happy man.

In the first, 'thus' is best taken to mark a conclusion, and in the second, to mean 'for example'. Again, in

Since no source of belief should ever be trusted if it is sometimes misleading, and since all of our external senses are sometimes misleading, our external senses should never be trusted.

'since' is used to mark the premises of an argument. But in

Since the fall of the Soviet Union, life in Moscow has been chaotic.

it is used in a temporal sense.

All of the examples of arguments given so far are relatively simple, in that they contain just one conclusion, together with a set of premises. In philosophy and other disciplines, however, it often happens that the conclusion of one argument serves as a premise for a further argument, and a number of arguments related in this way are presented in a single passage.[1] To understand such a passage, one must first determine what the main conclusion in the passage is. A good example is the atheistic argument that St. Thomas Aquinas considers at the beginning of the article in which he tries to prove that God exists:

It seems that God does not exist, for, if one of a pair of contraries were infinite, it would completely destroy the other. But God is something infinitely good. Therefore if God were to exist, there would be no evil in the world. But there is evil in the world. Therefore, God does not exist. (Summa Theologiae, part I, question 2, article 3, first objection).

St. Thomas is at pains to make clear to his readers what the main conclusion is. He states it first, then gives the rest of the argument, and then states it once again. In order to understand and evaluate a complex argument of this kind, it is useful to identify and number the propositions of which it is composed, setting extraneous matter aside in brackets, thus:

{It seems that} 1God does not exist, {for} 2if one of a pair of contraries were infinite, it would completely destroy the other. {But} 3God is something infinitely good. {Therefore} 4if God were to exist, there would be no evil in the world. {But} 5there is evil in the world. {Therefore,} 6God does not exist.

The main conclusion of the argument is proposition 1, which reappears as proposition 6. The basic premises or assumptions of the argument, that is, the premises that are not supported within the argument, are 2, 3, and 5.Proposition 4 is an intermediary conclusion, which is claimed to follow from 2 and 3.

Let us say that an argument is in explicit form when each proposition in the argument is set forth once in a separate numbered line as a grammatically complete proposition (with extraneous material omitted); and the main and any intermediate conclusions are marked with the word 'Therefore'. In addition, each conclusion must be preceded by the premises by which it is supposed to be supported. The above argument can be put into explicit form as follows:

2. If one of a pair of contraries were infinite, it would completely destroy the other.

3. God is something infinitely good.

Therefore,

4. If God were to exist, there would be no evil in the world.

5. There is evil in the world.

Therefore,

6. God does not exist.

Since explicit form requires that each proposition appear only once, and that each conclusion appear after the premises on which it rests, it was necessary to suppress

proposition 1. I have retained the numbering used in the above analysis, although normally, the lines would be numbered sequentially, beginning with '1'.

There is no simple set of rules, no simple procedure, for identifying and analyzing arguments. The skill has to be developed by practice. Two general guidelines can be offered, however. First, identifying and analyzing arguments requires going beyond the grammatical characteristics of the propositions involved. It requires getting at the thought expressed, and this often requires consideration of the content and the context of a given passage. Second, if there is any difficulty in deciding whether an argument is being presented by an author, a good rule is to begin by asking yourself what conclusion, if any, is being argued for. If you can spot a conclusion or conclusions, then you know that an argument is being presented.

EXERCISES

A. Say whether an argument is presented in each of the passages below. If you think an argument is presented, identify the conclusion and the premises.

1. If a person's desk is organized, her mind is organized. Sue's desk is organized. We can conclude that Sue's mind is organized.

2. Abortion raises serious moral questions, for abortion involves the taking of human life, and anything that involves the taking of human life raises serious moral questions.

3. All living things need some external source of energy. The sun is the only external source of energy for living things on earth. Thus, all living things on earth need the sun.

4. Since no man has any natural authority over his fellows, and since force produces no right to any, all justifiable authority among men must be established on the basis of conventions (Jean-Jacques Rousseau, The Social Contract).

5. If free-market capitalism is the best economic arrangement and free-market capitalism is based on greed, then the best economic arrangement is one based on greed.

6. The existence of God provides the only satisfactory explanation of the existence of changing things. Therefore God exists, since there must be a satisfactory explanation of whatever exists.

7. Believe me, officer, I wasn't driving 150 in a 100 kph zone, honestly. You've got to believe me; I really mean it.

8. Toronto is quite a multiracial city. Thus most of my neighbors on one side are Chinese, and my next-door neighbors on the other side are Sikhs.

9. Health is not the only desirable characteristic in a human being. If it were, then healthy people would always be happier than unhealthy people. But this is not always the case.

10. By utility is meant that property in any object, whereby it tends to produce benefit, advantage, pleasure, good, or happiness . . . or . . . to prevent the happening of mischief, pain, evil, or unhappiness to the party whose interest is being considered: if that party be the community in general, then the happiness of the community: if a particular individual, then the happiness of the individual (Jeremy Bentham, Introduction to the Principles of Morals and Legislation).

11. A prince, therefore, not being able to exercise this virtue of liberality without risk if it be known, must not, if he be prudent, object to being called miserly (Niccolo Machiavelli, The Prince).

II: DEDUCTIVE AND NON-DEDUCTIVE ARGUMENTS

The examples of arguments give above are all of the kind called "deductive arguments." In an argument of that kind, the premises are put forward as providing conclusive, or strictly sufficient, evidence for the conclusion. By contrast, the premises of an argument are sometimes offered as providing no more than some evidence for the conclusion, and in this case the argument is non-deductive.

When it is said that a set of premises give conclusive or strictly sufficient support for a conclusion, this means (1) that the premises are true and (2) that the conclusion strictly follows from the premises. To say that the conclusion strictly follows from the premises, in turn, means that an inference from the premises to the conclusion preserves truth with absolute necessity, that it is absolutely impossible that such an inference will take one from true premises to a false conclusion. To say that the premises of an argument provide good evidence for the conclusion, on the other hand, means (1) that the premises are true and (2) that the inference tends to be truth preserving, even though it remains possible that it should lead from true premises to a false conclusion. An example may help to make the distinction between two types of argument clear. Consider the argument:

1. Every student in PHL 100Y is under 30 years of age.

2. Susan is a student in PHL 100Y

Therefore,

3. Susan is under 30 years of age.

Now compare it with the following:

1. Most students in PHL 100Y are under 30 years of age.

2. Tom is a student in PHL 100Y.

Therefore,

3. Tom is under 30 years of age.

It would be normal to interpret the first argument as a deductive argument, because the conclusion does indeed strictly follow from the premises. By contrast, it would be normal to interpret the second argument as non-deductive, for if it were interpreted as a deductive, it would obviously be a defective argument, for it is obviously possible that the premises should both be true and the conclusion false. Taken as a non-deductive argument, however, it has some merit, for if the premises are true and one infers the conclusion from the premises, one will tend to arrive at the truth. So, following the principle of charity, which directs us to accept that interpretation of written or spoken language on which it is most reasonable, we would ordinarily interpret the second argument as non-deductive.

There are well-established methods for deciding whether the conclusion of a deductive argument strictly follows from its premises. The study of such methods is the core of the discipline known as logic, which has a long history reaching back to Aristotle and has undergone enormous development since the nineteenth century. Non-deductive arguments, by contrast, are less well understood. In these notes, I concentrate mainly on deductive arguments, and unless otherwise indicated, when I speak of "arguments," I mean deductive arguments.

III: EVALUATING DEDUCTIVE ARGUMENTS: Validity and Soundness

The next question is how to evaluate arguments, how to tell whether the premises of an argument really do support its conclusion. In a deductive argument, the conclusion is put forward as strictly following from the premises. If the conclusion of a deductive argument does strictly follow from the premises, the argument is said to be valid; otherwise it is said to be invalid. As I pointed out above, to say that a conclusion strictly follows from a set of premises means that it is absolutely impossible for all the premises to be true and the conclusion false. This provides the basis for a definition of 'valid' and 'invalid'.

Definition: A deductive argument is valid if and only if it is not possible for all of its premises to be true and its conclusion false.

Definition: A deductive argument is invalid if and only if it is not valid.

The terms 'valid' and 'invalid' apply only to deductive arguments, not to non-deductive ones.

It is very important to see that the validity or invalidity of an argument is independent of the truth or falsehood of the premises and conclusion. Most people, theists as well as atheists, will see that the conclusion in the following argument strictly follows from the premises, and hence that the argument is valid:

If there is no God, then life is meaningless.

There is no God.

Therefore,

Life is meaningless.

But both theists and atheists may deny the conclusion; for atheists and theists alike may reject the first premise, and theists certainly reject the second premise.

When an argument is valid and, in addition, all of its premises are true, then the argument is said to be sound:

Definition: A deductive argument is sound if and only if it is valid and all of its premises are true.

Definition: A deductive argument is unsound if and only if it is not sound.

When deciding whether or not to accept the conclusion of an argument, it is a good idea to begin by asking whether the argument is valid, before inquiring whether the premises are true. If it is not valid, there is no need to ask whether the premises are true, for in that case they give no support to the conclusion even if they are true. Furthermore, there are well-established and useful methods for deciding whether an argument is valid. The study of such methods is the core of the discipline known as logic, which has a long history reaching back to Aristotle and has undergone enormous development since the nineteenth century.

Aristotle (384-22 B.C.) was the first philosopher to make an explicit study of logic. He realized that it is useful to approach the question of whether an argument is valid by considering its logical form. The logical form of an argument is what remains if all the parts of the argument that tie it down to a particular subject matter are replaced with logical variables. Consider the argument:

If human beings are purely physical things, then they do not have the power of free choice.

But human beings are purely physical things.

Therefore,

Human beings do not have the power of free choice.

One way to abstract from the particular subject matter of the argument is to replace the proposition, 'human beings are purely physical things' throughout by one logical variable and the proposition, 'human beings do not have free choice' throughout by a second variable. Let us adopt the convention of using the lower-case letters of the alphabet as logical variables. The result is the logical form,

If p, then q.

p

Therefore

q.

What results from replacing each variable in a logical form consistently throughout by the same expression of the appropriate type (in the present case, with propositions) is called a substitution instance of the form. It is not hard to see that every substitution instance of the above form is a valid argument. If such an argument were not valid, then it would be possible for both premises to be true and the conclusion false. So, if such an argument were not valid, the following arrangement would be possible: p is true; q is false and if p then q is true. But of course that is not possible: That if p then q is true precisely rules out the arrangement in which p is true and q false. So every substitution instance of that form is a valid argument. The form is known as Modus Ponendo Ponens (Modus Ponens, for short). When one discovers that Modus Ponendo Ponens is a valid argument form, one has learned something that is very widely applicable. For an argument form is a valid form if and only if every substitution instance of it is a valid argument. Again, if an argument form has even one invalid substitution instance, then it is an invalid argument form.

There are three valid argument forms that, together with Modus Ponendo Ponens, form a set of four Modi, which provide the basic strategy of many philosophical arguments.

Modus Ponendo Ponens (for short, Modus Ponens)

If p, then q

p

Therefore,

q

Modus Tollendo Tollens (for short, Modus Tollens)

If p, then q

not-q

Therefore,

not-p

Modus Ponendo Tollens (in two forms)

not-(p and q) not-(p and q)

p q

Therefore, Therefore,

not-q not-p

Modus Tollendo Ponens (in two forms)

p or q p or q

not-p not-q

Therefore, Therefore,

q p

The fact that there are two forms of MPT and MTP reflects the fact that conjunctions and disjunctions are convertible: p or q is equivalent to q or p; and p or q is equivalent to q or p. Conditional propositions, in contrast, are not convertible: If p, then q is not equivalent to If q, then p. Hence there is only one form of MPP and MTT.

There are standard names for the parts of conditional propositions, disjunctive propositions, and conjunctive propositions. The first part of a conditional proposition is called the 'antecedent' and the second part the 'consequent'. Thus in the proposition "If God exists, then life has a meaning," the antecedent is 'God exists', and the consequent is 'life has a meaning'. Both parts of a disjunctive proposition are called its 'disjuncts'. Thus in the proposition "The butler is guilty or the maid is guilty," the two disjuncts are 'the butler is guilty' and 'the maid is guilty'. Finally, both parts of a conjunctive proposition are called its 'conjuncts'. Thus, in the proposition "Tom is fat and Tom is happy," the two conjuncts are 'Tom is fat' and 'Tom is happy'.

The exercises following this section will give you practice in identifying arguments of the above forms. This will help you to understand the forms and to remember them. It is also helpful to notice that there are forms of argument that resemble the above valid forms but are nevertheless invalid. As I pointed out above, an argument form is invalid if and only if it has at least one invalid substitution instance.

An invalid argument form that resembles Modus Ponendo Ponens is

If p, then q

q

Therefore,

p

That this is an invalid form can be brought out by constructing a substitution instance that has obviously true premises and an obviously false conclusion. Here is an example:

If Socrates was a woman, then Socrates was a human being.

Socrates was a human being.

Therefore,

Socrates was a woman.

Since arguments of this type resemble instances of Modus Ponendo Ponens, they are often called fallacies. In particular, arguments of this type are said to commit the Fallacy of Affirming the Consequent.

An invalid argument form that resembles Modus Tollendo Tollens is:

If p, then q

Not-p

Therefore,

Not-q.

An argument of this kind, with obviously true premises and an obviously false conclusion is:

If Socrates was a woman, then Socrates was a human being.

Socrates was not a woman.

Therefore,

Socrates was not a human being.

Arguments of this kind are said to commit the Fallacy of Denying the Antecedent.

Proceeding through the list of four Modi, the fallacy corresponding to Modus Ponendo Tollens is called The Conjunctive Fallacy. It is committed when one argues as follows:

Not both (p and q)

Not-p

Therefore,

q

An argument of this type is:

It is not both the case that Tom's house is large and expensive.

Tom's house is not large.

Therefore,

Tom's house is not expensive.

Finally, the fallacy corresponding to Modus Tollendo Ponens is called The Disjunctive Fallacy, and is committed by arguments of this form:

p or q

p

Therefore,

Not-q.

An argument of this type is:

Michael Jordan has been a professional basketball player or a professional baseball player.

Michael Jordan has been a professional basketball player.

Therefore,

Michael Jordan has not been a professional baseball player.

At first glance, some arguments of this kind may seem to be valid, because it is easy to assume that 'p or q' means 'p or q, and not both p and q'. But 'or', in its simplest form, does not have that meaning. Rather, 'p or q', taken simply, just means that at least one of p and q is true. That is the way 'or' is read in logic. If a proposition, in the context of a given argument, clearly means 'p or q, and not both', then that can be made explicit before evaluating the argument. When 'or' is restricted to the simple use, it should be clear that the above argument form is not valid.

EXERCISES

For each of the following arguments, say whether it can reasonably be interpreted as a substitution instance of one of the four modi, and if it is, identify the modus. If the argument is an example of one of the fallacies discussed above, name the fallacy.

1. If there is life after death, then self-sacrifice is sometimes wise. But self-sacrifice is never wise. Therefore, there is no life after death.

2. If there is life after death, then self-sacrifice is sometimes wise. But self-sacrifice is often not wise. Therefore, there is no life after death.

3. If Robin is a woman, then Robin is discriminated against in the workplace. Robin is discriminated against in the workplace. Therefore, Robin is a woman.

4. If Robin is a woman, then Robin is discriminated against in the workplace. Robin is not discriminated against in the workplace. Therefore, Robin is not a woman.

5. If Robin is a father, then Robin is a man. Robin is a man. Therefore Robin is a father.

6. John is either uninterested in the course or unable to do the work. But we know that he is able to do the work. We conclude that he is uninterested in the course.

7. If God is good, evil is just an illusion. But God is good. So evil is just an illusion.

8. Evil is just an illusion if God is good. But God is good. So evil is just an illusion.

9. Susan is not both a fool and a master criminal. She is a fool. So she is not a master criminal.

10. Either God is unable to eliminate disease and poverty from the world or He is unwilling to do so. But He is not unwilling to do so. So He is unable to do so.

IV: SHOWING THAT AN ARGUMENT IS INVALID

There are of course, many valid arguments that are not examples of the four modi explained above. For example, the following argument is clearly valid:

All material objects are things that can be exhaustively explained by physics.

But all human beings are material objects.

Therefore,

All human beings are things that can be exhaustively explained by physics.

This is an example of what are called "standard form categorical syllogisms," a type of argument that was studied by Aristotle. There are many other types of arguments that are studied in formal logic.

Now suppose someone knew all the logic there is to know, tried to show that a given argument is valid, and failed. This would still not show that the argument is not valid, for the person's failure may simply reflect the limitations of their skill in proving arguments valid; he or she may simply have failed to hit on the right strategy. How, then, can one show that a given argument is invalid? Fortunately, it is possible to set up a general, informal way of doing so. This technique capitalizes on the fact that an argument is valid only if it is impossible that its premises all be true and at the same time its conclusion false. Therefore, if it can be shown that it is possible that the premises of an argument are true and its conclusion false, then the argument is invalid. One way to show that is to describe a way in which the world might be such that if it were like that, the premises would all be true and the conclusion false. Consider the following argument:

If Smith is a hemophiliac, then Smith is in danger of contracting AIDS.

Smith is not a hemophiliac.

Therefore,

Smith is not in danger of contracting AIDS.

This argument is invalid, for the world might be like this: There are two groups of people who are in danger of contracting AIDS, hemophiliacs and librarians, and Smith is a non-hemophiliac librarian. If the world were like that, the premises of the argument would be true and the conclusion false. So the argument is invalid.

Three points about this technique must be emphasized. First, the point of the technique is not to develop a proof of the falsehood of the conclusion, but only to show that it does not follow from the given premises. Second, in describing a way in which the world might be, one can let one's imagination run wild, so long as the description does not involve any contradictions. Third, the description that is given must be of a world in which precisely the premises of the argument would be true and precisely the conclusion of the argument false.

EXERCISES

A. Show that each of the following arguments is invalid by giving a counter example.

1. All those who love Canada will vote "yes" in the constitutional referendum. Tom Smith will vote "yes" in the referendum. Therefore, Tom Smith loves Canada.

(Remember that the point is not to show that the conclusion is false, or to show that the premises are false, or to show that premises like those given may be true and yet a conclusion like that given may be false. Rather you must show that it is possible that the given premises should be true and at the same time the given conclusion false.)

2. It is always wrong to kill an innocent human being. Capital punishment is the killing of a human being. Therefore, capital punishment is always wrong.

3. Everyone loves someone. Therefore there is someone who loves everyone.

4. One can make a choice only if one has some reason to do so, and Susan had a good reason for choosing medicine as a career. Therefore she could not have chosen any other career.

B. Which of the following combinations are ruled out by the definition of validity? For example, (1) is the combination, valid argument with all true premises and a true conclusion. (The answer for (3) is given; it is an immediate consequence of the definition of 'valid argument'.)

Valid argument Invalid argument

All premises true 1. 2.

conclusion true

All premises true 3. not possible 4

Conclusion false

At least one premise 5. 6.

false

Conclusion true

At least one premises 7. 8.

false

Conclusion false

V : EVALUATING NON-DEDUCTIVE ARGUMENTS

The study of non-deductive arguments has not arrived at anything like the degree of perfection that has been attained in the treatment of deductive arguments. Indeed, non-deductive arguments come in a considerable variety of types, and it is not clear how much these types have in common. In Section II above, I gave the example:

1. Most students in Philosophy 100Y are under 30 years of age.

2. Susan is a student in Philosophy 100Y.

This tends to show that

3. Susan is under 30 years of age.

In what follows, I will continue to mark the conclusion of non-deductive arguments with 'This tends to show that' instead of 'Therefore', in order to make it clear that the conclusion is not put forward as strictly following from the premises. For want of a better expression, I will refer to arguments of this kind as "most" arguments.

Another type of non-deductive argument can be called inductive arguments. For example,

1. We have so far entered 55% of the houses in Sunshine Acres subdivision and they have all had the kitchen in front.

This tends to show that

2. Every house in Sunshine Acres subdivision has the kitchen in front.

Further types of non-deductive argument are arguments from analogy and arguments to an explanation. Here is an argument from analogy.

1. The last car I owned was a Volkswagen and it was economical to run.

2. The car my neighbor is selling is a Volkswagen.

This tends to show that

3. The car my neighbor is selling is economical to run.

Common sense arguments to an explanation are often employed by detectives in murder mystery novels. For example:

1. Of all those reasonably suspected of the murder, only the butler is left-handed.

2. The wound from which the murdered man died enters his chest from left to right.

This tends to show that

3. The butler committed the murder.

Much more important is the use of such arguments in natural science, in which certain theories are accepted because they provide the best explanation of natural phenomena. A famous example is the acceptance of the Copernican theory that the earth moves relative to the sun, as opposed to the earlier Ptolemaic astronomy, because the Copernican theory provided a simpler, hence better, explanation of the observed movements of the moon, the planets, and the stars.

Non-deductive arguments are not evaluated as valid or invalid, but rather as having some degree of strength. If a non-deductive argument has no strength whatever or a negligible degree of strength, it can be ignored without further consideration. Consider the argument,

1. The last car I owned was red and was economical to run.

2. The car my neighbor is selling is red.

This tends to show that

3. The car my neighbor is selling is economical to run.

This argument has no strength worth considering, because the color of a car is not relevant to whether it is economical to run. So the argument can be ignored without considering the truth or falsehood of the premises.

If a non-deductive argument has a non-negligible degree of strength, then the conclusion of the argument is (to some extent or other) probable relative to the premises. If the conclusion is more probable than its negation, relative to the premises, then the conclusion is said to be reasonable relative to the premises. If a non-deductive argument has a very high degree of strength, then the conclusion is said to be not only reasonable, but certain or nearly certain, relative to the premises.

A natural question at this point is how one can determine the degree of strength of a non-deductive argument. How can one determine to what extent the premises "tend to show" that the conclusion is true? Unfortunately, no general method has been developed for answering this question. Indeed, it is far from clear that a general method can be developed.

However, some suggestions can be made in connection with the various types I have distinguished. The conclusion of every "most" argument is reasonable relative to the premises. Inductive arguments depend for their strength on whether the group of individual cases from which the conclusion is a generalization is representative, in relevant ways, of the group about which the generalization is formed. For example, if the houses in Sunshine Acres referred to in the premise of the inductive argument given above are representative as regards architectural plan of the houses in Sunshine Acres, then the argument would be rather strong. But if Sunshine Acres contained houses built on several different plans, and the houses referred to in the premise represent just one plan, then the argument is weak.

Arguments from analogy are strong only if the given similarity between the two objects is relevant to the inferred similarity. Thus the argument comparing two Volkswagens with regard to economy was strong, while the argument comparing two red cars with regard to economy was weak because the make of a car is relevant to the economy of running it, while the color is irrelevant.

Arguments to an explanation are strong only if the explanation offered in the conclusion is a good explanation and there is not a better explanation available than the one offered in the conclusion. What makes an explanation a good one is, however, controversial.

VI: THE REQUIREMENT OF TOTAL EVIDENCE

Suppose that the conclusion of a non-deductive argument is reasonable, or even certain, relative to the premises. If, in addition, the premises are true, does it follow that it is reasonable for anyone who understands the argument to accept the conclusion? Here caution is in order. Consider the "most" argument about Susan and PHL 100Y. It is given that most students in PHL 100Y are under thirty years of age and that Susan is a student in PHL 100Y. This tends to show that Susan is under thirty years of age. And surely, relative to those premises, it is more probable that Susan is under thirty years of age than that she is not under thirty years of age. So it is reasonable, relative to those premises that Susan is under thirty years of age. Suppose that the premises are true. But suppose, in addition, that you happen to know that Susan is 45. In that case, it is not reasonable for you to accept the statement that Susan is under 30 years of age, even though you understand the argument, see that its conclusion is reasonable relative to the premises, and know that its premises are true.

Again, consider a case in which your friend Jane appears with a man who has the exact physical appearance and mannerisms of her husband. Relative to this premise, it is highly probable that Jane is accompanied by her husband. But you happen to know that Jane's husband has an identical twin who has many of the same mannerisms as her husband. Furthermore, you happen to know that Jane's husband is out of the country. In these circumstances it is not reasonable for you to accept the statement that Jane is accompanied by her husband.

Such examples suggest that the following is not sufficient to make it reasonable for a person to accept the conclusion of a non-deductive argument: (a) the person knows that the conclusion is reasonable relative to the premises of the argument, and (b) the person knows that the premises of the argument are true. A further condition must be met, which is sometimes called the condition of total evidence. This condition is that the conclusion not be less than reasonable relative to the total evidence available to the person.

Introducing the condition of total evidence makes it clear that it may be reasonable for one person to accept the conclusion of a non-deductive argument and not reasonable for another. An example adapted from an article we will study later in the course should make the point clear. Suppose that Mary is piloting a small, one-seater airplane that goes down over the Atlantic Ocean. She has notified the authorities that the plane is going down. She survives and is floating on a piece of wreckage, but has not yet been found, owing to bad weather. Mary speculates that her friends back home have reasoned as follows:

1. Most people who crash land in a small airplane in the Atlantic Ocean die in

the crash.

2. Mary has crash-landed in a small airplane in the Atlantic Ocean.

This tends to show that

3. Mary has died.

It would be reasonable for Mary's friends to accept the conclusion, but not reasonable for Mary to accept it.

The requirement of total evidence marks an important difference between deductive and non-deductive arguments. If a deductive argument is valid, then it is reasonable for anyone who knows the premises to be true to accept the conclusion. Nothing can upset this reasonableness. But in the case of an inductive argument, even if it is strong enough to make its conclusion certain relative to the premises and to make it reasonable for some people to believe the conclusion, it may not be reasonable for everyone to accept its conclusion.

We will encounter several non-deductive arguments in our study of philosophical problems. It is in general difficult to evaluate such arguments, and the difficulty is compounded when they come up in philosophy. We will have to study them individually and evaluate them on their individual merits.

EXERCISES

Say whether each of the following arguments is best interpreted as deductive or as inductive. Give reasons for your answers.

1. All the students in PHL100Y are under 60 years of age. Susan is a student in PHL100Y. I conclude that Susan is under 60 years of age.

2. Almost all of the students in PHL100Y are under 60 years of age. Tom is a student in PHL100Y. I conclude that Tom is under 60 years of age.

3. Toronto is to the north of Buffalo. But if Toronto is to the north of Buffalo, then Toronto is to the north of Philadelphia. I conclude that Toronto is to the north of Philadelphia.

4. The average temperature in Montreal is lower than in Toronto. I conclude that Montreal is to the north of Toronto.

5. Tom has contradicted himself. I conclude that he has made a false statement.

6. Tom is a well-known crook. I conclude that he will not repay the money you loaned him.

7. Mary and Susan are identical twins and I know that Mary did well in school. I conclude that Susan also did well in school.

8. VE Day was December 8, 1945. December 8, 1945 was the feast of the Immaculate Conception in the Catholic calendar. I conclude that VE Day was the feast of the Immaculate Conception in the Catholic Calendar.

9. None of the consequences of the tragic accident on highway 401 that are so far known would justify God in permitting the accident. I conclude that none the of consequences of that accident, known or unknown, would justify God in permitting it.

10. If the world were in the hands of an all-powerful and wholly good God, then bad things would not happen to good people. But bad things do happen to good people. I conclude that the world is not in the hands of an all-powerful and wholly good God.

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[1] The following treatment of complex arguments is largely borrowed from Bernard D. Katz, Logic Notes for PHI 236Y.

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