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PART ONE: PHILOSOPHY AND ARGUMENT
Copyright 2006 © by Elmar J. Kremer
What is philosophy? Perhaps the best answer is that philosophy is thinking hard
about questions like whether there is a God; what beliefs, if any, one ought to hold with
certainty; whether human beings have free will; and whether some actions are wrong in
all circumstances. What thinking hard about such questions amounts to will be illustrated
in the later parts of the course. But, first, a brief excursus into the history of philosophy
may help get us off on the right foot.
For a large part of the history of Western thought, indeed, up to the eighteenth
century, the term 'philosophy' had a much broader meaning than it has today. It included
virtually all careful, disciplined attempts to arrive at the truth about any subject matter.
Thus Isaac Newton's great work on physics, published near the end of the seventeenth
century, was entitled Mathematical Principles of the Philosophy of Nature. Again, what
is today referred to as Psychology broke off from philosophy even more recently; only in
the 1930's did it become the norm for universities to have special departments of
psychology.
The splitting off of specific scientific disciplines from philosophy has been
associated with the development of special techniques in the specific disciplines, for
example, the development of telescopes and microscopes in the sixteenth and seventeenth
centuries, the development of laboratory techniques in the physical sciences, and the use
of applied statistics in psychology. What has remained of the original discipline of
philosophy consists of questions that are very general and basic, like those mentioned at
the beginning of this chapter.
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Now for the study of these very general and basic questions, there are no special
methods. What philosophers do can be described as organizing their thought in such a
way as to help them decide which of the possible answers to the questions are true. The
philosopher’s thinking about the general and basic questions that remain as the subject
matter of philosophy frequently takes the form of arguments. So we are going to begin
our study of philosophical problems by trying to get clear about what an argument is and
how arguments can be 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 statements 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 statements in which one or more statements
(called the premises) are offered in support of another statement (called the
conclusion).
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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
statement is the conclusion and which statements 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 statement as the conclusion are 'so' (as used
in the third example), 'thus,' and 'hence'. Words that sometimes mark statements 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
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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).
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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 statements 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 statement 1, which reappears as statement
6. The basic premises or assumptions of the argument, that is, the premises that are not
supported within the argument, are statements 2, 3, and 5. Statement 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 statement in the
argument is set forth once in a separate numbered line as a grammatically complete
statement (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.
1
The following treatment of complex arguments is largely borrowed from Bernard D. Katz, Logic Notes
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5. There is evil in the world.
Therefore,
6. God does not exist.
Since explicit form requires that each statement appear only once, and that each
conclusion appear after the premises on which it rests, it was necessary to suppress
statement 1. I have retained the numbering used in the above analysis, although normally,
the lines would be numbered sequentially, beginning with '1'.
The above argument contains just one subordinate conclusion, but a complex
argument may contain more than one. For example:
Skepticism is either true or false. If skepticism is true, then no statement is known
to be true. But if no statement is known to be true, then skepticism is not known
to be true. Therefore, if skepticism is true, then skepticism is not known to be
true. On the other hand, if a statement is false, then it cannot be known to be true.
Therefore, if skepticism is false, skepticism is not known to be true. Therefore,
skepticism is not known to be true.
Numbering the constituent statements, we arrive at
1
Skepticism is either true or false. 2If skepticism is true, then no statement is
known to be true. {But} 3if no statement is known to be true, then skepticism is
not known to be true. {Therefore} 4If skepticism is true, then skepticism is not
known to be true. {On the other hand} 5if something is false, then it cannot be
known to be true. {Therefore} 6if skepticism is false, then skepticism is not
known to be true. {Therefore} 7Skepticism is not known to be true.
In order to put this argument into explicit form, we must note that the main
conclusion is 7, and that both 4 and 6 are subordinate conclusions:
for PHI 236Y.
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1. Skepticism is either true or false.
2. If skepticism is true, then no statement is known to be true.
3. If no statement is true, then skepticism is not known to be true.
Therefore,
4. If skepticism is true, then skepticism is not known to be true.
5. If a statement is false, then it cannot be known to be true.
Therefore,
6. If skepticism is false, then skepticism is not known to be true.
Therefore,
7. Skepticism is not known to be true.
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 statements 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.
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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
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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).
B. Each of the following passages presents an argument with one or more
subordinate conclusions. Put each argument into explicit form.
1. A wholly good and omnipotent being would eliminate all evil from the world.
Therefore if there is evil in the world there is no God. But there is evil in the world. So
there is no God.
2. All alcoholic beverages are good for people's health if consumed in
moderation. Wine is an alcoholic beverage. So wine is good for people's health if
consumed in moderation. Therefore, wine should not be heavily taxed, since nothing that
is good for people's health when consumed in moderation should be heavily taxed.
3. The so-called morning star is Venus. Venus is a planet. Hence, the so-called
morning star is a planet. But no planet is a star. Therefore the so-called morning star is
not a star.
4. Consciousness-altering drugs produce a bad state in their users, for they
produce illusions in their users and an illusion is a bad state. But any drug that produces a
bad state in its users ought to be illegal. So consciousness-altering drugs ought to be
illegal.
5. The argument from the progress of science for the conclusion that human
beings do not have free will assumes that scientists are sure to discover physical laws
governing human choices, and that is false. So the argument from the progress of science
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has a false assumption. But no argument that has a false assumption is sound. Therefore,
the argument from the progress of science is not sound.
6. There is no God. For there is a God only if there is a being that is both
omnipotent and good. But since there is much evil in the world, if there is a being that it
omnipotent, it is not good and if there is a being that is good it is not omnipotent.
Therefore there is not a being that is both omnipotent and good.
II: EVALUATING ARGUMENTS: Validity and Elementary Valid Argument
Forms
The next question is how to evaluate arguments, how to tell whether the premises
of an argument really do support its conclusion. In approaching this question, it is useful
to distinguish between deductive arguments and non-deductive arguments. In a deductive
argument, the premises are put forward as providing strictly sufficient, or conclusive,
evidence for the conclusion. In a non-deductive argument, on the other hand, the
premises are offered only as providing good evidence for the conclusion. When we say
that a set of premises give conclusive or strictly sufficient support for a conclusion, we
mean (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:
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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 the second argument 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. But if the second
argument is interpreted as a non-deductive argument, then 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. Later on, I will formulate nondeductive arguments by using the expression 'This tends to show that,' instead of
'Therefore'. But I will deal first with deductive arguments, taking up non-deductive
arguments toward the end of this chapter. Up to that point, I will use the word 'argument'
to refer only to deductive arguments.
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
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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.
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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 statement, 'human beings are purely physical things' throughout by one logical
variable and the statement, '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,
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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
statements) 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.
I will begin to develop ways of telling whether an argument is valid by
introducing several other elementary valid argument forms (EVAFs). First, 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.
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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 statements, 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 statements, disjunctive
statements, and conjunctive statements. The first part of a conditional statement is called
the 'antecedent' and the second part the 'consequent'. Thus in the statement "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 statement are called its 'disjuncts'. Thus in the
statement "The butler is guilty or the maid is guilty," the two disjuncts are 'the butler is
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guilty' and 'the maid is guilty'. Finally, both parts of a conjunctive statement are called its
'conjuncts'. Thus, in the statement "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
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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
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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 statement, 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.
Further development of techniques for showing that an argument is valid require
adding. three very simple argument forms to the list of EVAFs: Conjunction Introduction,
Conjunction Elimination., and Disjunction Introduction. Conjunction Introduction allows
one to infer a conjunction from a premise giving one of the conjuncts and another
premise giving the other conjunct, thus:
p
p
q
q
Therefore,
Therefore,
p and q
q and p
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Conjunction Elimination permits one to proceed from a conjunction to either of
its conjuncts, thus:
p and q
p and q
Therefore,
Therefore,
p
q.
The remaining new EVAF, Disjunction Introduction, allows one proceed from
any statement to any disjunction of which the premise is a disjunct, thus:
p
q
Therefore,
Therefore.
p or q
p or q
Once again, there are two forms of the EVAF because p and q is equivalent to q
and p, and p or q is equivalent to q or p.2
In dealing with arguments, it is also very useful to be aware of several logical
equivalences. Two statements are logically equivalent if and only if they necessarily have
the same truth value (true or false). If two statements are logically equivalent, then either
one can be inferred from the other. One important equivalence is:
The Principle of Double Negation:
This principle is formulated as an equivalence, rather than a simple argument
form: p is equivalent to not-not-p. This means that both of the following argument forms
are valid:
2
p
Not-not-p
Therefore,
Therefore,
Not-not-p
p.
Is there an EVAF called Disjunction Elimination, parallel to Conjunction Elimination? Modus Tollendo
Ponens is sometimes called "Disjunction Elimination," because it permits one to proceed in an inference
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Two other logical equivalences are forms of DeMorgan's Rule:
Not (p and q) is equivalent to Not-p or not-q.
Not (p or q) is equivalent to Not-p and not-q.
Because of these equivalences, all four of the following argument forms are valid:
Not (p and q)
Not-p or not-q
Not (p or q)
Not-p and not-q
Therefore,
Therefore,
Therefore,
Therefore,
Not-p or not-q
Not (p and q)
Not-p and not-q
Not (p or q)
Notice that in each case, one changes from 'and' to 'or' or vice-versa, and changes from an
external 'not' to a pair of internal negations, or vice-versa. One can say that an argument
of any of these forms is valid because it is an instance of DeMorgan's Rule.
It is important to note that an argument is one of the above forms just in case one
can arrive at the argument by replacing the variables consistently throughout by
statements. So the argument,
If God does not exist, life is meaningless.
But God does not exist.
Therefore,
Life is meaningless.
is of the form Modus Ponendo Ponens, for one can arrive at that argument by replacing 'p'
throughout MPP by 'God does not exist' and 'q' throughout by 'life is meaningless'. Of
course, the same argument is also of the form,
If not-p, then q
not-p
Therefore,
q
from a disjunction (p or q) to one of its disjuncts, for example to p, or again to q. Of course one needs the
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This tells us that a single argument can be an instance of more than one logical form. It is
also important to see that a simple logical variable, like 'p' or 'q' can take as its
substituend a statement of any degree of complexity. Hence the argument
If God does not exist, then if John believes in God he has a false belief.
But God does not exist.
Therefore,
If John believes in God he has a false belief.
is also an instance of Modus Ponendo Ponens.
EXERCISES
For each of the following arguments, say whether it can reasonably be interpreted
as a substitution instance of an EVAF, and if it is, identify the EVAF. 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.
negation of the other disjunct as an additional premise.
22
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.
11. Tom did not visit both Rome and Venice. Therefore, either Tom did not visit Rome or
Tom did not visit Venice.
12. Tom did not visit Rome or he did not visit Venice. Therefore, it is not the case that
Tom visited both Rome and Venice.
13. It is not the case that Tom visited Rome or Venice. Therefore Tom did not visit Rome
and he did not visit Venice.
14. If Toronto is to the north of Buffalo, then Toronto is to the north of Miami.. Toronto is to the
north of Buffalo. Therefore, Toronto is to the north of Miami.
15. If Toronto is to the north of Buffalo, then Toronto is to the north of Miami. Toronto is not to
the north of Miami. Therefore, Toronto is not to the north of Buffalo.
16. If Robin is a woman, then Robin is a human being. Robin is a human being. Therefore, Robin
is a woman.
17. If the sun is shining, then it is not pouring rain. It is not pouring rain. Therefore, the sun is
shining.
18. If the sun is shining, then it is not pouring rain. It is pouring rain. Therefore, the sun is not
shining.
19. Edward talked to Tom or Edward talked to Steve. Edward did not talk to Tom. Therefore,
Edward talked to Steve.
20. Pat did not visit both Rome and Venice. Pat visited Rome. Therefore, Pat did not visit Venice.
23
21. Pat did not visit both Rome and Venice. Pat visited Venice. Therefore, Pat did not visit Rome.
22. Pat did not visit both Rome and Venice. Pat did not visit Rome. Therefore, Pat visited Venice.
23. Tom is a student or a teacher. Tom is a teacher. Therefore, Tom is not a student.
24. Tom's idea came from the professor or from the book. Tom's idea did not come from the
book. Therefore, Tom's idea came from the professor.
III: Regimenting Arguments
As I pointed out in Section I above, the same argument can be expressed in a variety of
ways in English. The same point can be made about conditional, disjunctive, conjunctive, and
negative statements. For example, the statement,
1. If God does not exist, then life is meaningless.
Can also be expressed by saying,
1'. Life is meaningless, if God does not exist.
Once again, the same statement can be expressed by saying,
1". God does not exist only if life is meaningless.
(1") is apt to mislead the careless reader because the phrase 'life is meaningless' comes after the
word 'if'. But it should be clear on more careful consideration that (1") is just another way of
saying (1).
Now if one encounters an argument containing a premise or conclusion like (1') or
(1"), it is helpful to translate into the form exemplified by (1). The benefit of this sort of
rephrasing, whether carried out on paper or only in one's mind, is that it permits the
straightforward use of EVAFs like the four Modi in evaluating the argument in question.
Let us say that one regiments an argument when one rephrases the argument in such a
way that the only sentential connectives it contains are 'if . . . then', 'or', 'and', and 'not'.
Two other connectives that occur frequently in arguments and that are replaced
when one regiments arguments are 'if and only if' and 'unless'. Consider the argument
24
Susan has a right to an education if and only if someone has the obligation to
provide for Susan's education.
But someone does have the obligation to provide for Susan's education.
Therefore,
Susan has a right to an education.
In order to appreciate the force of this argument, it is necessary to realize that
p if and only if q
can be rephrased as
if p then q and if q then p.
If the first premise of the above argument is rephrased in this way, then the argument has
been regimented, and it becomes clear that the argument involves a valid instance of
MPP..
Premises and conclusions that include the connective 'unless' can be regimented in
several ways. But the simplest way to regiment them is to replace 'unless' with 'or'.
Consider the argument
The murderer was the butler unless the murderer was the maid.
But the murderer was not the maid.
Therefore,
The murderer was the butler.
Rephrasing the argument by replacing 'unless' with 'or' makes it clear that the argument is
a valid instance of Modus Tollendo Ponens.
The first statement in the above argument can also be rephrased as follows: If the
murderer was not the butler, then the murderer was the maid. This does not pose any
problem because in general p or q is equivalent to if not-p then q. Indeed p or q is also
25
equivalent to if not-q then p, and the later is equivalent to if not-p, then q. The reader will
shortly be in a position to prove all three of these equivalences.
EXERCISES
A. Regiment each of the following arguments and say whether it is a substitution instance
of an EVAF. If it is, say which EVAF it exemplifies.
1. God is good only if evil is just an illusion. But God is good. So evil is just an illusion.
2. There is such a thing as moral responsibility only if there is free will. But there is free
will. Therefore, there is such a thing as moral responsibility.
3. God is unable to eliminate disease and poverty from the world unless He is unwilling
to do so. But He is not unwilling to do so. So He is unable to do so.
4. The butler killed the master unless the cook did. The cook didn't kill the master. So the
butler did.
5. Tom is Susan's husband if and only if Susan is Tom's wife. Therefore, if Tom is
Susan's husband, then Susan is Tom's wife.
B. What follows is a simplified form of an argument given by Nelson Pike in “Divine
Omniscience and Voluntary Action." Put it into explicit form, and for each subordinate
argument, name the EVAF that it exemplifies.
God existed at t1 and Jones mowed his lawn at t2. If God existed at t1 and Jones mowed
his lawn at t2, then God believed at t1 that Jones would mow his lawn at t2. So God
believed at t1 that Jones would mow his lawn at t2, Furthermore, if God believed at t1 that
Jones would mow his lawn at t2, then it was within Jones’s power not to mow his lawn at
t2 only if it was within Jones’s power at t2 to bring it about that God held a false belief at
t1 or to bring it about that at t1 God did not hold the belief He held at t1. Therefore, it was
within Jones's power not to mow his lawn at t2 only if it was within Jones's power at t2 to
bring it about that God held a false belief at t1 or to bring it about that at t1 God did not
26
hold the belief He held at t1. But it was not within Jones’s power at t2 to bring it about
that God held a false belief at t1 and it was not within Jones’s power at t2 to bring it about
that at t1 God did not hold the belief He held at t1. So it was not within Jones’s power at t2
to bring it about that God held a false belief at t1 or to bring it about that at t1 God did not
hold the belief He held at t1. Therefore, it was not within Jones’s power at t2 not to mow
his lawn.
IV: ABBREVIATION AND SUBSTITUTION
In order to deal with arguments in logic, it is often helpful to block out the special
subject matter of the argument by using arbitrary shorthand symbols. Consider the
argument: If God does not exist, then if John believes in God, he has a false belief; but
God does not exist; Therefore, if John believes in God, he has a false belief.. If one
replaces the statements in the argument according to the scheme,
P: God does not exist
Q: If John believes in God then he has a false belief,
the result is the abbreviated argument,
If P then Q
P
Therefore,
Q
It is worth noting that the same argument can be abbreviated in more than one way.
Therefore, no one way of abbreviating an argument is the "right" way. At the same time,
some abbreviations will bring out the logical force of an argument more clearly than
others. For example, if the above argument is abbreviated using the scheme
P: God exists
Q: John believes in God
27
R: John has a false belief,
the result is:
If not-P then (if Q then R)
Not-P
Therefore
If Q then R.
Both of these abbreviations show that the argument is an instance of Modus Ponendo
Ponens. The first abbreviation shows this in a clearer and simpler way, but both
abbreviations are correct, in that they correctly apply the scheme of abbreviation on
which they are based, and both succeed in bringing out the logical form of the argument.
On the other hand, abbreviating the argument on the scheme
P: If God does not exist, then if John believes in God he has a false belief
Q: God does not exist
R: If John believes in God he has a false belief.
yields the abbreviated argument
P
Q
Therefore
R
which does not bring out the logical form of the argument.
To abbreviate an argument, first set up a scheme of abbreviations3 in which
symbolic letters are correlated with statements. By an abbreviation I mean an ordered pair
the first member of which is a symbolic letter and the second a statement. A scheme of
abbreviation is a set of abbreviations, with the restriction that no two pairs can have the
28
same first member. That is to say, a given symbolic letter may be correlated only with
one statement. Three examples of schemes of abbreviation were given just above.
It is important to be clear about the difference between the use of capital letters to
abbreviate arguments and the use of lower-case letters, as in the last section above, to set
forth argument forms. The meaning of a capital letter, in this context, is always relative to
a scheme of abbreviation. The capital letter is just a shorthand way of writing down the
associated statement. The lower case letters, on the other hand, are logical variables.
They can be replaced by any statement whatsoever, so long as each variable is replaced
uniformly throughout the argument form. This means that a substitution instance of a
form, say, of Modus Ponendo Ponens, can have a complex internal structure. Thus the
following argument (abbreviated in accordance with a scheme of abbreviation which is
not given here) is an instance of MPP:
If (P or Q), then (R an S)
P or Q
Therefore,
R and S
It will be convenient to use the upper case letters from P to Z as abbreviations for
statements, and the lower case letters from p to z as logical variables for statements.4
EXERCISES
For each of the examples on p. 25, Part A, that can be taken as a substitution instance of
an EVAF, set up a scheme of abbreviation that can be used to make clear that the
example is a substitution instance of an EVAF. Then abbreviate the argument.
3
I borrow the phrase from Donald Kalish, Richard Montague and Gary Mar, Logic, Techniques of Formal
Reasoning, second edition, New York: Harcourt, Brace, Jovanovich, Inc., 1980, p. 8.
4
There are just eleven letters in the alphabet from 'p' to 'z'. So what if we want a formula with more than
eleven variables in it? We can get around this difficulty by using as variables letters with subscripts, e.g., p 1,
p2 …q1, q2 . . .
29
V: USING EVAFs TO SHOW THAT ARGUMENTS OF OTHER FORMS ARE
VALID
If an argument form is valid, then every substitution instance of it is a valid
argument. So every argument that can be arrived at as an instance of one of the above
EVAFs is a valid argument. This is the first step toward a technique for deciding whether
deductive arguments are valid. But it does not take us very far, for there are many
arguments that are valid but are not substitution instances of EVAFs. The next step is to
explain how to use more than one EVAF, or to use an EVAF more than once, to show
that an argument is valid.
Suppose we want to show that the argument
If Tom is a student or Tom is a professor then Tom can use the library.
Tom is a student.
Therefore,
Tom can use the library.
is valid. It is not a substitution instance of any of our EVAFs. But it is obviously closely
connected to Modus Ponendo Ponens. One can capitalize on that connection by a
procedure that I will illustrate, and explain as I go along. I begin by writing down the
premises in numbered lines, in each case noting that it is an original premise, thus:
1. If Tom is a student or Tom is a professor then Tom can use the library. OP.
2. Tom is a student. OP
I then write down a further statement which follows from one or more of the
original premises by an EVAF, noting the original premise(s) and the EVAF involved,
thus:
3. Tom is a student or Tom is a professor.
from 2 by Disj. Introd.
30
I proceed in this way until the numbered lines include statements from which the
conclusion of the original argument follows by an EVAF. In the present, very simple,
case, I have already arrived at that point. I then write down the conclusion in a further
step, again noting the previous lines and EVAF involved, thus:
4. Tom has the right to a library card.
from 1, 3 by Modus Ponendo Ponens
Here is a slightly more complicated example of the same technique. We want to
show that the following argument is valid:
Tom committed the murder or Susan did it.
If Susan committed the murder, then she was in Toronto yesterday.
Susan was not in Toronto yesterday.
Therefore,
Tom committed the murder.
This argument can be shown to be valid as follows:
1. Tom committed the murder or Susan did it. OP
2. If Susan committed the murder, then she was in Toronto yesterday. OP
3. Susan was not in Toronto yesterday. OP
4. Susan did not commit the murder. from 2,3, by MTT
5. Tom committed the murder.
from 1,4 by MTP
The following argument can be shown to be valid by appealing to the same EVAF
more than once:
If Sudbury is north of Toronto, then it is north of Buffalo.
If Sudbury is north of Buffalo, then it is north of Philadelphia.
Sudbury is north of Toronto
Therefore
Sudbury is north of Philadelphia.
31
Showing that this is so is left as an easy exercise for the reader.
I will call the procedure explained in this section 'Proving that an argument is
valid by appeal to EVAFs', and refer to the series of numbered steps as a 'formal proof of
the validity of an argument'. It enables one to prove the validity of many arguments that
are not instances of EVAFs.
EXERCISES
A. Put each of the following arguments into explicit form and then give a formal proof of
its validity. (You may prefer to set up a scheme of abbreviation and to abbreviate the
argument before giving a formal proof of its validity.)
1. If the weather is bad, Mary will be late for the meeting. If Mary is late for the meeting,
Sue will be upset. The weather is bad. Therefore, Sue will be upset.
2. If Tom committed the murder, then the murderer is left-handed. If the murderer is lefthanded, then the wound enters the victim's chest from left to right. The wound does not
enter the victim's chest from left to right. Therefore, Tom did not commit the murder.
3. Either the professor lost Tom's essay or Tom did not submit it. If the secretary
recorded Tom's essay, then he submitted it. The secretary did record Tom's essay.
Therefore, the professor lost it.
4. Either Tom is in the library or he is in the Athletic Center. If Tom is in the Athletic
Center, then his clothes are in his locker. Tom's clothes are not in his locker. Therefore,
Tom is in the library.
5. Humility is not both a virtue and a vice. If Christianity is to be believed, humility is a
virtue. Christianity is to be believed. Therefore, humility is not a vice.
6. Humility is not both a virtue and a vice. Humility is a virtue or Christianity is a false
religion. Humility is a vice. Therefore, Christianity is a false religion..
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7. If humility is a virtue, the Nietzsche is a misguided philosopher. Humility is a virtue or
a vice. Humility is not a vice. Therefore, Nietzsche is a misguided philosopher.
8. If God exists, then God is omnipotent and God is good. God is not omnipotent or God
is not good. Therefore, God does not exist.
9. If Tom is honest, then he gave me the right change or he made a mistake. Tom did not
give me the right change and he did not make a mistake. Therefore Tom is not honest.
10. Either human beings have free will, or they cannot rightly be blamed for their actions.
If human beings cannot be rightly blamed for their actions, then there is no such thing as
injustice. There is such a thing as injustice. Therefore, human beings have free will.
VI: CONDITIONAL PROOF AND REDUCTIO AD ABSURDUM
Our ability to show that arguments are valid by the use of the EVAFs is greatly
enhanced by the use of two procedures known as Conditional Proof and Reductio ad
Absurdum.
(i) Conditional Proof
Conditional proof is a technique for showing that a conditional conclusion strictly
follows from a set of premises. Suppose we want to show that a conditional statement (if
p then q) strictly follows from some premises. The new technique directs us to take p as
an extra, assumed premise and to show that q strictly follows from the enriched set of
premises. If q does strictly follow from the enriched set of premises, then the original
argument is valid.
Let us apply this technique to the following argument: Either Dr. Jones was killed
by the butler or by the delivery man; if the butler killed Dr. Jones, then he was killed by a
trusted friend; therefore if Dr. Jones was not killed by a trusted friend, then he was killed
by the delivery man. It is relatively easy to show that this argument is valid by using
Conditional Proof. Consider the argument:
33
1. Either Dr. Jones was killed by the butler or he was killed by the delivery man.
2. If the butler killed Dr. Jones, then he was killed by a trusted friend.
3. Dr. Jones was not killed by a trusted friend.
Therefore,
4. Dr. Jones was killed by the delivery man.
The technique of Conditional Proof says that if this argument is valid, then so also
is the original argument. But it is easy to see that the new argument is valid. For from (2)
and (3), it follows by Modus Tollendo Tollens that (X) the butler did not kill Dr. Jones,
and (4) follows from (1) and (X) by Modus Tollendo Ponens.
The above example is easier to deal with if we begin by abbreviating the
argument in the way outlined in Section IV above. First we need a scheme of
abbreviation:
P: Dr. Jones was killed by the butler.
Q: Dr. Jones was killed by the delivery man
R: Dr. Jones was killed by a trusted friend.
On the basis of this scheme, the original argument is abbreviated as follows:
1. P or Q
2. If P, then R.
Therefore,
3. If not-R, then Q
The technique of conditional proof says that this argument is valid if the following is
valid:
1. P or Q
2. If P, then R
3. Not-R
34
Therefore,
4. Q
It is left as an easy exercise for the reader to write out a formal proof of the validity of the
last argument.
EXERCISES
Regiment the following arguments and put them into explicit form. Then show that they
are valid by using Conditional Proof and a series of EVAFs. (You may want to set up a
scheme of abbreviations and abbreviate the argument before using Conditional Proof.)
1. If I know that I exist, then there is at least one thing I know. But if there is at least one
thing that I know, then skepticism is false. Therefore, I know that I exist only if
skepticism is false.
2. Either some statements are objectively true or rational discourse is not possible;
therefore, rational discourse is possible only if some statements are objectively true.
3. Either the mind is not purely physical or Christian religious beliefs are false; but if
psychology is a physical science, then the mind is purely physical; therefore if
psychology is a physical science, Christian religious beliefs are false.
4. If the topic of moral responsibility is important and if moral responsibility
presupposes freedom of will, then the topic of freedom of will is important; but the topic
of moral responsibility is important; therefore, if moral responsibility presupposes
freedom of will, then the topic of freedom of will is important.
5. If Tom is a student, he is a member of the university. Therefore, if Tom is a student
and a library worker, he is a member of the university.
(ii) Reductio ad Absurdum
Conditional Proof can be applied to an argument only if the conclusion of the
argument is a conditional statement. The second new technique resembles Conditional
35
Proof in that it enables us to show that one argument is valid by showing that a different
argument with an enriched set of premises is valid. But it is more general than
Conditional Proof. Indeed, it can in principle be used to show that any argument is valid,
if the argument is indeed valid. Suppose we want to show that from the premises Pr(1)
and Pr(2), it follows that C. The new technique, known as Reductio Ad Absurdum,
directs us to assume not-C as an additional premise and then to show that a contradiction
follows from the enriched set of premises. A contradiction is a statement of the form, p
and not-p. Thus the statement that 2+2 equals 4 and 2+2 does not equal 4 is a
contradiction. Again, the statement that God exists and God does not exist is a
contradiction. In general terms, Reduction ad Absurdum rests on the fact that a
contradiction is necessarily false. But if a contradiction follows from a set of premises,
that means that it impossible for the contradiction to be false and the premises all true.
But the contradiction is false. So at least one of the premises must be false.
An example may help to make the technique clear. Suppose we want to show that
the following argument is valid: If I am going to die on the way home, then trying to
make the trip safe is pointless; but if am not going to die on my trip, then trying to make
the trip safe is pointless; therefore, trying to make the trip safe is pointless.
Our new technique directs us to consider the enriched set of premises
1 If I am going to die on the way home, then trying to make the trip safe is
pointless.
2. If am not going to die on my trip, then trying to make the trip safe is pointless.
3. Trying to make the trip safe is not pointless.
The technique directs us to ask whether this enriched set of premises implies a
contradiction. But it is fairly obvious that it does imply a contradiction. For (1) and (3)
imply, my Modus Tollendo Tollens, that I am not going to die on my trip, and (2) and (3),
36
by the same EVAF, imply that I am going to die on my trip. So (1), (2), and (3), taken
together, imply that I am going to die on my trip and I am not going to die on my trip.
Thus they imply a contradiction.
The example may be clearer if we go back and begin by abbreviating the
argument on the scheme of abbreviation,
P: I am going to die on the way home.
Q: Trying to make the trip safe is pointless.
The argument becomes,
1. If P, then Q
2. If not-P, then Q
Therefore,
3. Q
We can show that this argument is valid by showing that the following argument is valid:
1. If P, then Q
2. If not-P, then Q
3. Not-Q
4. P and not-P
The fact that not-Q, together with the two given premises, implies a contradiction (P and
not-P) shows that the original argument is valid. How does it show that? Well, it is
necessarily false that (I am going to die on my trip and I am not going to die on my trip).
Since this necessary falsehood follows from (1), (2), and (3), we know that not all three
of these premises can be true. So, given that (1) and (2) are true, (3) must be false. Hence,
given that (1) and (2) are true, the negation of (3) must be true. In other words, the
original argument is valid.
37
It is worth noting that the method of Reductio ad Absurdum can be applied in a
variety of ways to a given argument. For example, to prove the above argument valid, it
would also suffice to show that this argument is valid:
If P, then Q
If not-P, then Q
Not-Q
Therefore,
Q and not-Q
EXERCISES
Regiment the following arguments and put them into explicit form. Then show that they
are valid, using the technique of Reductio Ad Absurdum. Once again, you may prefer to
abbreviate the argument before using RAA.
1. If rational discourse is possible, then some statements are objectively true;
therefore, either some statements are objectively true or rational discourse is not possible.
2. If every truth is part of the science of physics, then it is a truth that every truth
is a part of physics. But if it is a truth that every truth is a part of physics, then not every
truth is a part of physics. Therefore, not every truth is a part of physics.
3. Either theists are in error or atheists are in error. But if theists are in error then
there is evil in the world. Furthermore, if atheists are in error then there is evil in the
world. Therefore, there is evil in the world.
4. If God cannot make a stone too heavy for God to lift, then God is not
omnipotent. But if God can make a stone too heavy for God to lift, then God is not
omnipotent. Therefore God is not omnipotent.
5. If God is good and there is evil in the world, then God is not omnipotent. There
is evil in the world. Therefore, either God is not omnipotent or God is not good.
38
6. Either (a) if Mayor Jones is offered a bribe of $1500 tomorrow to give the
contract to Smith Construction, then Jones will accept the bribe or (b) if Mayor Jones is
offered a bribe of $1500 tomorrow to give the contract to Smith Construction, then Jones
will not accept the bribe. Now if (a) then God cannot create a world in which Mayor
Jones is offered that bribe and does not accept it, and if (b) then God cannot create a
world in which Mayor Jones is offered that bribe and accepts it. Further, if God cannot
create a world in which Mayor Jones is offered that bribe and does not accept it, then
there is at least one possible world God cannot create, and if God cannot create a world in
which Mayor Jones is offered that bribe and accepts it, then there is at least one possible
world God cannot create. Therefore, there is at least one possible world God cannot
create [Adapted from Alvin Plantinga, God, Freedom, and Evil].
VI: SHOWING THAT AN ARGUMENT IS INVALID
Suppose Jane tries to show that some argument is valid by the techniques so far
outlined and fails. If she has tried quite hard, she may be tempted to conclude that the
argument is not valid. But that would be an unjustified conclusion. It possible that she
failed because she did not hit on the right strategy in using the techniques so far
developed. Furthermore, these techniques are demonstrably incomplete, in the sense that
they are not sufficient to establish the validity of all valid arguments. So her inability to
show that an argument is valid is in no case an adequate reason for saying that it is
invalid.
Fortunately, it is possible to set up a general, informal technique for showing that
a deductive argument is invalid. 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
39
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 nonhemophiliac 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.)
40
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. Judy has thought long and hard about whether capital punishment is wrong and has
concluded that it is wrong. Therefore, Judy's opinion ought to be taken seriously.
5. 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
All premises true
Invalid argument
1.
2.
3. not possible
4
5.
6.
7.
8.
conclusion true
All premises true
Conclusion false
At least one premise
false
Conclusion true
At least one premises
false
Conclusion false
41
VII. THE LOGIC OF CATEGORICAL STATEMENTS
The argument forms I have considered so far belong to a branch of logic known as
sentential logic. However, there are many arguments that are valid but cannot be shown
to be valid by the techniques of sentential logic. Consider the argument,
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 pretty clearly a valid argument, whatever one may think of the truth of the
premises. To see that it is valid, however, one needs to see it as involving not a pattern of
more or less complex statements made up of statements (as in sentential logic), but rather
a pattern of smaller parts, like the terms 'material object,' 'human beings'. One way to do
this is to use the techniques of the Logic of Categorical Statements.
By 'categorical statement' is meant a statement expressed in English in one of the
following ways:
(A) The word 'All' followed by a plural common noun (or common noun phrase)
followed by the word 'are' followed by a second, distinct, plural common noun (or
common noun phrase), e.g., 'All human beings are mortal beings'.
(E) The word 'No' followed by a plural common noun (or common noun phrase)
followed by the word 'are' followed by a second, distinct, plural common noun (or
common noun phrase), e.g., 'No human beings over the age of 60 are infallible beings'.
(I) The word 'Some' followed by a plural common noun (or common noun phrase)
followed by the word 'are' followed by a second, distinct, plural common noun (or
common noun phrase), e.g., 'Some human beings are married logicians'.
42
(O) The word 'Some' followed by a plural common noun (or common noun
phrase) followed by the words 'are not' followed by a second, distinct, plural common
noun (or common noun phrase), e.g., 'Some human beings are not logicians'.
The two common nouns or noun phrases that appear in a categorical statement are
called terms, the first being the subject term and the second the predicate term. The four
types of categorical statement are said to differ in quantity (whether they are universal or
particular) and quality (whether they are affirmative or negative). The four possible
combinations of quantity and quality are traditionally labeled as follows: If a categorical
statement is universal and affirmative it is said to be an A statement; if it is particular and
affirmative it is an I statement; if it is universal and negative, it is an E statement; and if it
is particular and negative it is an O statement. These traditional letters are derived from
the Latin words 'affirmo' (I affirm) and 'nego' (I deny).
Throughout the traditional study of categorical statements it is assumed that the
subject terms of the statements are "non-empty." For example, when the statement "All
human beings are material things" is treated as a categorical statement, it is assumed that
there are human beings. If examples with "empty" subject terms were permitted, such as
"All frictionless surfaces are things of interest to physicists," matters would become more
complicated. So I will limit attention to categorical statements with non-empty subject
terms, up to the last page of this section, where I explain briefly how things would change
if that assumption were dropped.
The logic of categorical statements is of special interest partly for historical
reasons. Aristotle developed a detailed theory of categorical statements that had an
enormous influence in philosophy, science, and literature up to the nineteenth century,
when there was an explosion of new work and new discoveries in logic. As a result of
these more recent developments, the logic of categorical statements is now treated as a
43
fragment of a larger and more powerful system called predicate, or quantifier, logic.
Nevertheless, because categorical logic was dominant in intellectual work for many
centuries, acquaintance with it continues to be helpful in studying the history of
philosophy and of other fields. It should be added that Aristotle was attracted to
categorical logic because people often find it natural to formulate their arguments as sets
of categorical statements, and many arguments of philosophical importance can be
formulated in this way.
i. Standard Form Categorical Syllogisms
The centerpiece of the Aristotelian logic of categorical statements is the theory of
the standard form categorical syllogism (SFCS). These are arguments of the following
type: They consist of three categorical statements (two premises and one conclusion).
They are composed of exactly three distinct terms, each occurring twice, as follows: The
predicate term of the conclusion, called the Major Term of the syllogism, also occurs in
the first premise, called the Major Premise. The subject term of the conclusion, called the
minor term, also occurs in the second premise, called the minor premise. The remaining
term, which must occur in both premises, is called the middle term. The following is a
standard form categorical syllogism, which happens to be valid:
All acts that can be explained by physics are causally determined acts.
No acts for which a person is morally responsible are causally determined acts.
Therefore,
No acts for which a person is morally responsible are acts that can be explained
by physics.
Note that the major term is 'acts that can be explained by physics,' the minor term is 'acts
for which a person is morally responsible,' and the middle term is 'causally determined
acts'.
44
The following is also a standard form categorical syllogism, which happens to be
invalid:
Some standard form categorical syllogisms are valid arguments.
All arguments on this page are standard form categorical syllogisms.
Therefore,
All arguments on this page are valid arguments.
It will be useful to have a set way of abbreviating SFCSs. We will always use the
letter 'M' for the middle term, 'F' for the major term (the second, or predicate, term in the
conclusion), and 'G' for the minor term (the first, or subject, term in the conclusion). The
first SFCS given just above would then be abbreviated on the scheme
M: causally determined acts
F: acts that can be explained by physics
G: acts for which a person is morally responsible
and the abbreviated argument is
All F are M.
No G are M.
Therefore,
No G are F.
Writing out a similar abbreviation of the second SFCS given just above is left as
an easy exercise for the reader.
I will also continue using lower case letters, in particular 'm', 'f', and 'g', as logical
variables. These variables take terms as their values, not statements. Thus the argument
abbreviated just above is a substitution instance of the form
All f are m.
No g are m.
45
Therefore,
No f are g.
If a SFCS is a substitution instance of a valid form, then I shall say that it is valid by
virtue of its form, or "formally valid." There are precisely 256 different forms of SFCS.
Anticipating results to be established below, 24 of the 256 forms are valid. All the rest are
invalid, that is, have substitution instances that are invalid arguments. If a SFCS is a
substitution of an invalid form, then I shall say that it is "formally invalid."
Various methods have been devised for deciding whether a form of SFCS is valid
or invalid. Here I will set forth two such methods: the method of mood and figure and the
method of rules for valid forms of SFCSs. The first method depends on the fact, first
noticed by Aristotle, that the formal validity of a SFCS depends on just two features of
the argument, which are called its mood and its figure. The mood of a syllogism is
determined by the quantity and quality of the three statements of which it is composed.
The moods are named by a string of three letters, thus: AAA, AEE, AII, etc. The figure of
a syllogism is determined by the order in which the major, minor, and middle terms make
their appearance. The four figures of syllogism can be set forth as follows:
Fig. I
Fig. II
Fig. III
Fig. IV
m
f
m f
f
m
m
f
g m
m
g m
g
m
g
Imagine four lines, each connecting two of the 'm's. These will form something like the
letter 'W,' This may help you to remember how the terms are arranged in each figure.5
Since the validity of a syllogism is determined by its mood and figure, an exhaustive
method for deciding the question of validity of syllogisms can be developed, simply by
5I
have borrowed this way of displaying the four figures from Stephen F. Barker, The Elements of Logic,
fourth edition, New York: McGraw-Hill Book Company, 1985, p. 58.
46
indicating which of the 256 mood-figure combinations are valid. For example, all AAA
syllogisms of the first figure are valid; they are traditionally called Barbara syllogisms.
There was a rhyme learned by medieval students, which gave the mood and figure of the
valid SFCS:
Barbara, Celarent, Darii, Ferioque prioris;
Cesare, Camestres, Festino, Baroco secundae;
Tertia Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison habet;
Quarta insuper addit Bramantip, Camenses, Dimaris, Fesapo, Fresison.
Deciphering the rhyme is left as an easy exercise for the reader, with the
following helps: 'Ferioque' means 'and Ferio,' 'prioris' can be translated by 'of the first
[figure]'; 'secundae' by 'of the second'; 'tertia . . .habet' by 'The third [figure] contains'; and
'Quarta insuper addit' by 'The fourth adds to the foregoing'.
Given our assumption that every term in a categorical statement corresponds to a
non-empty class, any form with a universal conclusion remains valid if the conclusion is
weakened to a particular conclusion of the same quality (an A to an I, or an E to an O).
Such valid forms with "weakened" conclusions are not mentioned in the rhyme. So, to
make it complete, five forms must be added: AAI and EAO in the first figure, EAO and
AEO in the second, and AEO in the fourth. Let us call the rhyme with these five forms
added the enriched rhyme.
You may well ask how it can be shown that all and only these mood-figure
combinations are formally valid. Since there is a finite number of such combinations,
they can be considered very carefully one at a time in order to determine whether are
valid or not. How this careful consideration of each mood and figure combination can be
carried out will be discussed shortly.
47
The second method is derivative of the first. If the mood-figure combinations
contained in the enriched rhyme are inspected, it turns out that they all have four features
in common. The presence of these features is, therefore, a sure sign that a syllogism is
formally valid, and the absence of any one of them that it is formally invalid. Two of the
common features are easy to understand: First, every valid syllogism has at least one
affirmative premise, second, a valid syllogism has a negative premise if and only if its
conclusion is also negative.
The third and fourth features involve a concept which less evident, and which
indeed is the subject of some controversy. These two rules can be put as follows: In every
valid syllogism, the middle term is distributed in at least one premise; and, the major or
minor term in a valid syllogism is distributed in the conclusion only if it is also
distributed in a premise. The controversial concept is that of distribution. The idea,
roughly, is that a term is distributed in a statement if the statement says something about
every object that exemplifies the term. It was thought that the subject term of any
universal statement is distributed in that statement, and that the subject term of any
particular statement is not distributed in that statement. It was also thought that the
predicate term of an affirmative statement is not distributed in the statement and that the
predicate term of a negative statement is. The notion of a statement saying something
about every object that exemplifies a given term has turned out to be difficult and
controversial. Fortunately, one can avoid the controversy, simply by stipulating a
definition of 'distributed in a statement'. Thus, to say that a term is distributed in a
statement means by stipulative definition, that the statement is universal and the term is
the subject, or the statement is negative and the term is the predicate. Given this
somewhat clumsy definition, there are four simple rules that are satisfied by a SFCS if
and only if it is valid:
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1. The middle term must be distributed at least once.
2. No term may be distributed in the conclusion unless it is distributed in a
premise.
3. There must be at least one affirmative premise.
4. The conclusion must be negative if there is a negative premise, and the
conclusion may be negative only if there is a negative premise.
However, I have not yet explained how one can tell, by careful scrutiny, that a
given mood and figure combination is valid or invalid. There are in fact several ways to
accomplish this. Here I will consider just one, which is derived from Aristotle. It takes
the forms Barbara, Celarent, Darii and Ferio (the forms in the first line of the rhyme) as
basic. The validity of these four forms is a direct consequence of what medieval logicians
called the dictum de omni and the dictum de nullo. The dictum de omni is the principle
that whatever is true of all members of a class is true of every sub-class of that class. For
example, if being mortal is true of every animal then it is true of every sub-class of
animals, no matter how specified (of cows and dogs, of male animals and female animals,
of furry animals and feathered animals, etc.). The dictum de nullo is the principle that
whatever is true of no member of a class is not true of any sub-class of that class. For
example if being immortal is true of no animals (if no animals are immortal) then it is not
true of any sub-class of animals.
It is not hard to see how these two principles apply to Barbara, Celarent, Darii and
Ferio. First consider Barbara:
All m are f.
All g are m.
Therefore,
All g are f.
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The dictum de omni says that since being an f is true of all members of the class
of m (according to the first premise), and since g is a sub-class of m (according to the
second premise), it follows that f is true of all g, in other words, that all g are f. Again,
consider Celarent:
No m are f.
All g are m.
Therefore,
No g are f.
The dictum de nullo says that since f is true of no member of the class of m
(according to the first premise), and since all members of g form a sub-class of m
(according to the second premise), it follows that f is true of no member of the class of g,
in other words, that No g are f. Applying the two dicta to Darii and Ferio is left as an easy
exercise for the reader.
Now the weakened forms of Barbara and Celarent (AAI and EAO in the first
figure) are valid because their conclusions follow directly from the conclusions of
Barbara and Celarent themselves. Furthermore, it turns out that all the other forms
mentioned in the enriched rhyme can be shown to be valid by the use of only the forms in
the first line, together with some rather elementary equivalences of forms of categorical
statement that I will explain in sub-section (iii) below. Consider, for example, Cesare,
EAE of the second figure:
No f are m.
All g are m.
Therefore,
No g are f.
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We can get this form simply by reversing the order of the terms in the major premise of
Celarent. This is an operation known as Conversion, and for any E statement, it yields an
equivalent statement. So Cesare is valid, given that Celarent is. The remaining forms in
the rhyme can be reduced to those of the first line by the use of similar equivalences.
Proof of all the forms in the first line of the rhyme by the dictum de omni and the
dictum de nullo and of all the other forms in the enriched rhyme by the use only of those
in the first line, shows that all the forms in the enriched rhyme are valid. In this way, it
can be shown that every syllogistic form in the enriched rhyme is valid. But how can it be
shown that all the other syllogistic forms are invalid? These can all be shown to be
invalid by the method of counterexamples. In each case, one can, without great difficulty,
find a substitution instance which is invalid by the method of counterexamples. Consider
for example AAE of the fourth figure: All f are m; all m are g; therefore, no f are g.; and
the substitution instance: All fathers are human beings; all human beings are mortal
beings; therefore, no fathers are mortal beings. The actual world provides a
counterexample, that is, a possible set-up in which the premises are both t rue and the
conclusion false. Indeed, this form is so obviously invalid that it almost seems a waste of
time to select a substitution instance and then construct a counterexample. It is important,
however, to have a method that can be used to show the invalidity of all of the 231
invalid forms, and this can be done by using the method of counterexamples.
EXERCISES
Translate each of the following arguments into a SFCS, if it is not an SFCS as it stands.
Then decide by the method of mood and figure whether it are valid. If the argument is
invalid, say which of the four rules for SFCSs it violates.
51
1. All cities to the north of Buffalo are to the north of New York City. All cities to the
north of Toronto are to the north of Buffalo. Therefore, all cities to the north of Toronto
are to the north of New York City.
2. All cities to the east of Toronto are to the east of Winnipeg. All cities to the east of
Winnipeg are to the east of Vancouver. Therefore, all cities to the east of Toronto are to
the east of Vancouver.
3. All cities to the east of Toronto are to the east of Winnipeg. All cities to the east of
Montreal are to the east of Winnepeg. Therefore all cities to the east of Toronto are to the
east of Montreal.
4. No satellites of planets are stars. But all moons are satellites of planets. Therefore no
moons are stars.
5. No satellites of planets are stars. But all members of constellations are stars. Therefore,
no members of constellations are satellites of planets.
6. No men are victims of gender-bias. But all men are voters. Therefore, no voters are
victims of gender-bias.
7. All voters are citizens. But some librarians are voters. Therefore, some librarians are
citizens.
8. All voters are citizens. But some librarians are citizens. Therefore, some librarians are
voters.
9. No voters are children. But some students are children. Therefore, some students are
not voters.
10. Only citizens are voters. But some criminals are voters. Therefore, some criminals are
citizens.
11. No moral principle is accepted by people of all cultures. All objective truths are
accepted by people of all cultures. Therefore, no moral principle is an objective truth.
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12. No voters are minors. But some students are voters. Therefore some students are not
minors.
13. No moral principle is accepted by people of all cultures. Some philosophical
principles are accepted by people of all cultures. Therefore, some philosophical principles
are not moral principles.
ii. The Square of Opposition for Categorical Statements
Before the above methods for testing the validity of syllogisms can be applied to
an argument, it must be regimented into a Standard Form Categorical Syllogism. This is a
matter of making sure (1) that the argument consists of three properly formed Categorical
Statements containing three distinct terms, each occurring twice; and (2) that the terms
are arranged properly: that the second term (predicate) of the conclusion, called, once
again, the major term, occurs in the first premise, that the first term (subject) of the
conclusion, called the minor term, is in the second premise, and that the remaining term,
called the middle term, occurs in both premises. Translating the premises and the
conclusion of an argument into properly formed categorical statements may take some
ingenuity. Consider the argument: If anyone is a citizen of Canada, then he or she has the
right to vote; but not everyone living in Canada has the right to vote; therefore, not
everyone living in Canada is a citizen of Canada. To regiment this argument into a SFCS,
we need to translate all three statements into properly formed categorical statements,
which we can do as follows:
All citizens of Canada are people having the right to vote.
Some people living in Canada are not people having the right to vote.
Therefore,
Some people living in Canada are not citizens of Canada.
53
Insight into the meaning of the four types of Categorical Statement, and how to
translated other statements in English into the four types, can be increased by studying
the Square of Opposition, which is a traditional representation of the logical relations
between an A, E, I, and O statement with the same subject and predicate terms. Let the
subject term be 'human beings' and the predicate term be 'logicians'. The logical relations
among the statements of the four types can then be represented as follows:
(A) All human beings are logicians
(E) No human beings are logicians.
(I) Some human beings are logicians
(O) Some human beings are not logicians.
In the above diagram,
(A) and (O) are CONTRADICTORIES.
(E) and (I) are CONTRADICTORIES.
(A) and (E) are CONTRARIES.
(I) and (O) are SUBCONTRARIES
When the (A) and (O) statements, and again the (E) and (I), are called Contradictories,
this means that if one is true it follows that the other is false, and if one is false it follows
that the other is true. When the (A) and (E) statements are said to be Contraries, this
means that if either one is true it follows that the other is false, but from the fact that one
is false is does not follow that the other is true. In other words, they cannot both be true,
but they can both be false. When the (I) and (O) are called Subcontraries, this means that
they cannot both be false, though they can both be true. Finally the (I) is said to be the
'subaltern' of the (A) and the (O) of the (E). This means that the (I) is implied by the (A),
and the (O) by the (E). It should be emphasized that these relations will hold among any
54
four categorical statements of the four types with the same subject term and same
predicate term. They do not depend on the particular subject matter of the statements;
rather they depend on the statements' being of the form All f are g, No f are g, Some fare
g and Some f are not g.
Since a statement and its contradictory are direct logical opposites, 'Some f are
not g' will from now on be taken to be the negation of 'All f are g,' and 'No f are g' to be
the negation of 'Some f are g'. The first of these two relations will, perhaps, be more
obvious if one thinks of the negation of 'All f are g' as being 'Not all f are g'. For 'Not all f
are g' (or again, 'Not every f is a g') is just another way of saying 'Some f are not g'.
Similarly, we might begin by thinking of the negation of 'Some f are g' as being 'Not any
f are g., which is just another way of saying 'No f are g'. Note that 'No f are g' is not the
negation of 'All f are g,' and that 'All f are g' is not the negation of 'No f are g'.
iii. The three traditional operations on categorical statements
Another way to get insight into the meaning of the four types of categorical
statement is to consider the results of three traditional operations that can be performed
on such statements to arrive at other statements. The simplest of these is conversion, in
which one simply reverses the two terms in the statement. The converse of 'No infants are
Prime Ministers' is 'No Prime Ministers are infants'. The second is obversion, in which
the quality of the statement is changed (from affirmative to negative, or vice-versa) and
the predicate term is replaced by its negation. Thus the obverse of 'All women are human
beings' is 'No women are non-human-beings'. (If you prefer: 'No women are things that
are not human beings'.) In contraposition, both terms are replaced with their negations
and then reversed. Thus the contrapositive of 'All women are human beings' is 'All nonhuman-beings are non-women'.
55
What is the logical relation of the converse, obverse or contrapositive of a
statement to the original? The answer can be summarized as follows:
Conversion of an E or an I yields an equivalent statement. Conversion of an A or
an O yields a statement which neither implies nor is implied by the original. (It is
sometimes added that if an A is converted "with limitation" the result is a statement
which does not imply, but is implied by, the original. Conversion with limitation is
conversion plus change of the quantity of the statement from universal to particular. The
converse with limitation of 'All women are human beings' is 'Some human beings are
women'.)
Obversion yields an equivalent statement in every case.
Contraposition of an A or an O statement yields an equivalent statement.
Contraposition of an E or an I yields a logically independent statement.
EXERCISES
1. Explain why the following argument is not an instance of Modus Tollens:
If all voters are wise, then no charletons are elected. Some charletons are elected.
Therefore no voters are wise.
2. Give the converse, obverse and contrapositive of each of the following statements, and
in each case say whether the result is equivalent to the original:
(a) All moral failures are evils.
(b) Some evils are moral failures.
(c) No foreigners are legitimate voters.
(d) Some men are not good cooks.
iv. Categorical Logic without the Assumption that All Subject Terms are Non-Empty
Finally, how would the logic of categorical statements change if the assumption
that the subject terms of all categorical statements are non-empty were dropped?
56
First, in the Square of Opposition, the statements on diagonally opposed corners
would still be contradictories, but all the other logical relationships would be cancelled.
Second, the operation of Conversion with limitation, performed on an A statement, would
no longer yield a result implied by the original. Third, the following syllogistic forms
would turn out to be invalid: In the first figure, the weakened forms of Barbara and
Celarent (AAI and EAO); In the second figure, the weakened forms of Cesare and
Camestres (EAO and AEO); in the third figure, Darapti; and in the fourth figure,
Bramantip and the weakened form of Camenses (AEO). The method of rules would have
to be changed by adding a fifth rule to the effect that a SFCS may not have a particular
conclusion unless it has a particular premise.
VIII: 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, nondeductive 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,
57
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.
58
IX: EVALUATING NON-DEDUCTIVE ARGUMENTS
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
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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.
X: 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
60
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
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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.
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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.