In this chapter we examine the concept of a good argument and,
relatedly, deductive validity and inductive strength. We learn
several valid and invalid logical forms, and we examine the
relationship between validity and truth.
We have defined an argument as a group of statements, one of
which is claimed to follow from the others. In this section we
examine the complex notion of a good argument, Identifying good
arguments is one of the most important objectives of logic and
is, in large part, what the rest of this book is about.
We know that some arguments are good and some are bad, but how
do we tell the difference? As a beginning, notice that whenever a
person offers an argument he or she is implicitly stating that
the premises are good reasons for inferring the conclusion. In
other words, two claims are presupposed in every argument: (1)
The conclusion does follow from the premises, and (2) the
premises are true. If either one of these claims is not the case,
then the argument has failed to establish its conclusion. For
example, if a premise is false, as in the following example,
Example 1
1. Cats are invertebrates.
2. Invertebrates are cuddly.
3. Therefore, cats are cuddly.
then the premise has no actual support to pass to the
conclusion. On the other hand, if the conclusion does not follow,
then whether or not the premises are true, they do not support
the conclusion drawn. For example,
Example 2
1. All physicists are good at mathematics.
2. All engineers are good at mathematics.
3. Therefore, all physicists are engineers.
Examples 1 and 2 are not good arguments for different reasons.
Example 1 has a false premise. Example 2, however, has true
premises but the conclusion does not follow.
An argument can fail to be good for one of two reasons: Either
the conclusion does not follow from the premises, or at least one
premise is false. A good argument, on the other hand, is one in
which the conclusion does follow and the premises are
true. These are the requirements for a good argument.
Now, not only are there two requirements for a good argument,
but these two requirements are logically separable, as we will
see. Requirement (1) has to do with the logical relationship
between the premises and the conclusion-in other words, the
inference; requirement (2) has to do with the actual
truth of the premises. An argument may meet one condition but not
the other. We return to this very important point in Section 2.5;
for the present, focus on requirement (1): whether, given
the premises, the conclusion follows.
What does it mean to say that "the conclusion follows
from the premises?" In logic this means one of two things:
Either the conclusion follows necessarily or the
conclusion follows probably. This gives us two different
notions of strong inferential support. The first we call deductive
validity, and the second we call inductive strength. Each
of these important concepts is developed thoroughly in later
chapters, but let's become familiar with them now. Consider these
two arguments:
Example 3
1. All snakes are poisonous.
2. The coachwhip is a snake.
3. Therefore, the coachwhip is poisonous.
Example 4
1. Most snakes are poisonous.
2. The coachwhip is a snake.
3. Therefore, the coachwhip is poisonous.
These arguments are alike except for their first premises.
Notice the difference it makes to reason from 'Most snakes
are poisonous' rather than 'All snakes are poisonous.'
For instance, in Example 3, if we are given as premises that all
snakes are poisonous and that the coachwhip is a snake, it must
follow that the coachwhip is poisonous. The conclusion
follows necessarily. On the other hand, if it is that most
snakes are poisonous, then the coachwhip, a snake, is likely
to be poisonous, but is not necessarily, The
conclusion in Example 4 follows probably.
Examples 3 and 4 illustrate two different senses of
"follows from" in the definition of a good argument. In
Example 3 the conclusion follows necessarily; in Example 4 the
conclusion follows probably. The inferential support in Example 4
is weaker than in Example 3. Yet notice that the support in
Example 4 is not to be dismissed; after all, if most snakes
are poisonous, then isnt it likely that the coachwhip is, too?
Comparing Examples 3 and 4 with a third, we see an additional
point about inference:
Example 9
1. Few snakes are poisonous.
2. The coachwhip is a snake.
3. The coachwhip is poisonous.
The conclusion in Example 5 certainly does not necessarily
follow; furthermore, it does not follow probably. Example
5 provides less support for the conclusion than does Example 4:
Given that few snakes are poisonous, it is even less likely that
the coachwhip is poisonous. We say that the conclusion in Example
5 does not follow.
Looking at all three examples and focusing just on the
relationships of support, we can see that they compare
differently on the degree of support each offers the
conclusion.
On the one hand, there is Example 3, an inference that could
be no stronger. Given its premises, it is certain that the
coachwhip is poisonous. On the other, there is Example 5 that,
given its premises, offers very little reason for the conclusion.
Example 4 sits in between, providing more reason than Example 5
but slightly less than Example 3.
As another example, consider the argument in Example 2:
1. All physicists are good at mathematics.
2. All engineers are good at mathematics.
3. Therefore, all physicists are engineers.
Here, we have premises that are as strongly worded as one
could want. Yet, if those premises are true, would it follow that
physicists are engineers? Would it follow necessarily?
Would it follow even probably? The answers are no. There is no
more reason to conclude that physicists are engineers from those
premises than that tomatoes are onions because both are
vegetables!
Whenever we have an argument in which, given the premises, the
conclusion follows necessarily, we will say that it is a deductively
valid argument. Whenever we have an argument in which, given
the premises the conclusion follows probably, we will say that it
is inductively strong. And when we have an argument in
which, given the premises, the conclusion is not even probable,
we will say that it is inductively weak-in other words, that
it does not follow. An argument in which the conclusion does
not follow cannot be a good argument.
If we think of inferential support as a descending scale of
strength from the strongest possible to the weakest, we can see
that the examples we've discussed rest at different places on
that scale ranging from logical certainty-as is the case with
deductive validity through degrees of inductive strength to
inductive weakness.
Given the premises ...
Deductively valid ----- the conclusion follows necessarily,
Inductively strong ---- the conclusion follows probably,
Inductively weak ----- the conclusion
does not follow.
We are now in a position to be more precise in our account of
a good argument and to appreciate some of the complexity of that
concept. A good argument, we said, is one in which the premises
are true and the conclusion follows from them. We have seen,
however, that premises vary in the degree of support they provide
for their conclusion. In a good argument the conclusion may follow
from the premises with necessity-that is, deductive
validity-or with probability-that is, inductive strength. Thus,
here is the more complete definition of a good argument:
Conversly, an argument may be bad for either of two
reasons: (1) It is neither deductively valid nor inductively
strong, or (2) at least one premise is false. Therefore, we can
conclude that the inferences of arguments are either deductively
valid, inductively strong, or inductively weak. In Chapter 8 we
will return to examine the concept of inductive strength. In
Section 2.3 we examine the concept of deductive validity more
closely.
The points in this section can be summarized as follows:
1. A good argument is one in which (1) the conclusion
follows from the premises and (2) the premises are true.
2. Since the conclusion may follow from the premises with
deductive validity or with a degree of inductive strength, it
follows that:
3. A good argument is one in which (1) the conclusion
follows from the premises either with deductive validity or with
inductive strength and (20 the premises are true.
4. A conclusion follows with deductive validity if,
given the premises, the conclusion follows necessarily.
5. Otherwise, a conclusion follows with inductive strength
if, given the premises, the conclusion follows probably. The
degree of support premises lend to an inductive conclusion varies
from little or none to very strong.
If the Ugly Sisters are older than Cinderella, it is (in
an iron and awful sense) necessary that Cinderella is
younger than the Ugly Sisters. There is no getting
out of it. If Jack is the sone of a miller, a miller is the
father of Jack. Cold reason decrees it from her awful
throne. (G. K. Chesterson, "The Logic of Elfland")
Another way to say it is this: If you assume the premises are
true, then it must follow that the conclusion is true, the
conclusion cannot be false. That is what is meant by
saying that the conclusion necessarily follows from
the premises. "Cold reason decrees it from her awful
throne." Consider what happens if we attempt to deny this
necessity:
Example 6a
1. Max is taller than Fred.
2. Fred is taller than Steve.
3a. Therefore, Max is taller than Steve.
To see how cold reason logically forces you to accept
the conclusion, try to affirm the premises yet
deny the conclusion, as we see in this modification of Example
6a:
Example 6b
1. Max is taller than Fred.
2. Fred is taller than Steve.
3b. Max is not taller than Steve.
Statement (3b) is the denial of 'Max is taller than Steve'.
When you try to assert all three statements
together, you encounter a contradiction. If premises (1) and (2)
are true, then (3b) must be false. Or if (1) and (3b) are true,
then (2) must be false, and so on. Thus, what the argument in
Example 6b asserts is logically impossible. It is like asserting
that the Ugly Sisters are older than Cinderella but that she is not
younger than they! What we are noticing is a logical fact
about deductively valid arguments: Namely, a valid argument
will yield a logical contradiction if you assert the
premises and deny the conclusion. This is why we say that,
given the premises, the conclusion must follow. It
cannot fail to follow on pain of logical nonsense!
On the other hand, an invalid argument does not logically
compel acceptance of the conclusion. For example,
Example 7
1. If Myles is a Frenchman, then he speaks French.
2. Myles speaks French.
3. Therefore, Myles is a Frenchman.
Given premises (1) and (2), the conclusion does not have to be
true. In this example it is easy to assume the premises are true
yet deny that Myles is a Frenchman. For example, assume it is
true that if he were a Frenchman he'd speak French. Assume also
that he does speak French, having studied it in school,
perhaps. But he happens not to be a Frenchman. No contradiction
results. Why is this? Notice that premise (1) states that if he's
a Frenchman he'll speak French. It does not state that only Frenchmen
speak French. The logical difference between those words makes
the difference in this case between an invalid and a valid
argument.
1. If you study, You will pass the test.
2. You do not study.
3. Therefore, you do not pass the test.
Is it conceivable that if you study, you will pass, and you do
not study, yet you do pass the test? Surely we can
imagine this. Here is a counterexample: You got a copy of the
test questions in advance and passed not by studying but by
copying the answers. So, given the premises, the conclusion does
not necessarily follow; its denial is possible.
Exercise 2.3 Does the Conclusion Follow? From
what you have learned so far, what is your judgment about the
inferential strength of the following arguments? For each
example, write whether the conclusion follows necessarily,
probably, or not at all, given the premises, and,
therefore, whether the inferential support is deductively
valid, inductively strong, or inductively weak. State
your reasons.
1. Every chemistry major must take one year of organic
chemistry. Max is a chemistry major, so he will take a year of
organic chemistry.
2. Most religions include a belief in a god. Buddhism contains
such a belief because it is a religion.
3. You shouldn't buy a foreign car. Patrick and Richard did,
and within a year they had to have the transmission replaced.
4. All life requires water. There is no water on the planet
Venus. Therefore, no life is possible on Venus.
5. Drinking coffee stunts your growth. Max's growth is
stunted. Therefore, Max drinks coffee.
6. Only movie stars live in Hollywood. Robert Redford is a
movie star. Therefore, he lives in Hollywood.
7. All movie stars Eve in Hollywood. Robert Redford is a movie
star. Therefore, he lives in Hollywood.
8. All movie stars live in Hollywood. Robert Redford lives in
Hollywood. Therefore, he is a movie star.
9. There are 367 students in my history class. I reason that
at least two of them have birthdays on the same day of the year.
10. There are 350 students in my philosophy class. I reason
that at least two of them have birthdays on the same day of the
year.
11. The right to life, which every person possesses, does not
give one the right to whatever one needs in order to live. So,
even though a person dying of, say, kidney disease has a right to
life, that person does not thereby have a right to use another's
kidneys.
12. The superior forms of art are those that capture reality
and display it for us. Film is one art form that most
successfully captures reality and displays it; thus, film is a
superior art form.
13. The human mind has no weight, no shape, and no size. The
human brain has weight, shape, and size. Two or more things are
identical only if they have all the same properties. Therefore,
the mind and the brain are not identical.
14. For the past twenty years researchers have been training
chimpanzees to use sign language. just recently, Washoe, the
first chimp to communicate with sign lan guage, began teaching a
ten-month-old chip named Loulis to use signs. She even went so
far as to mold Loulis's hands to form signs. Thus far, Loulis has
learned fifty-five words. This fact, that one chip can teach
another to use sign language, is the strongest evidence to date
that animals other than humans can learn and use language.
15. Most people who major in the humanities go on to teach the
humanities. People who teach the humanities are happy because the
humanities are always exciting and fun and they are about things
of human importance. Work that keeps you focused on being human
is hardly work at all. It comes to be a labor of love. Therefore,
people who major in the humanities love their work.
Let us develop our understanding of deductive validity by a brief examination of validity and logical form. In this section we learn about the concept of logical form and several recognizable forms, some deductively valid, some invalid. To begin with, consider these two important rules we will employ.
Let us begin by illustrating logical form. Consider the
following example:
Example 9
1. All cows are ruminating animals.
2. All ruminating animals are docile.
3. All cows are docile.
Now compare Example 9 with this example:
Example 10
1. All A are B.
2. All B are C.
3. Therefore, all A are C.
Letting the letters 'A' 'B', and 'C' stand for 'cows',
ruminating animals', and 'docile', respectively, we see that
Example 10 is the underlying form of Example 9. We say that
Example 10 is the logical form of Example 9 and (2)
Example 9 is a substitution instance of that form.
Example 9
1. All cows are ruminating animals.
2. All ruminating animals are docile.
3. All cows are docile.
Example 10
1. All Aare B.
2. All B are C
3. All A are C.
Example 10 is a well-recognized example of a valid logical
form called "Barbara" (a mnemonic device by which
medieval logicians could remember the forms of the statements and
their exact order in an argument). Since Example 10 is a valid
logical form and since Example 9 is a substitution instance of
Example 10, Example 9 is a valid argument. To put it differently,
Example 9 is an argument with a valid logical form; therefore, it
is valid by Rule 1 above.
1. All A are B,
2. All Bare C.
3. Therefore, all A are C.
just as Example 9 is an exact substitution of the form
Barbara, so other arguments may also have that valid logical
form. They, too, will be valid under Rule 1, which states that
any argument with a valid logical form is a valid argument.
Let's consider other examples and introduce several other
valid logical forms.
Example 11
1. Max wears glasses or contact lenses.
2. Max does not wear glasses.
3. Max wears contact lenses.
1. A or B.
2. Not A.
3. B.
The disjunctive syllogism is a valid logical form
according to which, if we are given that A or B is the
case, and that A is not the case, then it must follow
that B is the case. Since Example 11 has that form, it is a valid
argument. It can be demonstrated, as we will see in Chapter 7,
that the logical form
1. A or B
2. Not B
3. A
is also an instance of disjunctive syllogism: a disjunction,
the denial of one part and the inference to the other.
Disjunctive syllogism derives its name from the fact that it
consists of three statements (hence, "syllogism"): an
'or' statement called a disjunction, the denial of one part of
the disjunction, and an inference to the other part.
Here is a third valid logical form:
Example 12
1. If it rains, then your car is wet.
2. It rains.
3. Your car is wet.
1. If A then B
2. A
3. B
Briefly consider the form of modus ponens. Given that 'if A then B' and that A obtains, it must follow that B obtains. Notice that Example 12 is of the form modus ponens. Therefore, it is a valid argument.
A fourth valid logical form is modus tollens:
Example 13
1. If it rains, then your car is wet.
2. Your car is not wet.
3. It does not rain.
1. If A then B
2. Not B
3. Not A
We can intuitively convince ourselves of the validity of modus tollens by considering what it asserts: Premise (1) states that if A occurs, then B occurs; (2) states that B does not occur. It must follow then that A does not occur, since if (3) were false, then by (1), (2) would be false. Thus, given (1) and (2), (3) must follow. Since modus tollens is a valid form and since Example 13 is one of its instances, Example 13 is a valid argument.
Let's consider three invalid logical forms. First,
the fallacy of denying the antecedent, illustrated in this
example:
Example 14
1. If you study, then you pass,
2. You do not study.
3. You do not pass.
1. If A then B
2. Not A
3. Not B
As the name suggests, the fallacy of denying the antecedent
involves a premise, here premise (2), that denies the antecedent,
or first part, of a conditional statement, here premise (1).
The conclusion (3) does not necessarily follow, since both (1)
and (2) may be true yet (3) false, as we saw when we examined
Example 8.
A second invalid logical form involving the conditional is the
fallacy of affirming the consequent:
Example 15
1. If you study, then you pass.
2. You pass.
3. You study.
1. If A then B
2. B
3. A
Here the consequent, the second part of the
conditional, is asserted, as in premise (2); the argument
concludes that (3), the antecedent of the conditional must
follow. But this argument form is invalid, since, for example, it
does not follow that you studied from the facts that if you
study, you pass, and you did pass. Again, in our discussion of
Example 8 we saw that a counterexample is conceivable: Namely,
you pass by means other than studying, yet it may still be true
of you that if you study, you will pass.
A third and final invalid logical form is the fallacy of
undistributed middle:
Example 16
1. All ants are insects.
2. All beetles are insects.
3. All beetles are ants.
1 All A are B.
2. All C are B.
3. All C are A .
Notice that the term B in the form (and its counterpart
'insects' in the example) is the term common to the other two
terms, A and C. Yet being common to A and C doesn't entail that A
and C are related as the conclusion (3) asserts. The problem is
that neither premise attributes being an A or being a C to all
Bs. In the example, neither premise attributes being an ant or
being a beetle to all insects. In technical language, the middle
term, B, is not distributed over at least one of the
other terms-hence the name fallacy of undistributed middle.
1. All A are B.
2. All B are C.
3. All A are C.
1. A or B
2. Not A
3. B
1. If A then B
2. A
3. B
1. If A then B
2. Not B
3. Not A
1. If A then B
2. Not A
3. Not B
1. If A then B
2. B
3. A
1. All A are B
2. All C are B.
3. All C are A
We have reviewed four valid forms of arguments and three
invalid forms. We have illustrated how an argument may be
understood as having a form and, thus, how two or more arguments
may be said to have the same form. We have also illustrated these
two important rules about validity: (1) A substitution instance
of a valid logical form is a valid argument, and (2) a
substitution instance of an invalid logical form is an invalid
argument. Perhaps the most important point to emerge from this
discussion is that validity is a matter of the logical form or
structure of an argument, not a matter of its content. This
review also raises important questions. How do we determine the
form of an argument? What counts as form, and what counts as
content? How do we know that valid forms are indeed valid? Are
there ways to prove validity and invalidity? These are questions
answered by a study of the two logical systems we will take up.
We Will see that there are techniques for supplying the
form of an argument and procedures for proving validity and
invalidity.
Exercise 2.4A Validity and Logical Form State
which logical form the argument exhibits and whether it is a
valid or invalid argument.
1. If Webb is promoted, then Walters is transferred. Webb is
promoted; therefore, Walters is transferred.
2. There will be either sunshine or rain. It will not rain;
therefore, there will be sunshine.
3. Every fire official came to the conference and, since all
who came to the conference enjoyed the dinner, all the fire
officials enjoyed the dinner.
4. If she doesn't have a fever, then she doesn't have the flu.
She doesn't have a fever. So she doesn't have the flu.
5. All logicians have good manners, and all physicians have
good manners. Therefore, all logicians are physicians.
6. All Chinook winds have the f6hn effect, and the f6lin
effect can raise air temperatures by as much as 40'F. So all
Chinook winds are capable of raising the temperature as much as
40'F.
7. There is no need for surgery because if there is a tumor
then there is need for surgery, but there is no tumor.
8. If her argument is good, then all her premises are true.
But it's not the case that all her premises are true; thus, her
argument is not good.
9. If Shakespeare's works are histories, then they are not
science fiction. Shakespeare's works are histories; therefore,
they are not science fiction.
10. If Shakespeare's works are histories, then they are not
science fiction. They are science fiction. Therefore, they are
not histories.
11. If there is a tumor, then there is need for surgery. There
is need for surgery; therefore, there is a tumor.
12. Either the emergence of democracy is a cause for hope or
environmental problems will overshadow any promise of a bright
future. Since environmental problems will not overshadow any
promise of a bright future, it follows that the emergence of
democracy is a cause for hope.
13. If it is possible to keep people alive indefinitely, then
we face serious questions about the purpose and quality of such
life. Therefore, since it is not possible to keep people alive
indefinitely, we do not have to face those serious questions.
14. Every pediatrician is an M.D., and so is every podiatrist.
Hence, every pediatrician is a podiatrist.
15. If all elementary and secondary schools across the country
are reexamining their educational objectives, then major
educational reform will be a national goal. Since such
reexamination is the case, so is the national goal of educational
reform.
Exercise 2.4B More Logical Form What logical
form do you see at work in the following passages? If necessary,
write the argument in argument form to reveal the pattern.
1.
Cats like to gaze at the moon.
Animals that gaze at the moon are untrustworthy.
Untrustworthy animals are predators.
Predators are wily and unpredictable.
Wily and unpredictable animals make poor house pets.
Poor house pets are good only as laboratory test animals.
Therefore, cats are good only as laboratory test animals.
2. If any journalists learn about the invasion, then the
newspapers will print the news. And if the newspapers print the
news, then the invasion will not be a secret. If the invasion is
not a secret, then our troops will not have the advantage of
surprise. If we do not have the advantage of surprise, then the
enemy win be prepared. And if the enemy is prepared, then we are
likely to suffer higher casualties. But no journalists learned
about the invasion. Therefore, we are not likely to suffer higher
casualties.
3. To function as a citizen, you need to know a little bit about a lot of different sciences-a little biology, a little geology, a little physics, and so on. But universities (and, by extension, primary and secondary schools) are set up to teach one science at a time. Thus a fundamental mismatch exists between the kinds of knowledge educational institutions are equipped to impart and the kind of knowledge the citizen needs. (Robert M. Hazen and James Trefil, Science Matters: Achieving Scientific Literacy)
Hint: Rewrite the first premise to read:
If people get what they need to function as citizens, then
they would be taught a little bit about a lot of different
sciences.
4. In fact, there can be no such thing as a perfectly rigid
body in nature. If a golf ball were that rigid, and the entire
ball began moving at once, then the shock wave would have to
travel through the ball at an infinite velocity. This is
forbidden by Einstein's special theory of relativity, which
states that no signal or causal influence can travel at a
velocity greater than that of light. Thus it appears that if we
accept the strictures of relativity -which is one of the
best-confirmed theories in physics-then we must conclude that ...
no thing in nature can be perfectly rigid. (Richard Morris, The
Edges of Science)
5. When a bone is damaged, as part of Mr. Fuller's spine was
by the same bullet, it undergoes a series of sequential changes
before stabilizing, much as the skin does when it scars over. But
the process with the skin occurs rapidly, whereas the bone takes
its time, five years generally, from start to finish.
Technically, all I could say is that since Mr. Fuller's
bone-tissue exhibits having gone through this entire process, his
wound is at least five years old. (Archer Mayor, The
Skeleton's Knee)
6. How does one determine when a law is just or unjust? A just
law is a man-made code that squares with the moral law of the law
of God. An unjust law is a code that is out of harmony with the
moral law. To put it in the terms of Saint Thomas Aquinas, an
unjust law is a human law that is not rooted in eternal and
natural law. Any law that uplifts human personality is just.
Any law that degrades human personality is unjust. All
segregation statutes are unjust because segregation distorts the
soul and damages the personality. It gives the segregator a false
sense of superiority, and the segregated a false sense of
inferiority. (Martin Luther King, Jr., "Letter from
Birmingham jail") Hint: Focus on the inference that is
underlined. Write it in argument form and judge which logical
form it most exemplifies.
The strongest inferential relationship is deductive validity.
Recall that a deductively valid inference is one in which, given
the premises, the conclusion must follow. That property of an
argument-being deductively valid-is about the relationship
between premises and conclusion; it is about, to put it roughly,
the reasoning in the argument. What does it have to do with
truth? Furthermore, what does it have to do with whether the
argument is a good one? These are the issues we discuss. Let us
start by studying a number of points about validity, truth, and
good arguments.
Truth-value refers to a property of statements, including
premises and conclusions. They-along with beliefs, opinions, and
judgments-are the kinds of things that are either true or false.
Arguments and inferences, on the other hand, are either
deductively valid or invalid, inductively weak or strong.
Therefore, statements are not valid! Arguments are not true
Example 17
1. Smoking makes you stronger.
2. Being stronger makes you happier.
3. Therefore, smoking makes you happier.
Example 17 has at least one false premise, yet it is
deductively valid. Therefore, deductively validity does not
entail true premises.
Example 17 is deductively valid yet not a good argument
because it does not have all true premises. Do not think that an
argument is good and should be accepted merely because it is
valid. Validity is only part of the concept of a good argument.
Example 18
1. 5 is greater than 3.
2. 4 is greater than 3.
3. Thus, 5 is greater than 4.
Example 19
1. Mozart was a musician.
2. Composers are musicians.
3. Mozart was a composer.
True premises and conclusion do not make an argument valid.
Examples 18 and 19 exhibit the form we call the fallacy of
undistributed middle.
Given that an argument is deductively valid and the premises
are true, then it must follow that the conclusion is true. As we
have seen, to deny the conclusion entails asserting either that
the argument is not deductively valid or that at least one
premise is not true.
Example 20
1. All robins are thrushes,
2. All thrushes are passerines (perching birds).
3. All robins are passerines (perching birds).
Example 21
1. The only justification for a military invasion of another country is self-defense.
2. It is self-defense only if a nation faces imminent danger at its borders.
3. The United States was not threatened by imminent danger at its borders before or during the military invasion of Iraq.
4. Therefore, the United States was not acting in selfdefense in the military invasion of Iraq.
5. Therefore, the United States was not justified in the
military invasion of Iraq.
Example 21 is a deductively valid argument; it exemplifies the
valid logical form modus tollens. Many would argue that
the conclusion of this valid argument, statement (5), is false.
We were justified in invading Iraq, they say. Therefore, since
the argument is valid and (5) is denied, it follows that at least
one premise must be denied. This illustrates a useful strategy in
criticism: If you see that an argument is deductively valid
and yet you deny the conclusion, then at least one premise must
be denied. Locating such a premise and showing that it is
false is of course to show that the argument is not a good one.
There is perhaps no more important point in logic than the
point we have been discussing here from different angles. Validity
has to do with the logical connection between premises and
conclusion, not with the actual truth or falsity of the premises.
So do not confuse what you may know about the actual truth
or falsity of the premises with asking whether a particular
argument is or is not valid. To determine validity, always assume
the premises are true and ask. "Must the conclusion
follow?" Second, validity does not indicate a good argument.
As we have seen, an argument may be deductively valid yet not
good. Similarly, an argument may be good yet not deductively
valid, as is the case with inductively strong arguments.
Exercise 2.5 What, if Anything, Is Wrong with This
Argument? The following exercises test your
understanding of the concepts we have studied in this section,
the relationship between truth and validity. Read the passage
and, using the concepts of truth and validity, discuss the
argument.
Address three questions in particular: (1) To the best of your
knowledge, are the statements true? (b) Is the argument
deductively valid? If not, why not? (c) Is the argument good? If
not, how does the argument fail?
1.
1. All fish are swimmers.
2. All trout are swimmers.
3. All trout are fish.
2.
1. Health-care costs are declining.
2. If health-care costs are declining, then the federal deficit will decrease.
3. Therefore, the deficit will decrease,
3.
1 . If something is dangerous, then people should avoid it.
2. People should avoid hang gliding.
3. Therefore, hang gliding is dangerous.
4.
1. All composers are artists
2. Elton John is an artist.
3. Elton John is a composer.
5.
1. Abortion is the act of killing the fetus.
2. The fetus is a person.
3. Killing a person is morally wrong.
4. Therefore, abortion is morally wrong.
6. The predominant language in the United States is very
difficult to learn. That is because the predominant language is
German and German is difficult to learn.
7. Crows are birds because birds have feathers and crows have
feathers.
8. For this exercise, read Derek Gjertsen's comment (which
follows) on a deductive argument written by the philosopher
Spinoza. Given the concepts we have studied in this chapter, how
would you describe Gjertsen's analysis of Spinoza's argument?
Commenting on a deductive argument written by Spinoza, Derek
Gjertsen (Science and Philosophy: Past and Present) writes:
Thus, from the two axioms:
Axiom 4. The knowledge of an effect depends on and involves
the knowledge of a cause.
Axiom 5. Things which have nothing in common cannot be
understood, the one by means of the other,
Spinoza tries to deduce:
Proposition 3. Things which have nothing in cormmon cannot one
be the cause of the other.
The proof itself is obvious. Assume two things which have
nothing in common. Then, by Axiom 5, we cannot understand one in
terms of the other. Consequently, by Axiom 4, neither can be the
cause of the other. Therefore, we have proved Proposition 3.
But the proof depends upon the soundness of the axioms.
However impeccable the rigour of the logic employed, if
the axioms are at all doubtful, then the system itself will be
suspect. In this Spinoza has fared no better than many another
system-builder. Is, for example, the already quoted Axiom 4
really acceptable? To have some knowledge of an effect do I
really need to have knowledge of the cause? I know Newton had a
breakdown in 1693, and I also know that Vesalius died in
mysterious circumstances in 1564. The causes of these events are
unknown to me and, I fear, anyone else. Many things are known
about past catastrophes and present diseases without
their causes havin yet been identified.
In this chapter we have examined the concept of a good
argument. We saw that good arguments have both true premises and
strong inferential support. Inferential support may be one of
deductive validity or inductive strength. We were introduced to
the ideas of logical form and valid and invalid logical forms, in
particular. Next, we examined the very important distinction
between truth and validity. We saw that the validity of an
argument is independent of the truth-value of the premises.
Since assessing inferential support is a central objective of
logic, in the chapters to follow we will concentrate on
systematic techniques for determining inferential support. We
will study two systems of logic in Chapters 3 through 6,
categorical logic and truthfunctional logic. Each system
provides, among other things, a framework for examining
arguments or deductive validity. Then in Chapter 8 we will
explore methods for assessing inductive support. In Chapter 9 we
will look at failures in good reasoning, called informal
fallacies. Finally, Chapter 10 outlines an overall strategy for
analyzing arguments and applying the techniques we have learned.
I . What is a good argument?
2. According to the text, what is the definition of a
deductively valid argument?
3. According to the text, what is the definition of an
inductively strong argument?
4. Make up examples illustrating all those valid logical forms
we've studied that employ a conditional as a premise.
5. In what ways might an argument fail to be a good one?
6. Why is it that true premises do not make an argument
deductively valid?
7. 'If you were a chemist, you would be a scientist. But you
aren't a scientist.' What can we validly conclude from those
premises?
8. If an argument is deductively valid and yet the conclusion
is false, why must there be at least one false premise?
9. What does it mean to say that the validity of an argument
is logically independent of the truth-value of the premises?
Explain that idea.
10. In your own words, explain the claim of this text that statements
are not valid and arguments are not
true. Why is that?
True or False?
1. If an argument is not valid, then it is not a good argument.
2. If an argument is valid and you believe the conclusion is false, then you must conclude that at least one premise is false.
3. If the conclusion of an argument does not necessarily follow from the premises, then it must be an inductive argument.
4. Deductive validity is only part and not a necessary part of a good argument.
5. A statement may be valid or invalid depending on who judges it.
6. If the premises are false, then the argument is not valid.
7. If you assume the premises are true and get a contradiction when you deny the conclusion, then the argument must be valid.
8. Two arguments can have the same logical form yet one is valid and the other invalid.
9. The antecedent is the first part of a conditional statement.
10. Of the logical forms we studied, only Barbara employs an
'or' statement.
1. In describing the logical power of deductive validity, G. K. Chesterton writes:
If Jack is the son of a miller, a miller is the father of Jack Cold reason decrees it from her awful throne....
Why does Chesterton refer to reason as a cold ruler on an
awful throne? Is reason cold and awful, lacking in heart? It has
that reputation. Do you think it is well deserved? Argue for or
against the claim that logic makes us cold and unfeeling.
2. Imagine two possibilities for yourself. One is that you have great skill at constructing strong arguments but lack knowledge. The other is that you have knowledge but no ability to reason well with it. In the first case, all your arguments are deductively valid only; in the second, all your arguments have true premises but nothing else. Supposing that between knowledge and logical ability, you could have only one, which would you choose for yourself? Why?