One of the most important skills that a person can develop and refine is the ability to reason well. Each of us, because we are human beings, has the innate ability to construct arguments and draw inferences. However, many people never develop this ability beyond a very basic level. In your study of philosophy, (indeed, in many of the disciplines you might study in college and in any of the careers you might choose to pursue), you will need to develop and refine your ability to reason. Fortunately, humans have discovered and developed an art which directs the innate human ability to construct arguments and draw inferences so that the person constructs sound arguments and draws valid inferences. This art is logic–the art of critical thinking.
Logic: the art of critical thinking.
Each art operates upon some subject matter for the sake of some end; for example, the art of carpentry operates upon building materials such as wood and steel in order to construct things such as bookcases, cabinets, and houses. Logic also operates upon some subject matter. Without getting too complicated, we can identify three basic elements in the subject matter of logic: concepts, propositions, and arguments.
Concept: an idea which represents a collection of things with a nature or property in common.
Proposition: a statement which can be either true or false.
Argument: an ordered set of propositions, one of which is the conclusion, one or more of which are premises.
To explicate the definition of ‘argument,’ note the following additional definitions.
Premise: a proposition which supports another proposition (i.e., a conclusion).
Conclusion: a proposition which is supported by some other proposition(s) (i.e., premise(s)).
There are two main types of arguments–deductive and inductive.
Deductive Argument: an argument in which the conclusion is meant to necessarily follow from the premises. This kind of reasoning is demonstrative.
Inductive Argument: an argument in which the truth of the premises do not guarantee the truth of the conclusion but render the conclusion probable. This kind of reasoning is non-demonstrative.
With respect to deductive arguments, a good deductive argument is deductively valid.
Valid Argument: a deductive argument in which it is impossible to have true premises and a false conclusion (i.e., the truth of the premises demonstrate that the conclusion is true).
On the other hand, one way in which a deductive argument is defective is when it is invalid.
Invalid Argument: a deductive argument in which it is possible to have true premises and a false conclusion.
Later, we will say more about valid and invalid deductive arguments.
When confronted with a passage or when listening to a discussion, it is often difficult to determine whether or not it is an argument. Compare these two passages from Alexis de Tocqueville's Democracy in America:
[1] In America the principle of the sovereignty of the people is neither barren nor concealed, as it is with some other nations; it is recognized by the customs and proclaimed by the laws; it spreads freely, and arrives without impediment at its most remote consequences.
[2] In America the people appoint the legislative and executive power and furnish the jurors who punish all infractions of the laws. The institutions are democratic, not only in their principle, but in all their consequences. . . . The people are therefore the real directing power.
In order to distinguish arguments from other kinds of discourse,
it is helpful to distinguish words which frequently (although not always)
signify that an inference is being drawn. In English, we have inference
signs which indicate that a proposition is a premise and other inference
signs which indicate that a proposition is a conclusion. The following
list, which is not
exhaustive, indicates examples of each kind of inference sign:
Premise Indicators: Conclusion Indicators:
Since
Therefore
Because
So
For
Consequently
On account of
Thus
Given that
Hence
Assuming that
It follows that
Using these inference signs as guides, we can see that the second passage from Tocqueville is an argument. The conclusion, which is indicated by the word ‘therefore,’ is that the people are the real directing power. The first passage gives a description of America but the propositions are not ordered in such a way that one is supported and the others are supporting. Caveat: This is not to say that every argument has inference signs which indicate premises or conclusions.
In each of the following exercises, circle the inference sign and underline
the conclusion.
1. Defenseless herrings herd together because predators cannot focus
their attention on any one target.
2. The moon always has the same side towards the earth, since it rotates on its axis just once during a revolution around the earth.
3. Music can create a variety of moods in listeners, therefore it can enhance many dramatic productions.
4. Partly on account of their lacking a regular crystalline structure, ceramic materials are poor conductors of electricity.
5. Fish have no iris to control the amount of light that enters their eyes, hence they are found at different depths during a sunny day.
6. When good quality steel is deformed it recovers its original configuration, consequently it is a good example of an elastic material despite its inability to stretch and bend.
7. Dogs respond more to the emotional tones in the human voice than the words that are spoken, for when one says two contradictory sentences in the same emotional tone their behavior is the same.
8. There are twenty amino acids and sixty-four triplet codes; it follows that some of the amino acids are represented by more than one code.
9. Considering the fact that they are both very gentle people, they should have no trouble getting adjusted to one another after they are married.
10. Infrared waves have lower frequencies and therefore greater wavelengths than do red waves.
11. Brains of domesticated rabbits are considerably smaller than those of wild rabbits. This may be attributed to their lack of exercise of the intellect, instincts, senses and voluntary movements and also because they are closely confined.
12. Money is the most important thing in the world since it represents
health, strength, honor, generosity, and beauty as conspicuously as the
lack of it represents illness, weakness, disgrace, meanness, and ugliness.
1. We fancy that this din of religion, literature and philosophy,
which is heard in pulpits, lyceums, and parlors, vibrates through the universe
and is as catholic a sound as the creaking of the earth's axle; but if
a man sleep soundly, he will forget it all between sunset and dawn.
It is the three-inch swing of a pendulum in a cupboard, which the great
pulse of nature vibrates by and
through each instant (Henry David Thoreau).
2. Now each person judges well the things he knows, and of these he is a good judge. And so the person who has been educated in a subject is a good judge of that subject, and the person who has received an all-round education is a good judge in general (Aristotle).
3. I do not think that one should have children. [Because] I observe the acquisition of children has many risks and many griefs, whereas a harvest is rare, and even when it exists, it is thin and poor (Democritus).
4. There is widespread opinion that we do not need to pray, since God knows our needs better than we do. . . . But subjectively, prayer is indeed necessary, i.e., for our sakes. The reason is that to grasp and comprehend what he is thinking, a person needs to clothe his thoughts in words, and if he wants to produce for himself a lively representation of his affection and trust in God, he needs to have recourse to words (Immanuel Kant).
5. You express a great deal of anxiety over our willingness to break laws. This is certainly a legitimate concern. . . . One may well ask: “How can you advocate breaking some laws and obeying others?” The answer lies in the fact that there are two types of laws: just and unjust. . . . One has not only a legal but a moral responsibility to obey just laws. Conversely, one has a moral responsibility to disobey unjust laws (Martin Luther King).
6. The history of the present king of Great Britain is a history of repeated injuries and usurpations all having the direct object the establishment of an absolute tyranny over these states. To prove this, let the facts be submitted to a candid world. He has refused his assent to laws most wholesome and necessary for the common good. He has forbidden his governors to pass laws of immediate and pressing importance, unless suspended in their operation till his assent be obtained; and when so suspended, he has utterly neglected to attend to them. . . . He has dissolved representative houses repeatedly for opposing with manly firmness his invasions of the rights of the people. (Thomas Jefferson--Declaration of Independence).
7. There are three possible parts to a date, at least two of which
must be offered: entertainment, food, and affection. It is customary
to begin a series of dates with a great deal of entertainment, a moderate
amount of food, and the merest suggestion of affection. As the amount
of affection increases, the entertainment can be reduced proportionately.
When the affection is
the entertainment, we no longer call it dating. Under no circumstances
can the food be omitted (Judith Martin).
8. The power of acquiring and disposing private property is necessary
for human life, for three reasons. First, because everyone is more
careful to procure something that concerns oneself alone than something
that is common to all or many others. (For each one, escaping work,
leaves for others any common task, as happens when there are a great many
officials.) Second, because human affairs are handled in a more orderly
fashion when each one goes about one's own
business. There would be complete confusion if everyone tried to do
everything. Third, because this leads to a more peaceful condition
for man, while everyone is content with what he or she has. For we
see that among those possessing something in common, disputes arise more
often (Thomas Aquinas).
9. [One type of feminism is liberal feminism.] Liberal feminists find the roots of women’s oppression in those cultural constraints that hinder women from competing in the public world. . . . Because traditional society holds the false belief that women are, by nature, less mentally and physically capable than men, it largely excludes women from the academy, the forum, the marketplace. Excluded from the land of opportunity, women are unable to demonstrate their capabilities: This is a discriminatory state of affairs. Therefore, liberal feminists insist that society should give women the same educational and occupational opportunities that men have. . . (Rosmarie Tong).
10. If Protagoras is correct when he wrote that ‘the human is
the measure of all things: of things which are, that they are, and of things
which are not, that they are not’ then false judgment is impossible.
But false judgment is possible, for the following reasons: most people
believe that false judgment is possible, and they are either correct or
incorrect. If they are correct, then false judgment exists.
But if they are incorrect, then their judgment that false judgment exists
is itself false and so false judgment exists. Therefore, Protagoras’
position is incorrect. (Adapted from Plato.)
Syllogism: an argument in which the premises, if assumed true, necessitate a conclusion distinct from the premises.
A. Disjunctive Syllogism
The disjunctive syllogism is a valid argument with
two premises and a conclusion. One of the premises is a disjunction, i.e.,
of the form ‘either p or q.’ The other premise is the denial or negation
of one of the disjuncts. The other disjunct follows as the conclusion
of the disjunctive syllogism. The standard form of the disjunctive
syllogism is:
Either p or q
Either I will go to the movies or go to the library.
Not p
I will not go to the movies.
Therefore q
Therefore, I will go to the library.
It does not matter which of the disjuncts we deny; we can deny ‘q’ and thereby infer ‘p’. We can also use a series of disjunctive syllogisms when we are faced with several options and wish to eliminate all but one.
Either p or q or r or s
Not p
Therefore q or r or s
Not r
Therefore q or s
Not s
Therefore q
Frequently, when we use a disjunction we wish to convey that ‘p’ could be true or ‘q’ could be true but not both ‘p’ and ‘q’ could be true. For example, my sister-in-law’s newborn is either a boy or a girl. This is the exclusive sense of the disjunction. However, our use of disjunctions often conveys that ‘p’ or ‘q’ or both ‘p’ and ‘q’ could be true. For instance, in the above example I could go to the library and also (at a later time) go to the movies. This is the inclusive sense of the disjunction. It is often difficult to tell which sense of the disjunction is being used. Hence, we will assume the disjunction is being used in the inclusive sense unless the content of the disjunction clearly indicates otherwise. As a result, the following argument is invalid, i.e., it is possible for all of the premises to be true and the conclusion false.
Either p or q
Either I made a mistake in my checkbook or I am out of money.
p
I made a mistake in my checkbook.
Therefore not q
Therefore I am not out of money.
B. Hypothetical Syllogisms
There are two basic kinds of hypothetical syllogisms: (1) the pure hypothetical syllogism and (2) the mixed hypothetical syllogism. The pure hypothetical syllogism consists of two premises and a conclusion and all three propositions are hypothetical propositions. The basic structure of the pure hypothetical syllogism is:
If p then q
If the university raises tuition then I will take out another student loan.
If q then r
If I take out another student loan then I will be further in debt.
Therefore if p then r
Therefore, if the university raises tuition then I will be further in debt.
Notice that one of the component propositions (i.e., ‘q’) is the consequent in one premise and the antecedent in the other premise. It does not matter which order the premises appear in the argument. However, if the premises cannot be rearranged to fit into this basic pattern, the argument is invalid.
The mixed hypothetical syllogism consists of two
premises and a conclusion but one of the premises and the conclusion can
be non-hypothetical propositions. There are two valid forms of the
mixed hypothetical syllogism. In one form, one premise affirms the
antecedent of the hypothetical premise in order to infer the consequent
of that premise. This argument is called
‘modus ponens.’ The basic form is:
If p then q
If it rained last night then the grass would be wet.
p
It rained last night.
Therefore q
Therefore the grass is wet.
In the second form of a mixed hypothetical syllogism, one premise denies the consequent of the hypothetical premise in order to infer the denial of the antecedent of that premise. This argument is called ‘modus tollens.’ The basic form is:
If p then q
If it rained last night then the grass would be wet.
Not q
The grass is not wet.
Therefore not p
Therefore, it did not rain last night.
There are two kinds of mixed hypothetical arguments which are invalid-- affirming the consequent and denying the antecedent:
If p then q
If p then q
q
Not p
Therefore p
Therefore not q
It is easy to see how arguments of this form could have true premises and a false conclusion. Consider the following examples:
If Bill Clinton was born in Platteville then he was born in the
United States.
Bill Clinton was born in the United States.
Therefore, Bill Clinton was born in Platteville.
If Bill Clinton was born in Platteville then he was born in the
United States.
Bill Clinton was not born in Platteville.
Therefore, Bill Clinton was not born in the United States.
Either p or q
Not p
Therefore q
Examples:
1. Either the car battery is dead or
the starter is defective.
The car battery is
not dead.
Therefore, the starter
is defective.
2. Paul will major in either mechanical
or electrical engineering.
He will not major
in electrical engineering.
Therefore, he will
major in mechanical engineering.
If p then q
If q then r
Therefore if p then r
Example:
If the UW-Platteville
men’s basketball team wins the Conference title then they will get a bid
to the Division III national tournament.
If they get
a bid to the Division III national tournament then they will have a successful
season.
Therefore, if
the UW-Platteville men’s basketball team wins the Conference title then
they will have a successful season.
If p then q
p
Therefore q
Examples:
1. If Sarah is a chemist, then
she is a scientist.
Sarah is a chemist.
Therefore, she is
a scientist.
2. If the temperature drops below 32 F
then water will freeze.
The temperature dropped
below 32 F.
Therefore, water froze.
If p then q
Not q
Therefore Not p
Examples:
1. If Sarah is married then she
is someone’s wife.
She is not someone’s
wife.
Therefore, she is
not married.
2. If rain fell then the grass would
be wet
The grass is
not wet.
Therefore, rain did
not fall.
Notice that in each example the premises are true but the conclusion
is false.
If p then q
q
Therefore p
Example:
If Clinton was born in Wisconsin, he was born in the United States.
Clinton was born in the United States.
Therefore, he was born in Wisconsin.
If p then q
Not p
Therefore not q
Example:
If Clinton was born in Wisconsin, he was born in the United States.
He was not born in Wisconsin.
Therefore, he was not born in the United States.
Either p or q
p
Therefore Not q
Examples:
1. Either the car battery is dead or the
starter is defective.
The car battery is
dead.
Therefore, the starter
is not defective.
2. Paul will major in either mechanical
or electrical engineering.
He will major in electrical
engineering.
Therefore, he will
not major in mechanical engineering.
Notice that in each of these cases the premises could be true but the
conclusion false. Admittedly, sometimes this type of argument seems
valid but this is because of the content of the argument and not because
of the form.
1. If Sunday’s game is at Lambeau Field,
then the Packers will probably win.
Sunday’s game is at
Lambeau Field.
Therefore, the Packers
will probably win.
2. If the battery in my watch were dead,
the watch would stop.
My watch has stopped.
Thus, the battery
is dead.
3. If astrology had any credibility, the
predictions of astrologers would be more accurate than sheer guessing.
But their predictions
are not more accurate than sheer guessing.
Therefore, astrology
has no credibility.
4. There is either a mistake in my checkbook
or I am out of money.
I am not out of money.
Consequently, there
is a mistake in my checkbook.
5. If reindeer do not exist, then neither
does Santa Claus.
But reindeer do exist.
Hence, Santa Claus
also exists.
6. Used car dealers would not stay in
business if they were not aware of defects in their cars.
But they do stay in
business.
As a result, they
are aware of defects in their cars.
7. If Fred does not get a raise then Wilma
will not buy a new stove.
Fred did not get a
raise.
Thus, Wilma will not
buy a new stove.
8. If the show is over then the fat lady
has sung.
The fat lady has sung.
Hence, the show is
over.
9. Betty and Barney will use either ‘Joseph’
or ‘Michael’ in the name of their new son.
They used the name ‘Joseph.’
Thus, they did not use the
name ‘Michael.’
10. If a person can run for President, then that person
was born in the United States.
Brett Favre was born
in the United States.
Therefore, he can
run for President.
11. If a person is not over 35 years old, that person
cannot run for President.
Brett Favre is not
over 35 years old.
So he cannot run for
President.
12. If it rained last night then the grass would be
wet.
The grass is wet.
Hence, it rained last
night.
13. Eleanor went to the library or the movies last
night.
She went to the movies.
Therefore, she did
not go to the library.
14. If today is Saturday, then it is the weekend.
It is not the weekend.
Consequently, today
is not Saturday.
15. If the Packers win the Super Bowl then they will have
won 13 NFL titles.
If the Packers win 13 NFL
titles then they will have eight more NFL titles than the Cowboys.
Therefore, if the Packers
win the Super Bowl, then they will have eight more NFL titles than the
Cowboys.