A Brief Logic
Primer

**I. Arguments**

Logic
concerns the relationship between premises and conclusions of arguments. To
understand this, we need to understand what arguments are. “Argument” here does
not mean fights, squabbles, or even a disagreement. It has a technical meaning
in logic, namely:

An **argument **is a set of sentences where one or
more (the premises) are intended to provide evidence for, or support for,
another (the conclusion).

Here,
for example, is an argument in this sense:

Jason must be at the movies. He was going to either see the
movie “Spiderman” this evening with Sarah or stay home and study for his
philosophy exam. But I checked his
apartment, and he wasn’t there.

The
person who is offering this argument (Jason’s roommate Carl, say) is intending
to provide a reason to believe that the first sentence is correct, namely, that
Jason is at the movies. That claim—the claim for which evidence is
offered—is the **conclusion**. He
does this by citing two pieces of information as evidence: 1) that Jason is
either home or at the movies; and 2) Jason is not at home. These are the
premises. Carl is suggesting that if you accept these two pieces of information
then you ought to accept the conclusion, because they give you reason to
believe that the conclusion is true. (Notice that he hasn’t presented an
argument for the premises themselves, but is just expecting that his audience
will accept that they are true. If questioned, he might go on to give further
arguments for these premises; but those would be different arguments.)

Here is
the argument set out in what is called **standard
form**:

1.
Jason is either at home studying or at the movies.

2.
Jason is not at home.

3.
Therefore, Jason is at the movies.

An
argument in **standard form** is a
numbered list of the assertions in an argument, with the premises set out first
and the conclusion last. Some things to note:

-The first sentence in the
passage above was the conclusion, not the last sentence. So the first sentence
is listed last when presented in standard form. In general, the conclusion can
appear anywhere in the argument. It might be the first sentence. It might
appear in the middle. It might be the last sentence. It might even not be
explicitly stated; the arguer might be expecting her audience to draw the
obvious conclusion for her. In that case, when we present the argument in
standard form, we would present the conclusion explicitly as the last sentence.

-An
argument can have any number of premises (1 or more).

-An argument has only one
conclusion. If there is an argument presented to give reason to believe that
the premise of an argument is true, that is counted as another argument whose
conclusion happens to be the same sentence as the premise in the original.

**II. Evaluating Arguments: Two
Considerations**

When we
are trying to determine whether to believe the conclusion of an argument, there
are **two**, very different, questions
that we have to ask. The first question concerns the premises themselves. In
the argument above, for example, we have to accept that premises 1 and 2 are
true. If Jason was in fact considering a third option (going to the bar, say)
then premise 1 would be false. If he was studying on the balcony, but Carl
didn’t check the balcony and so assumed that Jason wasn’t home, then premise 2
would be false. If either premise is false then we should obviously not rely on
the argument as giving us a reason to believe the conclusion. So the first **requirement **in order for an argument to
be successful is that the question:

(a)
Are **all** the premises true?

must be
answered “yes”. (Notice the “all”; even if an argument has a thousand premises,
if even one of those is false, and that premise is required in order to give a
reason to believe the conclusion, then the argument is not successful.)

But
that’s not the only question we need to ask. The premises might be true and
still not provide a reason to believe the conclusion. For example, suppose that
I substitute the conclusion in the argument above with the conclusion below:

1.
Jason is either at home studying or at the movies with Sarah.

2.
Jason is not at home.

3.
Therefore, pigs can fly.

Obviously
these two premises give no reason to believe that the conclusion is true, even
if those premises are true. We need in addition that there be the right kind of
**relationship** between the premises
and the conclusion. That relation is that the premises, **if** true, must give reason to believe that the conclusion is true.
(This will be made more precise in a moment.) This relationship is obviously
missing here; how things are with Jason lends no reason to believe anything
about the aeronautical abilities of swine.

So the
second **requirement** in order for an
argument to be successful is that the question:

(b)
Do the premises, **if** true, give
reason to believe that the conclusion is true?

must
also be answered “yes.”

We have
seen that requirement (a) can be satisfied without satisfying requirement (b):
the premises can be true even though they don’t give reason to believe the
conclusion. The reverse is true as well: (b) can be satisfied without
satisfying (a). That is, it can be the case that the premises, if true, would
give reason to believe that the conclusion is true, even though the premises
are not, as a matter of fact, true. Here is an example:

1.
If pigs fly, then I’m a monkey’s uncle.

2.
Pigs fly.

3.
Therefore, I’m a monkey’s uncle.

The
premises are false (and the conclusion is false as well). But suppose the
premises were actually true—suppose that the ability of pigs to fly
somehow means that I had a monkey for a niece or nephew, and suppose that pigs
do fly—then we would have to accept the conclusion. When we ask question
(b), we are engaging in a kind of imagining game: imagine that the world is
such that the premises are true; would that then give you reason to believe
that the conclusion is true as well? We can do this even if we have no idea
whether the premises are true or false. For example:

1. The only extra-terrestrial
life forms in the universe live on planets revolving around the star Alpha
Centauri.

2. Mars does not revolve around
Alpha Centauri.

3. Therefore, there are no
extra-terrestrial life forms on Mars.

I’m
sure you know that 2. is true, since Mars revolves around our sun and our sun
is not Alpha Centauri. But you do not know (I bet!) whether 1. is true. But you
can imagine that they are true. If they were true, then the conclusion would
have to be true as well; the premises provide the right support for the
conclusion. So we can determine whether requirement (b) is satisfied without
paying any attention to the question whether requirement (a) is satisfied. We
can also determine whether requirement (a) is satisfied without paying any
attention to the question whether requirement (b) is satisfied; that is, we can
investigate the question whether the premises are true or false without at all
considering the question whether, supposing they were true, they would give
reason to believe that the conclusion is true as well. Requirement (a) and (b)
are **completely** independent issues:
whether the premises (or the conclusion) are true or not, and whether the
required relationship holds between the premises and the conclusion, are
entirely separate matters.

That is
what makes logic possible. Logic does not concern itself with the question
whether the premises are true; that is, it does not concern itself with
requirement (a). It’s zoologists, presumably, who would concern themselves with
the aeronautical abilities of pigs, astronomers who are concerned with the
question whether Mars revolves around Alpha Centauri, etc. Logic is, instead,** only** concerned with question (b): how,
in general, can we determine whether the premises of arguments support their
conclusions? That is, are there any general procedures that we can apply to
particular arguments to determine whether the premises, if true, would provide
reason to believe that the conclusion is true? *Why* do the premises support the conclusions of arguments and not
others? These questions—questions concerning the **evidential** **relationship **between
premises and conclusions of arguments—are the province of logic.

So
logic does not determine whether you should believe the conclusion of an
argument. That depends on the answer to both questions (a) and (b) being “yes”;
and logic can only help with the second question. But it does take care of the
second question, leaving you with the task of determining the answer to
question (a). Question (b) is the question that concerns **reasoning**, because reasoning is the rational process by which we
draw inferences from information—that is, conclusions from premises. So
logic concerns the processes of reasoning. And reasoning is different from
fact-gathering (requirement (a)). Reasoning is what we do with the facts we
have gathered; it determines what we can glean from those facts. Sherlock
Holmes does two things in his investigations: he gathers facts (by means of
remarkable powers of observation); and he draws conclusions (who is the
murderer, say) by reasoning from those facts. When Holmes does the second, he
is using logic (“deduction”, as he would say, although that’s not quite right;
see below).

**III. First Relationship:
Deductive Arguments**

What
does it mean to say that the premises, if true, would give reason to believe
that the conclusion is true? It turns out that there are two answers to this
question, and so two fundamentally different kinds of argument. The first
answer is: to say that the premises, if true, would give reason to believe that
the conclusion is true is to say that that conclusion *must* be true if the premises are true; that is, it is *impossible *for the premises to be true and the conclusion false. This is
the strongest possible relation between the premises and the conclusion; it
means that the premises *guarantee *that
the conclusion is true, in the sense that it would be a logical **contradiction **to assert the premises
and deny the conclusion.

Both
the first argument concerning Jason’s whereabouts and the argument concerning
extra-terrestrial life are examples of this sort of argument. In both cases, if
the premises are true, the conclusion must be true as well. Here is another
example:

1. All cats are mammals.

2. Some cats are black.

3. Therefore, some mammals are black.

It is
impossible for these premises to be true and the conclusion false. So if the
premises are true, then the conclusion must be true; the truth of the premises
guarantees the truth of the conclusion. Someone who asserted the premises but
denied the conclusion would be contradicting themselves. In this case, the
premises *are* true; and so, since the
premises guarantee the truth of the conclusion, we know that the conclusion is
true as well. But the premises can guarantee the truth of the
conclusion—in the sense that, *if*
the premises are true the conclusion must be true—even if the premises
are false. For example:

1.
All cats are dogs.

2.
Some cats are nuclear physicists.

3.
Therefore, some dogs are nuclear physicists.

The
premises (and the conclusion) of this argument are false. But still, if the
premises of the argument *were*
true—if all cats are in fact dogs, and if some of those cat-dogs are
nuclear physicists—then the conclusion would have to be true as well. The
question we are asking is not whether the premises are true (or whether the
conclusion is true); it is whether the premises, *if* true, would mean that the conclusion would have to be true.

Sometimes
someone presents an argument where they *intend*
that the premises guarantee the truth of the conclusion, but *in fact* they’re
wrong; even if the premises were true, that would not ensure that the
conclusion is true as well. Suppose, for example, someone were to argue as
follows:

1.
All cats are mammals.

2.
Some mammals are black.

3.
Therefore, some cats are black.

This *looks *very similar to the argument
before the last one; but in fact it is very different. The premises are true;
and the conclusion is true as well. Nevertheless, it is *possible* for the premises to be true and the conclusion to be
false. Suppose, for example, that no cats are in fact black, but that some dogs
are. Then premise 1 would be true; it is still true that all cats are mammals.
Premise 2 would be true; since dogs are mammals, and some dogs are black, it
would be true that some mammals are black. But the conclusion would be false;
no cats would be black. Now that is not how things are; but it is *possible*. And that means that the
premises, if true, do not ensure that the conclusion is true; it is *not* the case that the conclusion *must* be true if the premises are
true. We know that cats are black. But we could not know this because of this
argument. If someone believed that some cats are black because of this
argument, they would be wrong to do so, because the premises do not support the
conclusion in the required way.

Someone
who presents this argument has made a mistake. They think that the premises
guarantee the truth of the conclusion. But they’re wrong. They *intend*
to present an argument where the premises guarantee the truth of the
conclusion; that is their aim, the standard to which this argument is held. But
the argument fails to satisfy that aim, or meet that standard. So there are two
features of this argument to notice:

A1.
The premises are supposed to, or intended to, guarantee the truth of the
conclusion.

B1.
The premises do not, in fact, guarantee the truth of the conclusion.

With the previous arguments,
however, the following is the case:

A1.
The premises are supposed to, or intended to, guarantee the truth of the
conclusion.

B2.
The premises *do*, in fact, guarantee the truth of the conclusion.

Notice that these points are
different then the *further* question whether:

C.
The premises are true.

Logic is not concerned with C at
all, although in order to finally assess the argument, we would have to know
whether C is true as well.

Arguments for which A1 is true
are called **deductive** arguments.
Here’s the definition:

A **deductive**
argument is one where it is *intended* that the conclusion must be true
if the premises are true.

To
say that an argument is deductive is to specify what the relationship between
the premises and the conclusion is *supposed* to be; it does *not *say
whether the relationship between the premises *is* that way. So a deductive
argument might be one where the premises *do* guarantee the truth of the
conclusion (B2) *or* it might be one where the premises do *not*
guarantee the truth of the conclusion (B1). What makes an argument deductive is
that it is held to that standard; it is a *further* question whether it *meets* that
standard.

Of
course, someone who presents a deductive argument *hopes* that their argument will meet that standard. If it does, then
it is **valid**:

A deductive is **valid **if
the conclusion must be true if the premises are true; equivalently, if it is
impossible for the premises to be true and the conclusion false.

Such
arguments are called **valid** deductive
arguments (the above is the definition of “valid deductive argument”).

If the
argument does not meet that standard (but is intended to, and so is deductive)
it is called an** invalid** deductive
argument. Here’s the definition:

A deductive argument is **invalid
**if it is possible for the premises to be true and the conclusion false.

Notice
that the only issue here is whether it is *possible*
for the premises to be true and the conclusion false. As we saw, an argument
may have premises and a conclusion that are, as a matter of fact, all true, but
where it is possible for the premises to be true and the conclusion false (the
argument above whose conclusion was “All cats are black”). That argument was
therefore invalid. The question is not whether the premises, or the conclusion
is true. The question is instead whether it is possible for the premises to be
true and the conclusion false. Those are not the same questions.

So:
there are deductive arguments: the premises are intended to guarantee the truth
of the conclusion. Some of those are valid: the premises *do* guarantee the truth of the conclusion. Some are invalid: the
premises do *not* guarantee the truth
of the conclusion. We still haven’t asked whether the premises are true. Here
are the possibilities:

I. Valid deductive argument, premises are all true.

II. Valid deductive argument, at least one premise is false.

III. Invalid deductive argument, premises are all true.

IV. Invalid deductive argument, at least one premise is
false.

Arguments
that fall into category II are not to be relied on, because an argument should
not convince you if *any* of the premises are false (whether only one is false or
all of them are). Arguments that fall into category III should not be relied
on, because an argument should not convince you if the premises do not support
the conclusion—which, for deductive arguments, means that the argument is
invalid—even if the premises are true. Arguments that fall into
category IV. should not be relied on for both of these reasons. But arguments
in category I have true premises *and*
it is impossible for their premises to be true and their conclusions false
(because they are valid). Their conclusions must therefore be true as well.
These are, of course, of course, the sort of arguments we try to present. These
arguments—arguments in category I—are called **sound **arguments; the arguments in II, III, and IV are all called **unsound**. To be sound an argument must
therefore satisfy the following two conditions:

1.
The premises are all true.

2.
The argument is a valid deductive argument.

Arguments
in II violate condition 1; arguments in III violate condition 2; arguments in
III violate both conditions.

**IV: Second Relationship:
Inductive Arguments**

Not all
arguments are deductive. That is, not all arguments are held to the standard
that the premises must *guarantee* that
the conclusion is true. Meteorologists gather all kinds of evidence concerning
air pressure, the movement of the gulf stream, etc. and draw conclusions
concerning what the weather will be tomorrow. Their arguments are quite good;
notwithstanding various jokes at the meteorologists’ expense, meteorology is a
very sophisticated science, and meteorologists’ predictions are often right.
But not always; they do not guarantee accuracy, nor can they. That fact does
not, however, mean that we should ignore all the weather reports as just bad
reasoning.

The sun
has risen every day for millions of years; that gives us very good reason to
believe that it will also rise tomorrow. But it is *possible* that it doesn’t; the sun might explode, the earth
might be blown up by aliens, etc. That history nevertheless gives you a very
good reason to believe that the sun will be rise tomorrow.

If
you hold a brick in your hand four feet from the ground and then let go, it
will almost certainly fall to the ground. That is always what has happened
before, after all, and that is what the laws of gravity say will happen. But it
is *possible*
that it does not. An unexpected gust of wind might push it up; a meteorite
might hit it before it has a chance to fall; the laws of gravity might even be
wrong, or change, or for some reason not apply to this brick. This is all
extremely unlikely, however; the rational thing for you to expect is that the
brick will fall.

Such
arguments—and many, many others—are *not* intended to guarantee the
truth of their conclusions. That is not the standard to which they are to be
held. If it were—if we thought of such arguments as deductive—they
would all be counted as invalid, and so as irrational. But that’s not right;
these are perfectly good arguments. So not all arguments are deductive; not all
arguments are supposed to be valid. There must be another standard.

There
is. These arguments are **inductive**.
What they have in common is that the arguer intends that the premises, if true,
make it **likely**, or **probable** that the conclusion is true;
that is, it is **unlikely**, or **improbable**, that the premises are
true and the conclusion is false.

An **inductive **argument
is one where it is *intended* that the conclusion is probably true if the premises
are true.

The
premises of an argument might make it probable that the conclusion is true even
if the premises are true and the conclusion false. The meteorologist might have
very good reason to believe that it will rain tomorrow (she might say that
there is a “90% chance” of rain tomorrow), but it not in fact rain tomorrow.
The meteorologist was still correct; given the information she had, it was very
likely that it would rain, even though it didn’t. Inductive arguments are not
supposed to guarantee the truth of their conclusions; they are not deductive
arguments.

To
say that an argument is inductive, like saying that an argument is deductive,
is to specify the intended relationship between the premises and the
conclusion. It is not to say that that is, in fact, the relationship between
the premises and the conclusion. Just as some deductive arguments are valid and
some are invalid, some inductive arguments are **strong** and some are **weak**.
Here are the definitions:

An inductive argument is **strong **if the conclusion is probably true if the premises are true;
equivalently, it is unlikely for the premises to be true and the conclusion
false.

An inductive argument is**
weak** if the premises’ being true does not make it likely that the
conclusion is true.

Here is
an example of a strong inductive argument:

1. In a randomly chosen sample
of 1000 WMU students, 80% grew up in southwestern Michigan.

2. Therefore, approximately 80%
of the WMU student population grew up in southwestern Michigan.

This
argument is strong because the sample is **representative**
of the population (because it is a random sample from that population), and it
is a large sample. Here is an example of a weak inductive argument:

1. In a randomly chosen sample of 1000 WMU students, 80% are
under 30 years old.

2. Therefore, approximately 80% of the population of Kalamazoo
is under 30 years old.

This
argument is weak because the sample is *not*
representative of the population; it is **biased**.
WMU students are far more likely to be under 30 than are people in Kalamazoo in
general.

Notice
that the strong argument is still one where the premises *could* be false. It is *possible*, though very unlikely,
that the randomly chosen sample just happened to pick out *all *the
students from southwestern Michigan, and that in fact the rest of the
population is from elsewhere. Again, that is extremely unlikely; but it is
possible. So the premise could be true and the conclusion false. So if we apply
the valid/invalid distinction to it, we would have to say that it is invalid,
and so a bad argument. Nevertheless, it is a good argument. So we should not
apply the valid/invalid distinction to it; that distinction *only*
applies to *deductive* arguments, and this is not a deductive argument; it
is an inductive argument. **Only deductive
arguments are valid or invalid; only inductive arguments are strong or weak**.

So:
there are inductive arguments. some are strong inductive arguments; some are
weak inductive arguments. All these categories concern *only* the relationship between the premises and the
conclusion. We still haven’t said anything about the truth of the premises. As
with deductive arguments, there are four possibilities:

I. Strong inductive argument, premises are all true.

II. Strong inductive argument, at least one premise is
false.

III. Weak inductive argument, premises are all true.

IV. Weak inductive argument, at least one premise is false.

Arguments
that fall into category II are not to be relied on, because an argument should
not convince you if *any* of the premises are false (whether only one is false or
all of them are). Arguments that fall into category III should not be relied
on, because an argument should not convince you if the premises do not support
the conclusion—which, for inductive arguments, means that the argument is
weak—even if the premises are true. Arguments that fall into
category IV. should not be relied on for both of these reasons. But arguments
in category I have true premises *and*
it is unlikely for their premises to be true and their conclusions false
(because they are strong). Their conclusions is therefore probably true. These
are, of course, of course, the sort of arguments we try to present. These
arguments—arguments in category I—are called **cogent **arguments; the arguments in II, III, and IV are all called **uncogent**. To be cogent an argument must
therefore satisfy the following two conditions:

1.
The premises are all true.

2.
The argument is a strong inductive argument.

Arguments
in II violate condition 1; arguments in III violate condition 2; arguments in
IV violate both conditions. Notice that, unlike a sound argument, a cogent
argument might still have a false conclusion. What we do know is that the
conclusions of all cogent arguments are *probably*
true; but, as we saw, a conclusion can be probably true and still be false.

**V. Summary**

Here’s
a map of the distinctions:

Deductive
Arguments

Valid
Deductive Arguments

Valid
Deductive Arguments with all true premises: **sound**

Valid
Deductive Arguments with at least one false premise: **unsound**

Invalid
Deductive Arguments

Invalid
Deductive Arguments with all true premises: **unsound**

Invalid
Deductive Arguments with at least one false premise: **unsound**

Inductive
Arguments

Strong
Inductive Arguments

Strong
Inductive Arguments with all true premises: **cogent**

Strong
Inductive Arguments with at least one false premise: **uncogent**

Weak
Inductive Arguments

Weak
Inductive Arguments with all true premises: **uncogent**

Weak
Inductive Arguments with at least one false premise: **uncogent **

Notice
that this map does not allow inductive arguments to be either valid or invalid,
or inductive arguments to be either strong or weak.

**VI. Exercises**

Part I:
Identify whether the following arguments are deductive or inductive.

1) Because triangle A is congruent with triangle B, and
triangle A is isosceles, it follows that triangle B is isosceles.

2) The plaque on the leaning tower of Pisa says that Galileo
performed experiments there with falling objects. So Galileo must have
performed those experiments there.

3) The rainfall in Seattle has been more than 15 inches
every year for the past thirty years. Therefore, the rainfall next year will
probably be more than 15 inches.

4) Since Agatha is the mother of Raquel and the sister of
Tom, it follows that Tom is the uncle of Raquel.

5) Since Agatha is the mother of Raquel and the sister of
Tom, it follows that Tom is the cousin of Raquel.

6) Paying off terrorists in exchange for money is not a wise
policy, since such action will only lead them to take more hostages in the
future.

7) Either classical culture originated in Greece, or it
originated in Egypt. Classical culture did not originate in Egypt. So it
originated in Greece.

8) Most famous movie stars are millionaires. Leonardo Di
Caprio is a famous movie star. Therefore, he is probably a millionaire.

9) No ads for toothpaste are works of art. Some
Shakespearean sonnets are works of art. Therefore some Shakespearean sonnets
are not ads for toothpaste.

10) If you go to WMU you are a university student. You are a
university student. Therefore, you go to WMU.

Part
II: The following arguments are deductive. Determine whether they are valid or
invalid. Then determine if they are sound or unsound.

1) Since *Moby Dick *was written by Shakespeare, and

2) If Bill Clinton was impeached, then there was a scandal
during his presidency. There was a scandal during his presidency. Therefore, he
was impeached.

3) Since the Spanish American War occurred before the
American Civil War, and the American Civil War occurred after the Korean war,
it follows that the Spanish American War occurred before the Korean War.

4) Some fruits are green. Some fruits are apples. Therefore,
some fruits are green apples.

5) The United States Congress has more members than there
are days in the year. Therefore, at least two members of Congress have the same
birthday.

Part
III: The following arguments are inductive. Determine whether they are strong
or weak. Then determine if they are cogent or uncogent.

1) The ebb and flow of the tides has been occurring every
day for millions of years. But nothing lasts forever. So the tides will
probably die out in the next few years.

2) Every map of the United States shows that Canada is next
to Hawaii. Therefore, Canada is next to Hawaii.

3) Paleontologists have unearthed the fossilized bones of
huge reptiles, which we call dinosaurs. Tests suggest that these creatures
roamed the earth more than 50 million years ago. Therefore dinosaurs probably
did roam the earth then.

4) Constructing the great Pyramids at Giza required lifting
massive stone blocks to great heights. The ancient Egyptians probably used some
sort of antigravity device to do this.

5) Most professors at WMU live in Moscow, Russia. Marc lives
in Moscow, Russia. Therefore Marc is a professor at WMU.