Basic Logical Concepts
20. That all who are happy, are equally happy, is not true. Apeasant and a
philosopher may be equally satisfied, but not equally happy. Happiness
consists in the multiplicity of agreeable consciousness. Apeasant has
not the capacity for having equal happiness with a philosopher.
—Samuel Johnson, in Boswell’s Life of Johnson, 1766
5 Deductive and Inductive Arguments
Every argument makes the claim that its premises provide grounds for the truth
of its conclusion; that claim is the mark of an argument. However, there are two
very different ways in which a conclusion may be supported by its premises, and
thus there are two great classes of arguments: the deductiveand the inductive. Un-
derstanding this distinction is essential in the study of logic.
Adeductive argument makes the claim that its conclusion is supported by its
premises conclusively. An inductive argument, in contrast, does not make such a
claim. Therefore, if we judge that in some passage a claim for conclusiveness is
being made, we treat the argument as deductive; if we judge that such a claim is
not being made, we treat it as inductive. Because every argument either makes
this claim of conclusiveness (explicitly or implicitly) or does not make it, every
argument is either deductive or inductive.
When the claim is made that the premises of an argument (if true) provide in-
controvertible grounds for the truth of its conclusion, that claim will be either
correct or not correct. If it is correct, that argument is valid. If it is not correct (that
is, if the premises when true fail to establish the conclusion irrefutably although
claiming to do so), that argument is invalid.
For logicians the term validityis applicable only to deductive arguments. To
say that a deductive argument is valid is to say that it is not possible for its con-
clusion to be false if its premises are true. Thus we define validityas follows: A
deductive argument is validwhen, if its premises are true, its conclusion mustbe
true. In everyday speech, of course, the term validis used much more loosely.
Validity Although every deductive argument makes the claim that its premises guar-
A characteristic of any antee the truth of its conclusion, not all deductive arguments live up to that
deductive argument
claim. Deductive arguments that fail to do so are invalid.
whose premises, if they
were all true, would Because every deductive argument either succeeds or does not succeed in
provide conclusive achieving its objective, every deductive argument is either valid or invalid. This
grounds for the truth of
point is important: If a deductive argument is not valid, it must be invalid; if it is
its conclusion. Such an
not invalid, it must be valid.
argument is said to be
valid. Validity is a formal The central task of deductive logic is to discriminate valid arguments from
characteristic; it applies invalid ones. Over centuries, logicians have devised powerful techniques to do
only to arguments, as
this—but the traditional techniques for determining validity differ from those
distinguished from truth,
used by most modern logicians. The former, collectively known as classical logic,
which applies to
propositions. is rooted in the analytical works of Aristotle. Logicians of the two schools differ
(cid:21)(cid:23)
Basic Logical Concepts
in their methods and in their interpretations of some arguments, but ancients
and moderns agree that the fundamental task of deductive logic is to develop the
tools that enable us to distinguish arguments that are valid from those that are
not.
In contrast, the central task of inductive arguments is to ascertain the facts by
which conduct may be guided directly, or on which other arguments may be
built. Empirical investigations are undertaken—as in medicine, or social science,
or astronomy—leading, when inductive techniques are applied appropriately, to
factual conclusions, most often concerning cause-and-effect relationships of
some importance.
An illustration of the inductive process will be helpful at this point to con-
trast induction with deduction. Medical investigators, using inductive methods,
are eager to learn the causes of disease, or the causes of the transmission of infec-
tious diseases. Sexually transmitted diseases (STDs), such as acquired immune
deficiency syndrome (AIDS), are of special concern because of their great seri-
ousness and worldwide spread. Can we learn inductively how to reduce the
spread of STDs? Yes, we can.
In 2006 the National Institutes of Health announced that large-scale studies
of the spread of STDs in Kenya and Uganda (African countries in which the risk
of HIV infection, commonly resulting in AIDS, is very high) was sharply lower
among circumcised men than among those who were not circumcised. Circumci-
sion is not a “magic bullet” for the treatment of disease, of course. However, we
did learn, by examining the experience of very many voluntary subjects (3,000 in
Uganda, 5,000 in Kenya, divided into circumcised and uncircumcised groups)
that a man’s risk of contracting HIV from heterosexual sex is reduced by halfas a
result of circumcision. The risk to women is also reduced by about 30 percent.*
These are discoveries (using the inductive method called concomitant varia-
tion) of very great importance. The causal connection between the absence of cir-
cumcision and the spread of HIV is not known with certainty, the way the
conclusion of a deductive argument is known, but it is now known with a very
high degree of probability.
Inductive arguments make weaker claims than those made by deductive ar-
guments. Because their conclusions are never certain, the terms validity and
invalidity do not apply to inductive arguments. We can evaluate inductive argu-
ments, of course; appraising such arguments is a central task of scientists in every
sphere. The higher the level of probability conferred on its conclusion by the
premises of an inductive argument, the greater is the merit of that argument. We
can say that inductive arguments may be “better” or “worse,” “weaker” or
“stronger,” and so on. The argument constituted by the circumcision study is very
strong, the probability of its conclusion very high. Even when the premises are all
true, however, and provide strong support for the conclusion, that conclusion is
not established with certainty.
*So great is the advantage of circumcision shown by these studies that they were stopped, on 13 December
2006, by the Data Safety and Monitoring Board of the National Institutes of Health, to be fair to all partic-
ipants by announcing the probable risks of the two patterns of conduct.
(cid:21)(cid:24)
Basic Logical Concepts
Because an inductive argument can yield no more than some degree of prob-
ability for its conclusion, it is always possible that additional information will
strengthen or weaken it. Newly discovered facts may cause us to change our es-
timate of the probabilities, and thus may lead us to judge the argument to be bet-
ter (or worse) than we had previously thought. In the world of inductive
argument—even when the conclusion is judged to be very highly probable—all
the evidence is never in. New discoveries may eventually disconfirm what was
earlier believed, and therefore we never assert that the conclusion of an inductive
argument is absolutely certain.
Deductive arguments, on the other hand, cannot become better or worse.
They either succeed or they do not succeed in exhibiting a compelling relation
between premises and conclusion. If a deductive argument is valid, no addition-
al premises can possibly add to the strength of that argument. For example, if all
humans are mortal and Socrates is human, we may conclude without reservation
that Socrates is mortal—and that conclusion will follow from those premises no matter
what else may be true in the world, and no matter what other information may be discov-
ered or added. If we come to learn that Socrates is ugly, or that immortality is a bur-
den, or that cows give milk, none of those findings nor any other findings can
have any bearing on the validity of the original argument. The conclusion that
follows with certainty from the premises of a deductive argument follows from
any enlarged set of premises with the same certainty, regardless of the nature of
the premises added. If an argument is valid, nothing in the world can make it
more valid; if a conclusion is validly inferred from some set of premises, nothing
can be added to that set to make that conclusion follow more strictly, or more
validly.
This is not true of inductive arguments, however, for which the relationship
claimed between premises and conclusion is much less strict and very different
in kind. Consider the following inductive argument:
Most corporation lawyers are conservatives.
Miriam Graf is a corporation lawyer.
Therefore Miriam Graf is probably a conservative.
This is a fairly good inductive argument; its first premise is true, and if its
second premise also is true, its conclusion is more likely to be true than false. But
in this case (in contrast to the argument about Socrates’ mortality), new premises
added to the original pair might weaken or (depending on the content of those
new premises) strengthen the original argument. Suppose we also learn that
Miriam Graf is an officer of the American Civil Liberties Union (ACLU).
and suppose we add the (true) premise that
Most officers of the ACLU are not conservatives.
Now the conclusion (that Miriam Graf is a conservative) no longer seems
very probable; the original inductive argument has been greatly weakened by
(cid:21)(cid:25)
Basic Logical Concepts
the presence of this additional information about Miriam Graf. Indeed, if the
final premise were to be transformed into the universal proposition
No officers of the ACLU are conservatives.
the opposite of the original conclusion would then follow deductively—and
validly—from the full set of premises affirmed.
On the other hand, suppose we enlarge the original set of premises by
adding the following additional premise:
Miriam Graf has long been an officer of the National Rifle Association (NRA).
The original conclusion (that she is a conservative) would be supported by
this enlarged set of premises with even greater likelihood than it was by the orig-
inal set.
Inductive arguments do not always acknowledge explicitly that their conclu-
sions are supported only with some degree of probability. On the other hand, the
mere presence of the word “probability” in an argument gives no assurance that
the argument is inductive. There are some strictly deductive arguments about
probabilities themselves, in which the probability of a certain combination of
events is deduced from the probabilities of other events. For example, if the
probability of three successive heads in three tosses of a coin is 1> , one may infer
8
deductively that the probability of getting at least one tail in three tosses of a coin
is 7> .
8
In sum, the distinction between induction and deduction rests on the nature
of the claims made by the two types of arguments about the relations between
their premises and their conclusions. Thus we characterize the two types of argu-
ments as follows: Adeductive argument is one whose conclusion is claimed to Deductive argument
follow from its premises with absolute necessity, this necessity not being a matter One of the two major
types of argument
of degree and not depending in any way on whatever else may be the case. In
traditionally
sharp contrast, an inductive argumentis one whose conclusion is claimed to fol- distinguished, the other
low from its premises only with probability, this probability being a matter of de- being the inductive
argument. A deductive
gree and dependent on what else may be the case.
argument claims to
provide conclusive
grounds for its
6 Validity and Truth
conclusion. If it does
provide such grounds, it
Adeductive argument is valid when it succeeds in linking, with logical necessity, is valid; if it does not, it
the conclusion to its premises. Its validity refers to the relation between its is invalid.
propositions—between the set of propositions that serve as the premises and the
Inductive argument
one proposition that serves as the conclusion of that argument. If the conclusion One of the two major
follows with logical necessity from the premises, we say that the argument is types of argument
traditionally
valid. Therefore validity can never apply to any single proposition by itself, because
distinguished, the other
the needed relation cannot possibly be found within any one proposition.
being the deductive
Truth and falsehood, on the other hand, are attributes of individual proposi- argument. An inductive
tions. Asingle statement that serves as a premise in an argument may be true; the argument claims that its
premises give only some
statement that serves as its conclusion may be false. This conclusion might have
degree of probability, but
been validly inferred, but to say that any conclusion (or any single premise) is it-
not certainty, to its
self valid or invalid makes no sense. conclusion.
(cid:21)(cid:26)
Basic Logical Concepts
Truthis the attribute of those propositions that assert what really is the case.
When I assert that Lake Superior is the largest of the five Great Lakes, I assert
what really is the case, what is true. If I had claimed that Lake Michigan is the
largest of the Great Lakes my assertion would not be in accord with the real
world; therefore it would be false. This contrast between validity and truth is im-
portant: Truth and falsity are attributes of individual propositions or statements; validi-
ty and invalidity are attributes of arguments.
Just as the concept of validity cannot apply to single propositions, the con-
cept of truth cannot apply to arguments. Of the several propositions in an argu-
ment, some (or all) may be true and some (or all) may be false. However, the
argument as a whole is neither true nor false. Propositions, which are statements
about the world, may be true or false; deductive arguments, which consist of
inferences from one set of propositions to other propositions, may be valid or
invalid.
The relations betweentrue (or false) propositions and valid (or invalid) argu-
ments are critical and complicated. Those relations lie at the heart of deductive
logic. It devoted largely to the examination of those complex relations, but a
preliminary discussion of the relation between validity and truth is in order here.
We begin by emphasizing that an argument may be valid even if one or more
of its premises is not true. Every argument makes a claim about the relation be-
tween its premises and the conclusion drawn from them; that relation may hold
even if the premises turn out to be false or the truth of the premises is in dispute.
This point was made dramatically by Abraham Lincoln in 1858 in one of his de-
bates with Stephen Douglas. Lincoln was attacking the Dred Scottdecision of the
Supreme Court, which had held that slaves who had escaped into Northern
states must be returned to their owners in the South. Lincoln said:
I think it follows [from the Dred Scottdecision], and I submit to the consideration of
men capable of arguing, whether as I state it, in syllogistic form, the argument has any
fault in it:
Nothing in the Constitution or laws of any State can destroy a right distinctly and
expressly affirmed in the Constitution of the United States.
The right of property in a slave is distinctly and expressly affirmed in the Constitution
of the United States.
Therefore, nothing in the Constitution or laws of any State can destroy the right of
property in a slave.
I believe that no fault can be pointed out in that argument; assuming the truth of the
premises, the conclusion, so far as I have capacity at all to understand it, follows in-
evitably. There is a fault in it as I think, but the fault is not in the reasoning; the false-
hood in fact is a fault of the premises. I believe that the right of property in a slave is
not distinctly and expressly affirmed in the Constitution, and Judge Douglas thinks it is.
I believe that the Supreme Court and the advocates of that decision [the Dred Scott
decision] may search in vain for the place in the Constitution where the right of proper-
ty in a slave is distinctly and expressly affirmed. I say, therefore, that I think one of the
premises is not true in fact.21
(cid:21)(cid:27)
Basic Logical Concepts
The reasoning in the argument that Lincoln recapitulates and attacks is not
faulty—but its second premise (that “the right of property in a slave is . . . af-
firmed in the Constitution”) is plainly false. The conclusion has therefore not
been established. Lincoln’s logical point is correct and important: An argument
may be valid even when its conclusion and one or more of its premises are false. The va-
lidity of an argument, we emphasize once again, depends only on the relationof
the premises to the conclusion.
There are many possible combinations of true and false premises and conclu-
sions in both valid and invalid arguments. Here follow seven illustrative argu-
ments, each prefaced by the statement of the combination (of truth and validity)
that it represents. With these illustrations (whose content is deliberately trivial)
before us, we will be in a position to formulate some important principles con-
cerning the relations between truth and validity.
I. Some validarguments contain only true propositions—true premises and a
true conclusion:
All mammals have lungs.
All whales are mammals.
Therefore all whales have lungs.
II. Some validarguments contain only false propositions—false premises and
a false conclusion:
All four-legged creatures have wings.
All spiders have exactly four legs.
Therefore all spiders have wings.
This argument is valid because, if its premises were true, its conclu-
sion would have to be true also—even though we know that in fact both
the premises andthe conclusion of this argument are false.
III. Some invalidarguments contain only true propositions—all their premises
are true, and their conclusions are true as well:
If I owned all the gold in Fort Knox, then I would be wealthy.
I do not own all the gold in Fort Knox.
Therefore I am not wealthy.
The true conclusion of this argument does not follow from its true
premises. This will be seen more clearly when the immediately follow-
ing illustration is considered.
IV. Some invalidarguments contain only true premisesand have a false con-
clusion. This is illustrated by an argument exactly like the previous one
(III) in form, changed only enough to make the conclusion false.
If Bill Gates owned all the gold in Fort Knox, then Bill Gates would be wealthy.
Bill Gates does not own all the gold in Fort Knox.
Therefore Bill Gates is not wealthy.
(cid:21)(cid:28)
Basic Logical Concepts
The premises of this argument are true, but its conclusion is false.
Such an argument cannot be valid because it is impossible for the prem-
ises of a valid argument to be true and its conclusion to be false.
V. Some validarguments have false premises and atrue conclusion:
All fishes are mammals.
All whales are fishes.
Therefore all whales are mammals.
The conclusion of this argument is true, as we know; moreover, it
may be validly inferred from these two premises, both of which are
wildly false.
VI. Some invalidarguments also have false premises and atrue conclusion:
All mammals have wings.
All whales have wings.
Therefore all whales are mammals.
From Examples V and VI taken together, it is clear that we cannot
tell from the fact that an argument has false premises and a true conclu-
sion whether it is valid or invalid.
VII. Some invalidarguments, of course, contain all false propositions—false
premises and a false conclusion:
All mammals have wings.
All whales have wings.
Therefore all mammals are whales.
These seven examples make it clear that there are valid arguments with false
conclusions (Example II), as well as invalid arguments with true conclusions (Ex-
amples III and VI). Hence it is clear that the truth or falsity of an argument’s conclu-
sion does not by itself determine the validity or invalidity of that argument. Moreover,
the fact that an argument is valid does not guarantee the truth of its conclusion(Exam-
ple II).
Two tables (referring to the seven preceding examples) will make very
clear the variety of possible combinations. The first table shows that invalidar-
guments can have every possible combination of true and false premises and
conclusions:
Invalid Arguments
True Conclusion False Conclusion
True Premises Example III Example IV
False Premises Example VI Example VII
(cid:22)(cid:19)
Basic Logical Concepts
The second table shows that valid arguments can have only three of those
combinations of true and false premises and conclusions:
Valid Arguments
True Conclusion False Conclusion
True Premises Example I —
False Premises Example V Example II
The one blank position in the second table exhibits a fundamental point: If an
argument is valid and its premises are true, we may be certain that its conclusion is true
also. To put it another way: If an argument is valid and its conclusion is false, not all of
its premises can be true. Some perfectly valid arguments do have false conclusions,
but any such argument must have at least one false premise.
When an argument is valid and all of its premises are true, we call it sound.
The conclusion of a sound argument obviously must be true—and only a sound
argument can establish the truth of its conclusion. If a deductive argument is not
sound—that is, if the argument is not valid or if not all of its premises are true—
it fails to establish the truth of its conclusion even if in fact the conclusion is true.
To test the truth or falsehood of premises is the task of science in general, be-
cause premises may deal with any subject matter at all. The logician is not (pro-
fessionally) interested in the truth or falsehood of propositions so much as in the
logical relations between them. By logical relations between propositions we mean
those relations that determine the correctness or incorrectness of the arguments
in which they occur. The task of determining the correctness or incorrectness of
arguments falls squarely within the province of logic. The logician is interested
in the correctness even of arguments whose premises may be false.
Why do we not confine ourselves to arguments with true premises, ignoring
all others? Because the correctness of arguments whose premises are not known
to be true may be of great importance. In science, for example, we verify theories
by deducing testable consequences from uncertain theoretical premises—but we
cannot know beforehand which theories are true. In everyday life also, we must
often choose between alternative courses of action, first seeking to deduce the
consequences of each. To avoid deceiving ourselves, we must reason correctly
about the consequences of the alternatives, taking each as a premise. If we were
interested only in arguments with true premises, we would not know which set
of consequences to trace out until we knew which of the alternative premises
was true. But if we knew which of the alternative premises was true, we would
not need to reason about it at all, because our purpose was to help us decide
which alternative premise to make true. To confine our attention to arguments
with premises known to be true would therefore be self-defeating.
(cid:22)(cid:20)
Basic Logical Concepts
EXERCISES
For each of the argument descriptions provided below, construct a deductive ar-
gument (on any subject of your choosing) having only two premises.
1. Avalid argument with one true premise, one false premise, and a false
conclusion
2. Avalid argument with one true premise, one false premise, and a true
conclusion
3. An invalid argument with two true premises and a false conclusion
4. An invalid argument with two true premises and a true conclusion
5. Avalid argument with two false premises and a true conclusion
6. An invalid argument with two false premises and a true conclusion
7. An invalid argument with one true premise, one false premise, and a
true conclusion
8. Avalid argument with two true premises and a true conclusion
chapter Summary
The most fundamental concepts of logic are introduced in this chapter.
In Section 1 we explained what logic is and why it is necessary, and we de-
fined it as the study of the methods and principles used to distinguish correct
from incorrect reasoning.
In Section 2 we gave an account of propositions, which may be asserted or
denied, and which are either true or false, and of arguments, which are clusters
of propositions of which one is the conclusion and the others are the premises of-
fered in its support. Arguments are the central concern of logicians.
In Section 3we discussed difficulties in the recognition of arguments, arising
from the variety of ways in which the propositions they contain may be ex-
pressed, and sometimes even from the absence of their express statement in ar-
guments called enthymemes.
In Section 4 we discussed the differences between arguments and explana-
tions, showing why this distinction often depends on the context and on the in-
tent of the passage in that context.
In Section 5we explained the fundamental difference between deductive ar-
guments, whose conclusions may be certain (if the premises are true and the rea-
soning valid), and inductive arguments, aiming to establish matters of fact,
whose conclusions may be very probable but are never certain.
In Section 6we discussed validity and invalidity (which apply to deductive
arguments) as contrasted with truth and falsity (which apply to propositions).
We explored some of the key relations between validity and truth.
(cid:22)(cid:21)
