A Brief Introduction to Logic

A Brief Introduction to Logic

logic–the study of arguments

argument–a collection of statements consisting of one or more
premises and a conclusion

valid argument–an argument for which it is impossible
(inconceivable) that all of its premises are trueand (at the
same time) its conclusion is false

example—

Brad will pass the test only if he studies. (premise)
If Brad goes to the fraternity party, then he will not
study. (premise)
Therefore, if Brad goes to the fraternity party he will
not pass the test. (conclusion)

Two types of arguments:

1.deductive–are intended to be valid
2.inductive–are not intended to be valid

invalid argument–a deductive argument that is not valid

example—

Brad will pass the exam only if he studies. (premise)
If Brad goes to the fraternity party, then he will not
study. (premise)
Therefore, if Brad doesn’t go to the fraternity, he will
pass the exam. (conclusion)

sound argument–a valid argument with true premises

Therefore, to determine whether an argument is sound, we must answer two questions:

(1)Is the argument valid?
(2)Are all of the premises true?

validity, soundness, and truth—

Even if the premises and conclusion of a deductive argument are all true, the argument may be invalid and unsound.

example—

All mammals are animals. [true]
Some animals are primates. [true]
Therefore, all primates are mammals. [true]

[argument is invalid (Why?) and unsound (Why?)]

Even if the premises of a valid argument are false, the conclusion may be true.

example—

Some mammals are reptiles. [false]
All reptiles are cats. [false]
Therefore, some mammals are cats. [true]

[argument is valid (Why?) and unsound (Why?)]

The conclusion of a sound argument must be true. (Why?)

To analyze an argument is to present it in premise–conclusion form, listing the premises and the conclusion.

General Procedure for Testing the Validity of a Deductive Argument:

1.Identify the form of the argument.

2.Try to find an argument of the same form with true premises and a false conclusion.

3.If such an argument can be found, then the original argument is invalid.

4.If no such argument can be found, then either (1) the original argument is valid or (2) the tester is incompetent.

inductive arguments—

not intended to be valid
intended to convince us to accept conclusion

examples—

I.Carol has observed 10,000 crows in the wild and all,
without exception, were black.
Therefore, all crows everywhere are black.

II.96% of all philosophy professors are underpaid.
Lockhart is a philosophy professor.
Therefore, Lockhart is probably underpaid.

Some types of inductive arguments:

generalization argument
causal argument
analogical argument

generalization argument—

x1 has characteristics C1 and C2
x2 has characteristics C1 and C2
. . .
xn has characteristics C1 and C2
Therefore, everything that has characteristic C1 also
hascharacteristic C2.

causal argument—

Event a1 of type X preceded event b1 of type Y.
Event a2 of type X preceded event b2 of typeY. . . .

Event an of type X preceded event bn of type Y. Therefore, events of type X are caused by events of typeY.

analogical argument—

Each of x and y has characteristics C1, C2, …, Cn
x has characteristic Cn+1
Therefore, y also has characteristic Cn+1

Evaluating inductive arguments—

1.Are the premises true?
2.If the premises were all true, would they give us
sufficient reason to accept the conclusion?

Steps in Analyzing and Evaluating Arguments—

1.Analyze the argument:

Identify the main argument and the supporting arguments.
List the premises and conclusion of each.
Add any unstated premises and conclusions.

2.Classify each argument (main and supporting) as deductive or inductive.

3.Evaluate each deductive argument by determining
whether the argument is valid
whether the premises are all true

[This may require considering prior evaluations of supporting deductive and inductive arguments.]

4.Evaluate each inductive argument by determining—

Whether the premises, if true, would adequately support the
conclusion
Whether the premises are all true

The No Nitpicking Rule—

In evaluating someone else’s argument, avoid nitpicking:

Give the author’s argument the most sympathetic
interpretation possible.
Add plausible premises that would strengthen the argument,
even if not stated by the author.
Remove any implausible premises that are not essential to
the argument.
Try to think of supporting arguments (not provided by the
author) for any questionable premises.

“prove”: a very tricky word; it may mean—

to establish (as true) with complete certainty
to establish (as true) with a very high probability
to draw as the conclusion of a sound deductive argument
to draw as the conclusion of a successful inductive
argument

Advice: Do not use “prove” in discussing or writing about philosophy unless you explain exactly what you mean!