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 true

and (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?)

Most philosophical arguments are deductive arguments

example—

1.    If God exists, God is all-powerful, all-knowing, and perfectly good.

2.    If God is all-powerful, all-knowing, and perfectly good, then God can prevent evil, knows how to prevent evil, and wants to prevent evil.

3.    If God can prevent evil, knows how to prevent evil, and wants to prevent evil, then God does prevent evil.

4.    If God does prevent evil, then evil does not exist.

5.    Evil does exist.

6.    Therefore, God does not exist.

To analyze an argument is to present it in premise–conclusion

form, listing each premise and the conclusion.

# Common Deductive Argument Forms

## Categorical Argument—consists of two premises (plus conclusion), where all three statements are categorical statements

categorical statement—a statement of any of the following

forms:

All X are Y.

No X are Y.

Some X are Y.

Some X are not Y.

example of a categorical argument—

All engineers are expert mathematicians.

Some lawyers are engineers.

Therefore, some lawyers are expert

mathematicians.

Hypothetical Argument—includes at least one premise that is a

hypothetical (conditional) statement

hypothetical (conditional) statement—a statement of the

form “If P then Q

## Two Types of Hypothetical Arguments—

1.    hypothetical syllogism—an argument consisting of two premises (plus conclusion), where one of the premises is a hypothetical statement, and each of the other premise and the conclusion is either the antecedent (“if” part–P) or the consequent (“then” part–Q) of the hypothetical statement, or their denials

example

If it rains, then the picnic will be cancelled.

The picnic will not be cancelled.

Therefore, it will not rain.

2.    hypothetical chain argument—an argument consisting entirely of hypothetical statements

example

If Sue studies regularly, then she will understand the

material.

If Sue understands the material, then she will do well

on the exams.

If Sue does well on the exams, then she will get an A in

the course.

If Sue gets an A in the course, then she will graduate

with honors.

Therefore, if Sue studies regularly, then she will

Disjunctive Argument—an argument that contains at least one disjunctive statement (as

premise or conclusion)

disjunctive statement—a statement of the form “Either P or Q

### One type of disjunctive argument:

disjunctive syllogism—contains two premises (plus

conclusion) where (1) one of the premises is a

disjunctive statement, (2) the other premise affirms or

denies one of the disjuncts (P or Q), and (3) the

conclusion affirms or denies the other disjunct

example

Either Tom will stay in engineering or he will switch to

Tom will not switch to business.

Therefore, Tom will stay in engineering.

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.

chain arguments: multiple arguments that are interlaced; i.e., the

conclusions of some arguments are premises of

others

“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

philosophy unless you explain exactly what you mean!

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 has

characteristic C2.

causal argument

Event a1 of type A preceded event b1 of type B.

Event a2 of type A preceded event b2 of type B.

. . .

Event an of type A preceded event bn of type B.

Therefore, events of type B are caused by events of type

A.

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?

informal fallacy—an error in reasoning that is not based on the

form of the argument

Two Categories of Informal Fallacies:

1.    fallacies of relevance–fallacies in which the premises are not directly relevant to the conclusion

example—

Most Americans favor a competitive, free-market

economy.

Therefore, the best economic system is a

competitive, free-market system.

2.    fallacies of ambiguity–fallacies resulting from the ambiguous or unclear use of words, phrases, clauses, etc.

example—

Power tends to corrupt.

Knowledge is power.

Therefore, knowledge tends to corrupt.

[example from Irving M. Copi, Keith

Burgess-Jackson, Informal Logic, 3rd

edition (Upper Saddle River, NJ: Prentice-

Hall, 1995), p. 99.]

## 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.