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.
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”
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
graduate with honors.
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”
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
business.
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
Advice: Do not use “prove” in
discussing or writing about
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.]
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
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.