Philosophy 101
(2/22/11)
• HW #2 to be returned today (end of class)
• I will be grading on a “curve” after all. [more soon]
• Solutions to HW #2 posted (later today)
• HW #3 assigned last week (due next Thursday)
• Quiz #2 on this Thursday (on rational belief)
• Today: Chapter 3, Continued
• Two subtle aspects of formal validity
• Cogency (of invalid arguments)
• Next: Chapter 4 — Strong Arguments
Chapter 3: Well-Formed Arguments 32
• Validity — Predicate Logic (some invalid forms)
74
Chapter 3 Well-Formed Arguments
Table 3.4
Some Patterns of invalid Arguments in Predicate
logic
Pattern I Example
L All
As
are
Bs.
1.
All
men
are mortaL
1.
All
As
are
Bs.
1.
All
men are mortal.
2. Fido
is
mortal.
3. Fido
is
a man.
As
are
Bs
is
called a quantifier. Many different quantifiers are used in generalizations,
including "lots of," "nearly all," "hardly any," "few,"
and
countless others.
Generalizations
figure
prominently in valid arguments from predicate
logic.
Table
3.3
displays some
of
the more common patterns. The common patterns in
Table
3.4
are
invalid.
EXERCISES
AND
STUDY
QUESTIONS
Each
of
the
following arguments follows one
of
the patterns identified in Tables 3.3
and
3.4. For each argument, use circles and boxes to identifY its key parts. Then state
the pattern for each argument
and
state whether
or
not
it
is
valid.
*1. 1.
All
logicians are dull.
2.
Irving
is
a logician.
3.
Irving
is
dull.
2.
1.
All
logicians are dull.
2. Irving
is
not
a logician.
3. Irving
is
dulL
*3. 1.
All
logicians are dull.
2. Irving
is
dull.
3. Irving
is
a logician.
4. 1.
All
logicians are dulL
2. All (who are) dull are party animals.
3. All logicians are party animals.
*5. 1. No logicians are dull.
2. Irving
is
a logician.
3. Irving
is
not
dull.
6. 1. All logicians are dull.
2.
Irving
is
not dull.
3. Irving
is
not
a logician.
7.
1.
All
bearded logicians wear glasses.
2. Irving
is
a bearded logician.
3. Irving wears glasses.
II.
Well-Formed Arguments 75
AS. A Modification
of
the Dtifinition tif Validity
Arguments such
as
the following one raise a question about our definition
Argument
3.12
1.
Jones
is
a mother.
2. jones
is
female.
Is
Argument
3.12 valid?
You
might
think
it
is.
Since there
is
no
way the premise
could
be true
and
the conclusion false, it seems
that
the
truth
of
the
premise
does guarantee the
truth
of
the conclusion.
On
the
other
hand,
you
might
think
that
Argument
3.12
is
not
valid. There
is
no
recognizable valid
pattern
of
argu-
ment
here,
and
we've said
that
validity has to
do
with
the
pattern
or
form
of
argument.
The correct answer to this question
is
somewhat complicated, for there are two
very different ways in which the premises
of
an argument can be said to guarantee
the truth
of
the argument's conclusion.
One
way depends
only
on
the form
or
pat-
tern
of
the argument. Argument 3.4, for example,
is
valid
no
matter who
the
term
"Boris" refers
to
and
no
matter what
is
meant by "student"
or
"State U." In contrast,
the premise
of
Argument 3.12 appears to guarantee the truth
of
its conclusion,
but
this depends in
part
on
the fact that "mother" means "female parent," so anything
that
is
a
mother
is
also female. Consequently,
if
(1)
is
true, then
(2)
will be true
as
well. There are, then, two
ways
to think about validity:
one
concerns the form
of
arguments alone
and
a second takes the meanings
of
the key terms
of
the argument
into account.
For our purposes, we
will
interpret validity in the first way, that is, the validity
of
an argument will
not
depend
on
extra assumptions
about
the meanings
of
terms.
Valid arguments are ones whose pattern
or
structure all
by
itself assures that the
premises are properly related to the conclusion.
To
avoid confusion,
we
can refine
our
earlier definition
of
validity
as
follows:
03.1
b:
An
argument
is
valid
if
and
only
the
argument
follows
a
pattern such that it is
impossible
for
any
argument
following
that pattern to have true premises
and
a false
conclusion.
According to this definition,
if
an argument
is
valid, then it follows a pattern such
that all arguments following that same pattern are also valid.
On
this new understanding
of
validity, Argument 3.12
is
not
valid. However,
Argument 3.12
is
very closely connected to another argument that
is
valid. The
premise
of
Argument 3.12 could be replaced by
la.
Jones
is
a female and Jones
is
a parent.
Notice
that
and
are equivalent.
When
is
put
into the argument,
we
get
Another important Example:
Most As are Bs.
x is an A.
--------------------
x is a B.
Chapter 3: Well-Formed Arguments 33
• Validity — A Clarification of the Definition
• Our initial definition of validity was a bit unclear. You can see
this unclarity in certain more subtle examples, such as:
74
Chapter 3 Well-Formed Arguments
Table 3.4
Some Patterns of invalid Arguments in Predicate
logic
Pattern I Example
L All
As
are
Bs.
1.
All
men
are mortaL
1.
All
As
are
Bs.
1.
All
men are mortal.
2. Fido
is
mortal.
3. Fido
is
a man.
As
are
Bs
is
called a quantifier. Many different quantifiers are used in generalizations,
including "lots of," "nearly all," "hardly any," "few,"
and
countless others.
Generalizations
figure
prominently in valid arguments from predicate
logic.
Table
3.3
displays some
of
the more common patterns. The common patterns in
Table
3.4
are
invalid.
EXERCISES
AND
STUDY
QUESTIONS
Each
of
the
following arguments follows one
of
the patterns identified in Tables 3.3
and
3.4. For each argument, use circles and boxes to identifY its key parts. Then state
the pattern for each argument
and
state whether
or
not
it
is
valid.
*1. 1.
All
logicians are dull.
2.
Irving
is
a logician.
3.
Irving
is
dull.
2.
1.
All
logicians are dull.
2. Irving
is
not
a logician.
3. Irving
is
dulL
*3. 1.
All
logicians are dull.
2. Irving
is
dull.
3. Irving
is
a logician.
4. 1.
All
logicians are dulL
2. All (who are) dull are party animals.
3. All logicians are party animals.
*5. 1. No logicians are dull.
2. Irving
is
a logician.
3. Irving
is
not
dull.
6. 1. All logicians are dull.
2.
Irving
is
not dull.
3. Irving
is
not
a logician.
7.
1.
All
bearded logicians wear glasses.
2. Irving
is
a bearded logician.
3. Irving wears glasses.
II.
Well-Formed Arguments 75
AS. A Modification
of
the Dtifinition tif Validity
Arguments such
as
the following one raise a question about our definition
Argument
3.12
1.
Jones
is
a mother.
2. jones
is
female.
Is
Argument
3.12 valid?
You
might
think
it
is.
Since there
is
no
way the premise
could
be true
and
the conclusion false, it seems
that
the
truth
of
the
premise
does guarantee the
truth
of
the conclusion.
On
the
other
hand,
you
might
think
that
Argument
3.12
is
not
valid. There
is
no
recognizable valid
pattern
of
argu-
ment
here,
and
we've said
that
validity has to
do
with
the
pattern
or
form
of
argument.
The correct answer to this question
is
somewhat complicated, for there are two
very different ways in which the premises
of
an argument can be said to guarantee
the truth
of
the argument's conclusion.
One
way depends
only
on
the form
or
pat-
tern
of
the argument. Argument 3.4, for example,
is
valid
no
matter who
the
term
"Boris" refers
to
and
no
matter what
is
meant by "student"
or
"State U." In contrast,
the premise
of
Argument 3.12 appears to guarantee the truth
of
its conclusion,
but
this depends in
part
on
the fact that "mother" means "female parent," so anything
that
is
a
mother
is
also female. Consequently,
if
(1)
is
true, then
(2)
will be true
as
well. There are, then, two
ways
to think about validity:
one
concerns the form
of
arguments alone
and
a second takes the meanings
of
the key terms
of
the argument
into account.
For our purposes, we
will
interpret validity in the first way, that is, the validity
of
an argument will
not
depend
on
extra assumptions
about
the meanings
of
terms.
Valid arguments are ones whose pattern
or
structure all
by
itself assures that the
premises are properly related to the conclusion.
To
avoid confusion,
we
can refine
our
earlier definition
of
validity
as
follows:
03.1
b:
An
argument
is
valid
if
and
only
the
argument
follows
a
pattern such that it is
impossible
for
any
argument
following
that pattern to have true premises
and
a false
conclusion.
According to this definition,
if
an argument
is
valid, then it follows a pattern such
that all arguments following that same pattern are also valid.
On
this new understanding
of
validity, Argument 3.12
is
not
valid. However,
Argument 3.12
is
very closely connected to another argument that
is
valid. The
premise
of
Argument 3.12 could be replaced by
la.
Jones
is
a female and Jones
is
a parent.
Notice
that
and
are equivalent.
When
is
put
into the argument,
we
get
• Is this argument valid? One might think it is, because it might
seem that it would be a logical contradiction for the premise of
this argument to be true while its conclusion is false.
• But, strictly speaking, we will not classify this argument as valid.
➡This is because we have no logical theory (sentential or
predicate) according to which this argument has a valid form.
• This leads to an important clarification of our definition.
Chapter 3: Well-Formed Arguments 34
• Validity — A Clarification of the Definition
• Here is a clarified defintion of validity:
• D3.1b: An argument is valid iff the argument has some
logical form such that it is impossible for any argument with
that form to have true premises and a false conclusion.
• That is, all valid arguments must have valid logical forms.
• The following two (“equivalent”) arguments are valid:
1. Jones is a female and Jones is a parent.
---------------------------------------------------
2. Jones is a parent.
1. Jones is a mother.
2. All mothers are females.
----------------------------------
3. Jones is a female.
