Formal Criteria for Evaluating
Arguments
Validity and Soundness
The Form Content Distinction
1.
Some students are athletes.
2.
Some athletes are political.
3.
Some students are political.
1.
Some young adults are in high
school.
2.
Some high school students are
in the 10th grade.
3.
Some young adults are in the
10th grade.
Both arguments have the
following form:
1.
Some A are B.
2.
Some B are C.
3.
Some A are C.
However, the arguments
differ in content.
Formal Evaluation of Arguments
The informal evaluation of an argument has to do with
determining whether the argument commits an informal fallacy,
which is an error in reasoning that has to do with the arguments
form, content, and context. In general, an informal fallacy cannot
be determined solely in virtue of the form of an argument.
The formal evaluation of an argument has to do solely with the
form of the argument, and not with the content or context of the
argument.
While there are disagreements as to what the form of a particular
argument is. Once the form has been determined and the system
for evaluation determined, assessing the argument requires using
techniques of formal logic.
Validity
The central criteria for assessing arguments formally is validity.
An argument form is valid when it is impossible for the premises to be
true and the conclusion false.
An argument form is invalid when it is possible for the premises to be
true and the conclusion false.
It is impossible for an argument form to have true premises and a false
conclusion just in case there is no instance of the form in which there
are true premises and a false conclusion.
It is possible for an argument form to have true premises and a false
conclusion just in case there is an instance in which the premises are
true and the conclusion is false.
Invalidity
Invalid argument with true
premises and a true conclusion.
Invalid argument with true
premises and a false conclusion.
1.
Some politicians are
conservative. (true)
1.
Some even numbers are whole
numbers. (true)
2.
Some conservatives are against
public health care. (true)
2.
Some whole numbers are odd
numbers. (true)
3.
Some politicians are against
public health care. (true)
3.
Some even numbers are odd
numbers. (false)
NOT AN INVALIDATING CASE
INVALIDATING CASE
Variations on Validity
Since validity is about form, and not content, there are various
combinations of true premises, false premises, true conclusion,
and false conclusion that can actually occur in a valid or
invalid argument.
Premises
Conclusion
Validity
All True
True
Valid or Invalid
Some True, Some False
True
Valid or Invalid
Some True, Some False
False
Valid or Invalid
All True
False
Only Invalid
Validity and the Counterexample Method
When thinking about validity think about the form of the
argument, not the actual premises and conclusion of the
argument.
Ask yourself the question: If the premises were true, could the
conclusion be false?
If your intuition is that this could occur, try to come up with an
example in which the premises are true and the conclusion is false.
If the example you come up with is coherent, then you have a
counterexample that proves the invalidity of the form of the
argument.
An Example of the Counterexample Method
Original Argument:
1.
2.
3.
All democrats are people that
might vote.
All liberals are people that might
vote.
All democrats are liberals.
Sounds okay, maybe you don’t know if
all the premises are true. But it looks
safe. Now think about the form, and
delete the content. What form of
reasoning is the person using?
Counterexample:
1.
All A are B.
2.
All C are B.
3.
All A are C.
Now put in new terms to make
true premises and a false
conclusion.
A = even numbers
B = whole numbers
C = odd numbers
Soundness
Although formal evaluation of arguments has to do primarily with
the form of the arguments, there are occasions when we need to
pay attention to the content as well. The occasions on which this
is true, is when we are assessing soundness.
An argument is sound when it has a valid form and true premises.
From the definition of soundness it follows that all sound
arguments have a true conclusion. If the form of the argument is
such that it is impossible for the premises to be true and the
conclusion false, because the form is a valid form, and then we
add actually true premises, it has to be the case that the conclusion
is also true.
Variations on Soundness
Since soundness includes validity, and validity is about the form of
an argument, there are variations of how these two ideas can
interact. Basically, if an argument is not valid, it cannot be sound.
Validity
Soundness
Yes
No
No
No
Two Cases to Keep in Mind
True premises, True conclusion,
invalid argument, and unsound.
1.
2.
3.
Some students are philosophy
majors.
Some philosophy majors are
double majoring in political
science.
Some students are double
majoring in political science
Not a good argument because it is
unsound because it is invalid.
False premise, False conclusion,
valid argument, and unsound.
1.
All cats are dogs.
2.
All dogs are rats.
3.
All cats are rats.
Valid Form:
All A are B.
All B are C.
All A are C.
Goodness ≠ Soundness
Many people often confuse the goodness of an argument with its
soundness.
One cannot infer from the validity or soundness of an argument that it is
good.
Anand is the teacher of this class. So, Anand is the teacher of this
class.
The argument is valid, since its form P, therefore P is valid, because
one and the same statement P cannot be both true and false.
The argument is sound, since it is true that Anand is the teacher of
this class.
However, the argument is clearly not good. It is circular.
Formal Evaluation by Formal Methods
Although it is possible to evaluate the form of an argument by thinking
about it and trying to come up with a counterexample, the method has
limitations.
It does not follow from the fact that Adam cannot come up with a
counterexample to argument form A, that A is valid.
The failure to find a counterexample to an argument does not prove that
it is valid. (sounds like appeal to ignorance –can’t find it so it doesn’t exist,
therefore valid.)
Thus, we need a formal method involving abstract reasoning and methods
of proof to test for validity.
Translation
One of the best ways to test to see whether an argument is valid is
to translate it into another language.
Just as one can translate English into French, most arguments
which occur in natural languages such as English, can be
translated into a formal language, such as the language of
propositional logic, categorical logic, predicate logic, modal logic,
or tense logic.
Our goal is to learn to translate arguments into the language of
propositional logic for the purposes of applying some formal
system of proof for checking validity.
Checking for Validity
Steps from here:
Learn the syntax and semantics of propositional logic.
Learn to translate propositional arguments into the language of
propositional logic.
Learn how to use a truth-table to check for various properties of
statements and groups of statements.
Learn how to construct a truth-tree to check for various properties of
statements and groups of statements.
Learn how to construct a natural deduction proof for propositional
arguments.
Learn how to use conditional and indirect proof techniques.