Refutation, Part 1:
Counterexamples & Reductio
Kareem Khalifa
Philosophy Department
Middlebury College
Overview
I. What is a refutation?
II. Counterexamples
III. Reductio ad absurdum
I. What is a refutation?
A. Refuting
arguments
B. Refuting
propositions
C. Deep and shallow
refutations
I.A.Refuting arguments
An argument is refuted if it is shown to be unsound or
circular.
Refutations of arguments come in two flavors:
1. Invalidity (Irefutations): an
argument that the
author has reasoned
invalidly.
2. Circularity (Crefutations): An
argument that the
author has reasoned
circularly.
I.B.Refuting propositions
• A proposition/claim is refuted if it
is shown to be false. (This is used
to show that an argument is
unsound.) Two flavors:
3. False Premises: An argument
that some of the author’s
premises are dubious/false.
4. False conclusion: An argument
that the author’s conclusion
leads to absurd results.
Refuting a conclusion….
…tells you that
something is wrong
with the argument…
…but it doesn’t tell
you precisely what
is wrong with the
argument.
I.C.Deep and shallow refutations
Otherwise, it’s shallow.
A refutation is deep if it can only be
answered by fundamentally revising
its intended argument/proposition.
II. Counterexamples
• A single counterexample refutes a universal claim
This is the best way to use a counterexample.
• Ex. “All F’s are G’s” is refuted by the claim that “At least one
F is not a G.”
What are “counterexamples” to the
following?
Some F’s are G’s?
Most F’s are G’s?
Why do counterexamples work?
Often, our judgments about particular cases are on
firmer ground than our ability to provide perfect
definitions.
Responses to alleged counterexamples
• Suppose that you claim that all F’s are G’s, and
someone offers a counterexample, i.e. they
claim that a is an F but not a G.
• What can you do in response?
– You can argue that a is not an F; or
– You can argue that a is a G.
• If neither of these responses is plausible, then
the counterexample is decisive.
Is there a counterexample to the
following?
“You can never get too much
of a good thing.”
• If so, how would you
respond to this
counterexample?
Homework
I.1. No prime number is even.
• Counterexample: 2.
I.12. If it would be horrible for everyone to do
something, then it would be morally wrong for anyone
to do something.
• Counterexample: It would be horrible if everybody lied,
but it would not be morally wrong if someone lied (e.g.
to a murderer about the whereabouts of his next
victim.)
I.13. If it would not be horrible for everyone to do
something, then it not be morally wrong for anyone to
do it.
• Counterexample: ???
More Homework
II.2. Killing is usually wrong.
• If the statement were that killing is always wrong, then
there would be exceptions/counterexamples (e.g.
involving self-defense). However, because of the
guarding term “usually,” these counterexamples don’t
apply, since presumably, most killings don’t involve
these special exceptions.
II.7.Everything that is green has a shape.
• If something is green, then it is extended in space. If it
is extended in space, then it has a shape. So,
everything that is green has a shape.
III. Reductio ad absurdum
• This is basically ~I: you show that if a claim is
true, something absurd follows.
Three questions to ask about reductios
1. Is the result really absurd?
2. Does the refuted claim really imply the
absurdity?
3. Can the refuted claim be modified in some
minor way so that it no longer implies the
absurdity?
HW, Continued
III.1. Claim to be refuted: even the worst of
enemies can become friends.
Reductio: If people are enemies, then they are not
friends. If they do become friends, then they are
not enemies. So it is absurd to think that enemies
become friends.
• Is the claim really absurd? No. If people become
friends, then they are friends in the future. But
people can be friends in the future while being
enemies in the present.
III.5. Some things are inconceivable.
Reductio: Consider something that is inconceivable. Since you are
considering it, you are conceiving it. But then it is conceivable as well
as inconceivable. That is absurd. So nothing is inconceivable.
• Is the claim really absurd? Yes. Something cannot both be
conceivable and inconceivable.
• Does the refuted claim imply the absurdity? No. It seems possible to
consider something without conceiving of it. For instance, if I invite
you to consider a married bachelor, this doesn’t mean you have a
coherent concept of it.
• Can the refuted claim be tweaked? Yes. Suppose that you don’t buy
my distinction between consideration and conceivability above.
Then you can still claim that there are some things about which we
can’t form a coherent concept.
IV.6. I know that I do not know
anything.
• Reductio: If you do not know anything, then you
do not know that you don’t know anything. But
this would mean that you both know and don’t
know that you don’t know anything. This is
absurd.
• If it’s easier: Let p = “You do not know anything.”
So, according to the refuted claim, you know that
p. Since knowledge entails truth, p is true. But if
you know that p, then you know something, i.e.
~p. So both p and ~p are true. This is absurd. So
you cannot know that you do not know anything.
IV.10. Most of the sentences in this exercise are true.
Reductio: Suppose that most of the sentences in this exercise are true. Then a majority of the
following is true:
1.
Some sisters are nephews,
2.
Some fathers were never children.
3.
Most students scored better than the median grade on the last test,
4.
Almost everyone in this class is exceptional,
5.
There is an exception to every universal claim,
6.
I know that I don’t know anything,
7.
Some morally wrong actions are morally permitted,
8.
God exists outside of time, and we will meet him someday
9.
There is a male barber who shaves all and only the men in this town who do not shave
themselves, and
10. Most of the sentences in this exercise are true.
However, at least half of these claims are false (1,3,5,6,9 are obviously false; the rest can be
debated). This means that:
• It’s not the case that most of the sentences in this exercise are true.
So 10 is both true and false. This is absurd. So, most of the sentences in this exercise are not true.
As a result, we can now make a stronger claim:
• Most of the sentences in this exercise are not true.