Refutation, Part 1: Counterexamples & Reductio

Kareem KhalifaPhilosophy Department

Middlebury College

Overview

I. What is a refutation?II. CounterexamplesIII. Reductio ad absurdum

I. What is a refutation?

A. Refuting arguments

B. Refuting propositions

C. Deep and shallow refutations

I.A.Refuting arguments

1. Invalidity (I-refutations): an argument that the author has reasoned invalidly.

2. Circularity (C-refutations): An argument that the author has reasoned circularly.

An argument is refuted if it is shown to be unsound or circular. Refutations of arguments come in two flavors:

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

A refutation is deep if it can only be answered by fundamentally revising its intended argument/proposition.

Otherwise, it’s shallow.

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?

Most F’s are G’s?

Some 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 someday9. There is a male barber who shaves all and only the men in this town who do not shave

themselves, and10. 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.