Validity and Counterexamples Kareem Khalifa Department of Philosophy Middlebury College.

Validity and Counterexamples

Kareem KhalifaDepartment of Philosophy

Middlebury College

Overview

• Why this matters• Validity and counterexamples: their relation• How to construct effective counterexamples• Exercises

Why this matters

• We want to be critical of arguments.• Showing an argument to be invalid—showing

that the conclusion doesn’t follow from the premises—is an important form of criticism.

• Counterexamples are often an intuitive way of showing an argument to be invalid.

Relation between validity & counterexamples

• A deductive argument is valid when, if all of its premises are true, its conclusion must be true.

• In other words, there is no possible situation in which all of the premises are true and the conclusion is false.

• So, in an invalid argument, there are possible situations in which all of the premises are true yet the conclusion is false.

• These are called counterexamples to the argument. • Thus, valid arguments admit of no counterexamples.

Caution

• While valid arguments admit no counterexamples, the fact that you can’t construct a counterexample doesn’t make an argument valid.

• It may simply suggest that you need to be more creative.

Valid or invalid?

• Argument 1:– The U.S. should import Chinese products only if China

strictly regulates those products.– China does not strictly regulate its products.– The U.S. should not import those products.

• Argument 2:– If China strictly regulates its products, then the U.S. should

import those products.– China does not strictly regulate its products.– The U.S. should not import those products.

VALID!!

INVALID!!

Every counterexample should…

1. Affirm of all the argument’s premises.

2. Deny the argument’s conclusion.

3. Explain how this is possible—that is, how the conclusion can still be false while the premises are all true.

An example of a counterexample

• Argument 2:– If China strictly regulates its products, then the U.S. should

import those products.– China does not strictly regulate its products.– The U.S. should not import those products.

• Suppose that goods produced in China are inspected and regulated in the U.S. (Explains the How possible? question).

• In this case, China does not regulate its products (Affirms Premise 2). However, it would nevertheless be true that if the Chinese were to perform their own regulations, the U.S. should import their products (Affirms Premise 1).

• Nevertheless, this would provide a good reason to import their products, on the assumption that production costs remain low (Denies Conclusion).

Tip #1 for constructing explanations in counterexamples

• The explanation need not be true; it only needs to be possible.

• Even if there has been talk of U.S. companies being held accountable for safety-checks of Chinese parts and products that they import, it need not actually be the case.

Tip #2• An explanation that is closer to reality usually illustrates the

force of a counterexample more vividly than one that requires extravagant leaps in imagination.

• Ex. Suppose that goods produced in China are inspected and regulated by little green men inhabiting the moon…(etc.)

• Logically speaking, this counterexample is just as good as the U.S. regulation one.

• But a typical person will start focusing on how improbable it is that little green men inhabit the moon rather than on the invalidity of their argument.

• So be kind to your audience!

Tip #3• Stay as close to the premises and conclusion as you can in

constructing a counterexample.• Ex. Suppose that Chiang Kai-shek, rather than Mao Zedong,

seized control of the Chinese Mainland, and that as a result of renouncing Communism, China had more extensive and collegial relations with the West during the Cold War. As a result, they formed a partnership with a French regulation company named Régulateur…

• Logically speaking, this works just as well as our example, but just like with the little green man example, you’ve bogged down your audience with details that distract from the main point: that there’s a bad argument being offered.

A schema for writing counterexamples

• Suppose that [INSERT EXPLANATION OF HOW POSSIBLE? HERE]. Then it could still be the case that [AFFIRM PREMISES HERE], but nevertheless [DENY CONCLUSION HERE].

Example of the schema

• Suppose that the most affordable goods are produced in China and are inspected and regulated in the U.S. Then it could still be the case that China does not regulate its products and that if the Chinese were to perform their own regulations, the U.S. should import their products, but the U.S. should nevertheless import Chinese products.

Exercise 1

• No plants are sentient.• All morally considerable

things are sentient.• No plants are morally

considerable.• VALID.

Exercise 2

• All mathematical truths are knowable.• All mathematical truths are eternal.• All that is knowable is eternal.• INVALID. • Suppose there is something knowable that is not a

mathematical truth and that is not eternal, e.g., that I am experiencing a headache. Then it could still be the case that all mathematical truths are knowable and eternal, but that something knowable is not eternal.

Exercise 3

• Most geniuses have been close to madness.• Blake was a genius.• Blake was close to madness.• INVALID. • Suppose that Blake was one of the few geniuses that

was well-adjusted. Then it could still be the case that most geniuses have been close to madness and Blake was a genius, but that Blake was nevertheless not close to madness.

Exercise 6

• Some angels are fallen.• Some angels are not fallen.• INVALID. • Suppose that all angels are fallen. Then it

would still be the case that some angels are fallen, but nevertheless not be the case that some angels are not fallen.

Exercise 7

• To know something is to be certain of it.• We cannot be certain of anything.• We cannot know anything.• VALID.

Exercise 10• The witnesses said that either one or two shots

were fired at the victim.• Two bullets were found in the victim’s body.• Two shots were fired at the victim.• INVALID. Suppose that one shot was fired at the

victim, and another was the victim’s own bullet that struck him due to a misfire of his gun. Then it could still be the case that the witnesses said that either one or two shots were fired at the victim, two bullets were found in the victim’s body, but two shots were not fired at the victim.

Exercise 12

• Some fools are greedy.• Some fools are lecherous.• There are some fools who are both greedy and

lecherous.• INVALID: Suppose that all the fools who are greedy

are not lecherous and vice versa. Then it would still be the case that some fools are greedy, some lecherous, but nevertheless, none are both greedy and lecherous.

Exercise 14• DNA contains the code of life.• Life is sacred.• It is wrong to manipulate DNA.• INVALID. Suppose that it is proper to manipulate certain

sacred things, as we do when we build or fix an altar. Then it could still be the case that DNA contains the code of life and that life is sacred, but that it is correct to manipulate DNA.

• ALTERNATIVELY, suppose that, because DNA contains the code of life, manipulating it is the way to preserve the sacredness of life, e.g., by saving people from hereditary diseases.