1. Validity and Counterexamples
Kareem Khalifa
Department of Philosophy
Middlebury College

2. Overview Why this matters Validity and counterexamples: their relation
How to construct effective counterexamples
Exercises
Any questions?

3. 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.

4. 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.

5. 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.
2 Points: Logic and reading are creative acts. Counterexamples are intuitive, but unreliable.

6. Valid or invalid? Argument 1: Argument 2: VALID!!
If polluting does more harm than good, then it is wrong.
Polluting disproportionately does more harm than good.
 Polluting is wrong.
Argument 2:
Polluting does not do more harm than good.
Polluting is not wrong.
VALID!!
INVALID!!
Whiteboard!!

7. Every counterexample should…
Affirm of all the argument’s premises.
Deny the argument’s conclusion.
Explain how this is possible—that is, how the conclusion can still be false while the premises are all true.
I frequently rearrange the order: explanation first.

8. An example of a counterexample
Argument 2:
If polluting does more harm than good, then it is wrong.
Polluting does not do more harm than good.
Polluting is not wrong.
Suppose that there are other reasons that polluting is wrong, e.g. that it also wrong disproportionately harms those who are not responsible for it (Explains the How possible? question).
In this case, polluting may not do more harm than good (Affirms Premise 2) and if polluting did more harm than good, it is wrong (Affirms Premise 1).
Nevertheless, polluting would still be wrong (Denies Conclusion).

9. Tip #1 for constructing explanations in counterexamples
The explanation need not be true; it only needs to be possible.
Even if, in actuality, there are no other reasons to think pollution is wrong than its net harm, it is still possible that such reasons exist.
Physical possibility: that which is compatible with the laws of nature; practical possibility: that which is compatible with our current technology; logical possibility: that which does not entail a contradiction.

10. 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 little green men inhabiting the moon are the authors of the moral law, and they decree that pollution is wrong.
Logically speaking, this counterexample is just as good as the “disproportionality” 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!

11. Tip #3
Stay as close to the premises and conclusion as you can in constructing a counterexample.
Ex. Along with amazing technological advances, the Industrial Revolution of the mid-19th century introduced new sources of air and water pollution. By the middle of the 20th century, the effects of these changes were beginning to be felt in countries around the world. In the 1960s, an environmental movement began to emerge that sought to stem the tide of pollutants flowing into the planet’s ecosystems. Out of this movement came events like Earth Day, and legislative victories like the Clean Air Act (1970) and the Clean Water Act (1972). [Then insert your how-possibly explanation, etc.]
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.

12. 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].

13. Example of the schema
Suppose that there other reasons that pollution is wrong Then it could still be the case that pollution does not do more harm than good, and if it were to do so, it would be wrong, but that pollution is nevertheless still wrong.

14. Exercise 1 No plants are sentient.
All morally considerable things are sentient.
 No plants are morally considerable.
VALID.

15. 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.

16. 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.

17. 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.

18. Exercise 7 To know something is to be certain of it.
We cannot be certain of anything.
 We cannot know anything.
VALID.

19. 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.

20. 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.

21. 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.