I have already described formal logic and explained why it’s important for proper reasoning. One of the best ways to learn formal logic is to take a logic class. However, we don’t have to take a class just to learn the basics and improve our intuitive grasp of formal logic. What I want to do here is explain how to use counterexamples to prove an argument to be logically invalid. This can help improve our understanding of logic and help us prove arguments to be logically invalid.
What are formal counterexamples?
Whenever someone asserts something false, we can attempt to give a counterexample. For example, someone who claims that all animals are mammals can be proven wrong when we give an example of an animal that’s a reptile rather than a mammal, such as a lizard. Formal counterexamples prove that an argument is logically invalid rather than that beliefs are false.
An argument is logically valid if it’s impossible for the argument structure to have true premises and a false conclusion at the same time. Any argument that’s not logically valid is invalid—the argument structure can have true premises and a false conclusion at the same time. A counterexample proves that a logical form is invalid because it can have true premises and a false conclusion at the same time.
Consider the following argument:
- All dogs are mammals.
- All cats are animals.
- Therefore, all dogs are animals.
This argument has true premises and a true conclusion, but it’s logically invalid. The argument form is the following:
- All A are B.
- All C are D.
- Therefore, all A are D.
A counterexample is the following:
- All dogs are mammals.
- All lizards are reptiles.
- Therefore, all dogs are reptiles.
We kept the same argument form, replaced C and D, and the result is that both premises are still true, but the conclusion is false.
How do we create formal counterexamples?
To create a counterexample, you should (a) find the argument structure, and (b) find content for the argument form that will have true premises and a false conclusion by replacing the variables (letters).
For example, consider the following invalid argument:
- If a human fetus is a person, then it’s wrong to have an abortion.
- It’s wrong to have an abortion.
- Therefore, a human fetus is a person.
The argument form can be revealed when we remove all the content until we are left with logical connectives and variables. In the case of this argument the content of the argument are statements—various truth claims. In this case the argument form is the following:
- If A, then B.
- Therefore, A.
We can then replace these variables with new content (statements). A counterexample could use the following schema (content for the variables):
A: Dogs are reptiles.
B: Dogs are animals.
The counterexample using this schema is the following:
- If dogs are reptiles, then dogs are animals.
- Dogs are animals.
- Therefore, dogs are reptiles.
The first two premises are true, but the conclusion is false. If dogs are reptiles, then they are animals, even though they aren’t reptiles because “all reptiles are animals.”
Understanding logical form and validity is important for proper argumentation. Although we have an intuitive grasp of logical form and validity, we can learn more about it and improve our understanding. Learning formal counterexamples not only helps us to improve our understanding of logical form, but it also helps us learn how to prove that certain arguments are logically invalid.