“Chapter 4. Putting Validity into Practice” in “Critical Thinking, Logic, and Argument”
Chapter 4 Putting Validity into Practice
4.1 Using Counter-Examples
We introduced validity and invalidity in the previous section. There are certain thought techniques that we can use to demonstrate when an argument is invalid. Recall the suggestion that you live in Canoe Lake Cree Nation in Saskatchewan. This was to offer a possible counter-example to the claim that if you live in Saskatchewan, then you live in Saskatoon. If a deductive argument form has a counter-example, then it is invalid.
A counter-example is when we imagine a circumstance or possible situation in which the premises are true and the conclusion is false. We’re sure you have done this before even if you haven’t specifically called it a counter-example.
A valid argument has no counter-examples, so the presence of a single counter-example refutes an argument.
For example, if your friend says, “Don’t eat at Joy’s Diner because all the food there is terrible,” all it takes is one counter-example to prove false that all the food is terrible. Keep in mind that your friend said all the food is terrible. This is quite different than saying “most” or “some” or “many” or even “almost all.” If you use definitive statements like “all” or “only” or “never,” then a single counter-example will refute it. In this case, if you have been to Joy’s Diner, you could say, “I had one good sandwich there” and refute the statement. This is very different from requiring you to say, “All the food at Joy’s Diner is good.” You don’t need another “all” statement here, you just need a single counter-example to prove it false that “all the food is terrible.” When we talk about generalizations and fallacies, we will talk a lot about how to construct good generalizations.
In the real world, it matters if the examples we use are true—it matters if the person saying they had a good sandwich at Joy’s Diner is telling the truth. But for a deductive argument, we are searching for the possible situation that could exist—like it is possible you live in Canoe Lake Cree Nation. But what we imagine must be consistently thinkable. This means it can’t be paradoxical or contradictory, such as “this sentence is false” or “triangles are round.”
The validity of an argument doesn’t depend merely on what is actually true or false—a valid argument must work in every possible circumstance. That means that every possible circumstance that makes the premises true must also make the conclusion true.
Finding a single counter-example refutes an argument’s claim to validity. To see this more clearly, let us compare modus ponens, modus tollens, and affirming the consequent.
4.2 Modus Ponens
Modus Ponens has two propositions (P and Q are the letters we have been using so far) within them. If we think of all the possible combinations of truth values, or combinations of truth and falsity that generates, then we can better understand how counter-examples help us understand validity. Consider two propositions:
- A = The cat is on the mat.
- B = It is raining.
We can compare them using a graphic diagram of the four possible situations or truth values generated by two sentences and their negations. Since each sentence is either true or false, there are four possibilities for truth and falsity for combining the two sentences.
Figure 4.1 Four possible truth combinations. Artwork by Jessica Tang.
Sentence | Possibility 1 | Possibility 2 | Possibility 3 | Possibility 4 |
|---|---|---|---|---|
A (cat is on the mat) | true | true | false | false |
B (it is raining) | true | false | true | false |
One way of reading this chart is that in the case of possibility 1, both A and B have a positive truth value in the argument. In the case of possibility 2, A is positive and B is negative, represented as “not B.” In possibility 3, it is “not A” and B, and in possibility 4, it is “not A” and “not B.” These are all the possible combinations of the possible truth values of the sentences.
Why is this valid? Since premise 1 asserts “if A, then B,” then possibility 2 is not possible because the cat is on the mat (A is true) but it isn’t raining (B is false), so that cannot represent “if A, then B.” Premise 2 essentially asserts A as true, which rules out situations 3 and 4, where A is false.
The only possible remaining truth values, which is to say the only possibility that is not ruled out by the premises, is situation 1. In situation 1, the conclusion (B) is true. Here’s the result of this whole discussion. The argument is valid because the only possible assignment of truth values is one in which the truth of the premises guarantees the truth of the conclusion (B). This is why validity is seen as a method of truth preservation. When the premises are true, the conclusion is also true. This proves modus ponens is valid.
4.3 Modus Tollens
We can also use this method to prove modus tollens is valid.
Recall our four possible truth-value assignments:
Sentence | Possibility 1 | Possibility 2 | Possibility 3 | Possibility 4 |
|---|---|---|---|---|
A (cat is on the mat) | true | true | false | false |
B (it is raining) | true | false | true | false |
Premise 1 states that “if A is true, then B is true as well,” which rules out possibility 2, “A is true, but B is false.” Premise 2 states “not B,” which rules out 1 and 3 because they require B to be true. So the two premises rule out all the possibilities except possibility 4, which states that A and B are both false. So, why is modus tollens valid? Because if the premises are true, there’s no way for the conclusion to be false, which includes possibility 4 where the premises are false.
This one is a bit harder to grasp given the setup we have. Consider: “If you are a poodle, then you are a dog.” “You are not a dog, therefore you are not a poodle.” Even though cats and rain are not connected in the same way as poodles and dogs, the logical structure is the same. So when we say that if the cat is on the mat, then it is raining, we are saying at the same time that if it is not raining, the cat is not on the mat. Why? Because premise 1 clearly states that if the cat is on the mat, then it is raining.
4.4 Affirming the Consequent
This method also allows us to demonstrate why affirming the consequent is invalid.
Sentence | Possibility 1 | Possibility 2 | Possibility 3 | Possibility 4 |
|---|---|---|---|---|
A (cat is on the mat) | true | true | false | false |
B (it is raining) | true | false | true | false |
Possibility 2 is ruled out because it would mean that it is true that the cat is on the mat, but it is false that it is raining. This contradicts premise 1 (if A is true, then B is true). Premise 2 asserts B is true, so that rules out possibilities 2 (again) and 4 (because B is false on those possibilities).
We are left with two situations: possibility 1 and possibility 3. In possibility 3, it is raining, but the cat is not on the mat. But the conclusion asserts the cat is on the mat. What we are seeing, then, is that the conclusion is false in situation 3, and thus the truth of the premises is consistent with the falsity of the conclusion, making the argument invalid.
4.5 Denying the Antecedent
Sentence | Possibility 1 | Possibility 2 | Possibility 3 | Possibility 4 |
|---|---|---|---|---|
A (cat is on the mat) | true | true | false | false |
B (it is raining) | true | false | true | false |
Remember that with denying the antecedent, you cannot derive “not B” from premises 1 and 2. This is because B has enough independence from A that it can occur on its own. We only know “if A, then B.” We do not know enough from “not A” to conclude “not B.” From the truth-value possibilities, possibility 2 is ruled out because it would mean that it is true that the cat is on the mat, but it is false that it is raining, which denies premise 1.
Another way of stating that is to say that premise 1 asserts that if A is true, then B is also true, so possibility 2 can’t be true (since B is false). Premise 2 asserts that the cat is not on the mat, so there has to be a possibility where A is false, which rules out possibility 1. Now we are left with possibilities 3 and 4. In this case, we cannot rule out either one, which means that it is possible for the premises to be true while the conclusion false (possibility 3 has B as true, which contradicts the conclusion, which is “not B”).
Key Takeaways
- • A counter-example is a possible instance where the premises are true but the conclusion is false.
- • Valid arguments do not have counter-examples. If they are valid, they are valid in every possible circumstance.
- • Valid arguments can have false premises.
- • Validity is a method of truth preservation: when the premises are true, the conclusion must also be true.
- • Affirming the consequent and denying the antecedent are invalid because the truths of their premises are consistent with the falsities of their conclusions.
- • Mapping all truth-value possibilities allows you to look for counter-examples to argument forms.
Exercises
Part I. Validity Practice
Complete the following exercises on validity.
- 1. Write down the argument forms modus ponens, modus tollens, and disjunctive syllogism. Give an example of each pattern that has true premises (and so has a true conclusion) and an example that has false premises.
- 2. For each example in question, one of modus tollens and disjunctive syllogism, make a square representing the four possibilities for truth and falsity of the component sentences of your examples, and show that each is valid by crossing out each possible situation that is ruled out by a premise and determining that the conclusion is true in any possibility consistent with the truth of the premises.
- 3. Write down some examples of the invalid forms denying the antecedent and affirming the consequent. Give an example of each that has true premises and a true conclusion and using the four-possibility method to show why it is still invalid.
Part II. Validity True and False
Here is a set of assertions involving soundness and validity. Use a T or an F to indicate whether these statements are true or false.
- 1. No sound argument has false premises.
- 2. If an argument has true premises and a false conclusion, then it is invalid.
- 3. Every valid argument has a true conclusion.
- 4. If an argument has a counter-example, it is invalid.
- 5. Some unsound arguments have true premises.
- 6. No valid argument has false conclusions.
- 7. Some invalid arguments have true premises.
- 8. Every sound argument has a true conclusion.
- 9. No sound argument has a counter-example.
- 10. All valid arguments have true premises.
- 11. If an argument has false premises, it cannot be sound.
- 12. No valid argument has a counter-example.
- 13. All unsound arguments have false premises.
We use cookies to analyze our traffic. Please decide if you are willing to accept cookies from our website. You can change this setting anytime in Privacy Settings.