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[Tutorial A03] Validity

§ A03.1 Definition of validity

One desirable feature of arguments is that the conclusion should follow from the premises. But what does it mean? Consider these two arguments :

Argument #1 :
  • Barbie is over 90 years old. So Barbie is over 20 years old.
Argument #2 :
  • Barbie is over 20 years old. So Barbie is over 90 years old.

Intuitively, the conclusion of the first argument follows from the premise, whereas the conclusion of the second argument does not follow from its premise. But how should we explain the difference between the two arguments more precisely? Here is a thought : In the first argument, if the premise is indeed true, then the conclusion cannot be false. On the other hand, even if the premise in the second argument is true, there is no guarantee that the conclusion must also be true. For example, Barbie could be 30 years old.

So we shall make use of this idea to define the notion of a deductively valid argument, or valid argument, as follows:

An argument is valid if and only if there is no logically possible situation where all the premises are true and the conclusion is false at the same time.

The idea of validity provides a more precise explication of what it is for a conclusion to follow from the premises. Applying this definition, we can see that the first argument above is valid, since there is no possible situation where Barbie can be over 90 but not over 20. The second argument is not valid because there are plenty of possible situations where the premise is true but the conclusion is false. Consider a situation where Barbie is 25, or one where she is 85. The fact that these situations are possible is enough to show that the argument is not valid, or invalid.

§ A03.2 Validity and truth

What if we have an argument with more than one premise? Here is an example :

All pigs can fly. Anything that can fly can swim. So all pigs can swim.

Although the two premises of this argument are false, this is actually a valid argument. To evaluate its validity, ask yourself whether it is possible to come up with a situation where all the premises are true and the conclusion is false. (We are not asking whether there is a situation where the premises and the conclusion are all true.) Of course, the answer is 'no'. If pigs can indeed fly, and if anything that can fly can also swim, then it must be the case that all pigs can swim.

So this example tells us something :

The premises and the conclusion of a valid argument can all be false.

Hopefully you will now realize that validity is not about the actual truth or falsity of the premises or the conclusion. Validity is about the logical connection between the premises and the conclusion. A valid argument is one where the truth of the premises guarantees the truth of the conclusion, but validity does not guarantee that the premises are in fact true. All that validity tells us is that if the premises are true, the conclusion must also be true.

§ A03.3 Showing that an argument is invalid

Now consider this argument :

Adam loves Beth. Beth loves Cathy. So Adam loves Cathy.

This argument is not valid, for it is possible that the premises are true and yet the conclusion is false. Perhaps Adam loves Beth but does not want Beth to love anyone else. So Adam actually hates Cathy. The mere possibility of such a situation is enough to show that the argument is not valid. Let us call these situations invalidating counterexamples to the argument. Basically, we are defining a valid argument as an argument with no possible invalidating counterexamples. To sharpen your skills in evaluating arguments, it is therefore important that you are able to discover and construct such examples.

Notice that a counterexample need not be real in the sense of being an actual situation. It might turn out that in fact that Adam, Beth and Cathy are members of the same family and they love each other. But the above argument is still invalid since the counterexample constructed is a possible situation, even if it is not actually real. All that is required of a counterexample is that the situation is a coherent one in which all the premises of the argument are true and the conclusion is false. So we should remember this :

An argument can be invalid even if the conclusion and the premises are all actually true.

To give you another example, here is another invalid argument with a true premise and a true conclusion : "Paris is the capital of France. So Rome is the capital of Italy." . It is not valid because it is possible for Italy to change its capital (say to Milan), while Paris remains the capital of France.

Another point to remember is that it is possible for a valid argument to have a true conclusion even when all its premises are false. Here is an example :

All pigs are purple in colour. Anything that is purple is an animal. So all pigs are animals.

Before proceeding any further, please make sure you understand why these claims are true and can give examples of such cases.

  1. The premises and the conclusion of an invalid argument can all be true.
  2. A valid argument should not be defined as an argument with true premises and a true conclusion.
  3. The premises and the conclusion of a valid argument can all be false.
  4. A valid argument with false premises can still have a true conclusion.

§ A03.4 A reminder

The concept of validity provides a more precise explication of what it is for a conclusion to follow from the premises. Since this is one of the most important concepts in this course, you should make sure you fully understand the definition. In giving our definition we are making a distinction between truth and validity. In ordinary usage "valid" is often used interchangeably with "true" (similarly with "false" and "not valid"). But here validity is restricted to only arguments and not statements, and truth is a property of statements but not arguments:

So never say things like "this statement is valid" or "that argument is true"!

§ A03.5 Exercises

Question 1 Are the following arguments valid? Why or why not?

1.

Someone is sick.
Someone is unhappy.
So someone is unhappy and sick.

Invalid.

2.

If he loves me then he gives me flowers.
He gives me flowers.
So he loves me.

Invalid.

3.

Beckham is famous.
Beckham is a football player.
Therefore, Beckham is a famous football player.

Invalid. Beckham might be a famous chef who is a football player but not a famous football player.

4.

If it rains, the streets will be wet.
If the streets are wet, accidents will happen.
Therefore, accidents will happen if it rains.

Valid.

5.

John was in Britain when Mary died in Hong Kong.
So Mary could not have been killed by John.

Invalid. Perhaps John shot Mary on Monday, and flew to Britain on Tuesday, but Mary died on Friday.

6.

If there is life on Pluto then Pluto contains water.
But there is no life on Pluto.
Therefore Pluto does not contain water.

Invalid.

7.

There were two rabbits in the room last week.
No rabbit has left the room since then.
Therefore there are two rabbits in the room now.

Invalid. As you know, rabbits can reproduce!

8.

All whales have wings.
Moby does not have wings.
So Moby is not a whale.

Valid.
Question 2

Consider this argument :

  • If there is a square in the picture then there is a circle as well.
    Therefore, if there is a circle in the picture there is a triangle in the picture.

Now look at these four pictures below. Which of them constitute invalidating counterexamples to the argument, and which do not?

Only the second one from the right.
Question 3 Are these arguments valid?
  1. John shot himself in the head. So John is dead.
    Not valid. Although the conclusion is very likely to be true given the premise, it is not a logical consequence of the premise. Perhaps a brillant doctor managed to save John.
  2. John shot himself in the head. So John shot himself in the head.
    Valid. This is a circular argument since the conclusion is also a premise. But it is nonetheless valid since it is impossible for the premises to be true while the conclusion is false.
  3. All management consultants are bald. Peter is bald. So Peter is a management consultant.
    Not valid. Perhaps Peter is a monk who happended to have shaved his hair.
  4. If time travel is possible, we would now have lots of time-travel visitors from the future. But we have no such visitors. So time travel is not possible.
    Valid.
  5. Jen is either in San Diego or in Tokyo. Since she is not in Tokyo, she is in San Diego.
    Valid.
  6. Some people are nice. Some people are rich. So some people are rich and nice.
    Not valid. Those who are rich might not be the same as those who are nice.
  7. If I drink then I will be happy. If I am happy then I will dance. So if I drink then I will dance.
    Valid.
  8. Every red fish is a fish.
    A trick question! This is a necessarily true statement, but it is not an argument and so not a valid argument.
  9. The services of mobile phone companies are getting worse as there has been an increasing number of complaints against mobile phone companies by consumers.
    Not valid, since the increase in complaints might only be due to increase in the number of mobile phone users.
  10. All capitalists exploit the weak and the poor. Property developers exploit the weak and the poor. So property developers are capitalists.
    Not valid. Perhaps there are non-capitalists who also exploit.
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Module:
Argument analysis

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There are two ways to slide easily through life: to believe everything or to doubt everything. Both ways save us from thinking.

Alfred Korzybski

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