﻿ [A06] Validity and relevance

Warning: This tutorial talks about some puzzling features in the definition of validity. This is more for those who are interested in logic, and you need not worry too much about this for the purpose of learning critical thinking. So feel free to skip this part if you want.

Recall our definition of validity:

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.

This definition is supposed to capture what is meant by one thing following from another. However, there are some consequences which might seem counterintuitive.

§1. Circularity

The first puzzling feature is that all circular arguments are actually valid. Here, we might take a circular argument as an argument where the conclusion also appears as a premise. Here are two examples:

God exists.
Therefore, God exists.

The moon is made of cheese.
The sun is made of tofu.
Therefore, the moon is made of cheese.

These arguments are valid because it is simply not possible for the premises to be true and the conclusion to be false at the same time. If all the premises are true in some situation, then the conclusion will also be true in the same situation since it is one of the premises!

This might seem weird, but if we think about this, perhaps this is acceptable. We can argue that given any statement, if it is true, then obviously it does follow that it is true. So we should accept that every statement follows from itself. We just need to remember that valid arguments need not be good arguments.

§2. Necessarily true conclusions

A more counterintuitive consequence of our definition of validity is that any argument with a necessarily true conclusion is valid, and it does not matter what the premises are and whether they are true or false:

The moon is made of cheese.
Therefore, either it is raining in Tokyo now or it is not.

1+1=3.
Therefore, if God exists, then God exists.

These examples seem weird because the premises have nothing to do with the consequences. They talk about completely different things. So how can these arguments be valid?

But note that their conclusions are all necessarily true. The first conclusion is true whether or not it is raining in Tokyo, and the second conclusion is true whether or not God exists. They are logical truths and it is impossible for them to be false. But this means in both cases, it is not possible for the premises of the argument to be true and the conclusion to be false at the same time. So they are valid!

This might seem a bit strange to you. You might even think there is something wrong with the definition of validity. Some philosophers and logicians have indeed argued we need a better definition of validity that can ensure that the conclusion is relevant to the premises. But this is quite controversial and very complicated, so we will stick with our simpler definition here. If you are interested you can read about how relevance logic is supposed to deal with this problem.

§3. Necessarily false premises

A related feature about the definition of validity is that any argument with inconsistent premises will be valid, regardless of the conclusion:

The moon is made of cheese.
The moon is not made of cheese.
Therefore, I am the most intelligent person in the universe.

The number 1 is not the number 1.
Therefore, Paris is the capital of France.

It is easy to easy why these arguments are valid. It is not possible for the premises to be true at the same time. So of course it is also not possible for the premises to be true and the conclusion to be false at the same time!

For the purpose of critical thinking in everyday life, we do not need to be too concerned about these cases since we seldom have to seriously deal with such arguments. But thinking about these cases is important in the development and study of logic and philosophy. We discussed these cases not in order to confuse you, but to show that even very simple concepts, like the idea of one thing following from another, can raise unexpected issues when we think about them more deeply.

Are these arguments valid?

1. John loves Mary. So John loves Mary.
2. John loves Mary. So Mary is loved by John.
3. 1>2. So 2>1.
4. All triangles have three sides. So all pigs have four legs.
5. All pigs have four legs. So all triangles have three sides.
6. All squares have three sides. So pigs have four legs.
7. All squares have four sides. So all triangles have three sides.