Today I spent some time reviewing my Formal Logic course for my up coming exam. I came across a section that I have never really explored in any proper depth… the difference between a valid argument and a sound argument. Here go some notes I made…

What is an argument? In this case we are not referring to a verbal fight, but more what we call a set of premise followed by a conclusion.

Before we go further we need to understand what a premise is… a premise is a statement that an argument claims will induce or justify a conclusion. Think of a premise as an assumption that something is true.

So, an argument can consist of one or more premises and a conclusion…

When is an argument valid? An argument can be either valid or invalid.

An argument is valid if, and only if, it is impossible for there to be a situation in which all it’s premises are TRUE and it’s conclusion is FALSE.

It is generally easier to determine if an argument is invalid. Do this by applying the following…

Assume that all the premises are true, then ask yourself if it is now possible for the conclusion to be false. If the answer is “yes,” the argument is invalid. If it’s “no,” the argument is valid.

Example 1…

P1 – Mark is Tall

P2 – Mark is a boy

C – Mark is a tall boy

Walkthrough 1…

Assume Mark is Tall is true and also assume that Mark is a boy. Based on these two premises, the conclusion is also true – Mark is a tall boy, thus the it is a valid argument.

Let’s make this an invalid argument…

Example 2…

P1 – Mark is Tall

P2 – Mark is a boy

C – Mark is a short boy

Walkthrough 2…

This would be an invalid argument, since from the premises we assume that Mark is tall and he is a boy, and then the conclusion goes against this by saying that Mark is short. Thus an invalid argument.

When is an argument sound? An argument is said to be sound when it is valid and all the premises are indeed true (not just assumed to be true).

Rephrased, an argument is said to be sound when the conclusion will follow from the premises and the premises are indeed true in real life.

In example 1 we were referring to a specific person, if we generalized it a bit we could come up with the following example.

Example 3

P1 – All people called Mark are tall

P2 – I know a specific person called Mark

C – He is a tall person

In this instance, it is a valid argument (we assume the premises are true, which leads to the conclusion being true), but the argument is NOT sound. In the real world there must be at least one person called Mark who is not tall.

Something also to note, all invalid arguments are also unsound – this makes sense, if an argument is not valid, how on earth can it be true in the real world.

What happens when the premises contradict themselves? This is an interesting one…

An argument is valid if, and only if, it is impossible for there to be a situation in which all it’s premises are TRUE and it’s conclusion is FALSE.

When premises are contradictory, the argument is always valid because it is impossible for all the premises to be true at one time.

Lets look at an example..

P1 - Elvis is dead

P2 – Elvis is alive

C – Laura is a woolly mammoth

This is a valid argument, but not a sound one.

Think about it. Is it possible to have a situation in which the premises are true and the conclusion is false? Sure, it’s possible to have a situation in which the conclusion is false, but for the argument to be invalid, it has to be possible for the premises to all be true at the same time the conclusion is false. So if the premises can’t all be true, the argument is valid. (If you still think the argument is invalid, draw a picture in which the premises are all true and the conclusion is false. Remember, there’s only one Elvis, and you can’t be both dead and alive.)

For more info on this I suggest reading the following blog post.