Validity

For the rest of this chapter we are going to be talking specifically about evaluating deductive arguments – non-deductive arguments will come later, in Chapter 4.

A deductive argument is one which is intended to guarantee the truth of its conclusion.  The terms we use in evaluating deductive arguments are validity/invalidity and soundness/unsoundness.

First, validity.  A valid argument is one in which if its premises were all true, its conclusion would have to be true as well.  It doesn’t matter (for validity) whether in fact the premises are true.  All that matters for validity is that there should be a connection between the premises and the conclusion such that if the premises were true, the conclusion would also have to be true.   A valid argument is an argument in which it is impossible for the premises to be true and the conclusion false.

The validity of an argument is independent of whether the premises are in fact true.  Thus you don’t need to know anything about the subject that the argument is about in order to judge whether or not it is valid.  To say that an argument is valid is to say something about its structure, not to say anything about its content.  When we talk about validity, we are talking about the first of the two argument evaluation tasks above: we are talking about what the connection is between the premises and the conclusion.

A valid deductive argument is one in which if all the premises were true, the conclusion would also have to be true.

For example

P1. No men are mothers.
P2. Some students are men.
                                      
C. Some students are not mothers.

and

P1. All rugby players sing opera.
P2. Kiri Te Kanawa is a rugby player.
                                     
C. Kiri Te Kanawa sings opera.

Remember that when what you are considering is an argument’s validity, it doesn’t matter whether the premises are actually true.  So it doesn’t matter, for the moment, whether it’s true that no men are mothers or that all rugby players sing opera.  What matters is what connection (if any) there is between the premises and the conclusion.  A valid argument has the strongest possible connection between premises and conclusion – so strong that if the premises were all true, the truth of the conclusion would be guaranteed.

So in the first example above, to see why the argument is valid, think: suppose it’s true that no men are mothers and that some students are men.  Then, must it also be true (on that supposition) that some students are not mothers?

The answer is that supposing those premises to be true, it must also be true that some students are not mothers.  So the argument is valid.

Doing the same for the Kiri Te Kanawa example: Suppose that it was true that all rugby players sang opera and that Kiri was a rugby player.  Then the conclusion would have to be true as well: it would have to be true that Kiri sings opera.  The argument is valid.

You may be thinking at this point, “But that’s stupid! We all know that it’s not true that all rugby players sing opera! So how can the argument be valid?”

Bear in mind that validity is not the only thing you have to take into consideration in deciding whether an argument is a good argument or not: it also matters whether the premises are true. The Kiri argument is valid, but it is still not a good argument. We will get onto this issue in a bit.

In everyday language, the word ‘valid’ is often used to mean ‘true’ or ‘reasonable’. In philosophy generally, and in this course, ‘valid’ has a technical meaning. An argument which is valid is one where it is impossible for the premises to all be true and the conclusion false.

Here are a number of different ways of saying what a valid argument is.  They all amount to the same thing – you can use whichever one or ones help you to understand validity.

A valid argument is one in which:

• it is impossible to have all the premises true and the conclusion false at the same time.
• the conclusion logically follows from the premises.
• if the premises were all true, the truth of the conclusion would be guaranteed.
• if the premises were all true, the conclusion would also have to be true.

​When you ask whether or not a particular argument is valid, you are talking about its structure, not about its content.  Consider the following argument.

P1. All adlers are bobkins.
P2. All bobkins are crockers.
                                        
C. All adlers are crockers.

You can tell it’s valid even if you don’t know what adlers, bobkins or crockers are.  I don’t know what they are, so I have no idea whether it’s true that all adlers are bobkins or that all bobkins or crockers.  Nevertheless, I know the argument is valid: because of its structure, if the premises were true, the conclusion would have to be true too.

You cannot usually tell, from the truth or falsity of the premises and conclusion of an argument, whether it is valid or invalid.  A valid argument can have false premises and a false conclusion, or false premises and a true conclusion, or true premises and a true conclusion.   An invalid argument can too.  There’s just one kind of case in which you can tell about an argument’s validity or invalidity from the truth or falsity of its premises and conclusion, and that is when you have an argument which has true premises and a false conclusion.  A deductive argument with true premises and a false conclusion has to be invalid, because a valid argument by definition can never have true premises and a false conclusion.

So when you are deciding whether or not an argument is valid, don’t think about whether the premises and conclusion are true. Instead, imagine or suppose that the premises are true, and then think about whether that would mean that the conclusion also had to be true.

You might be wondering at this point why we should care about validity.  Since validity has nothing to do with whether or not the premises are true, what’s the point of it?  We shouldn’t accept the conclusion of an argument on the basis of its premises if its premises are obviously false, as they are in the Kiri Te Kanawa argument or in the “sisters and brothers” argument above. So why bother pointing out that the argument is valid, since it’s obviously a really bad argument?

In those cases, perhaps in real life contexts you wouldn’t need to.  There are two conditions a deductive argument needs to satisfy in order to be any good: it has to be valid, and it has to have true premises.  Once we notice that this one has false premises, we can already tell it’s a bad argument, whether or not it’s valid.

But it is important to be able to tell whether an argument is valid in other cases. One kind of case in which it matters is when a deductive argument has all true premises. That’s not enough to make it a good argument: you need to check whether or not it’s valid as well. For example, suppose someone argued like this:

​P1. February is the next month after January.
P2. Grass is green
                          
C. Snow is white.

Although the premises are true, they don’t connect properly to the conclusion – in fact they don’t connect to the conclusion at all.  So they provide no reason whatsoever why you should believe the conclusion.  This shows that just having true premises is not enough (not nearly enough!) to make an argument a good argument.

Another kind of case in which being able to assess validity is of practical importance is when others disagree with you about the falsity of the premises. In some contexts it is useful to be able to point out that, even though you think the premises are false, that even if they were true the argument would be no good.