2.5: Deductive Validity

Once we determine that something is an argument, we can move on to identifying the type of argument it is. The taxonomy of arguments that we will explore over the next couple of chapters is going to be very useful. When we know what type of argument we are looking at, we are able to determine relatively quickly how to evaluate the success of the argument.

Definition of Deductive Validity

In this section, we will learn about deductive validity. It is easy to define this notion, but it is deceptively abstract and slippery. It takes practice to master it.

When the premises of an argument support its conclusion in the strongest possible way, we say that the argument is deductively valid. There are several different, but equivalent, ways to define deductive validity:

1. A deductively valid argument is one such that, if all its premises are true, its conclusion must be true.
2. A deductively valid argument is one such that it is impossible for its conclusion to be false when all its premises are true.

The most common mistake to make regarding validity is to think that this definition says more than it actually does. It does not say anything whatsoever about the premises (taken in isolation from the conclusion) or about the conclusion (taken in isolation from the premises). Deductive validity speaks only to the relationship between the premises and the conclusion. It says that a certain combination of the two, all true premises and a false conclusion, is impossible.

Deductively valid arguments are truth preserving; if all the premises are true, then the conclusion must be true as well. True premises in, true conclusion out. Since there is only one sort of validity, namely deductive validity, we will often speak simply of ‘validity’.

Understanding Deductive Validity

If you spend a few minutes thinking about the following examples, you will begin to get a feeling for what validity really means. Suppose someone makes the following two claims, and that you believe they are true:

1. If it is raining, then the parking lot will be full.
2. It is raining.

Now ask yourself: what can we conclude from 1 and 2? The answer isn’t difficult, but it is important to reflect on it.

3. The parking lot will be full.

Note that you do not need to know whether premises 1 and 2 are true to see that the claim that the lot will be full follows from them. It is this notion of following from that means that this argument is deductively valid.

Now ask yourself: is there any consistent, coherent story that we could imagine in which 1 and 2 were both true, but in which the claim that the lot is full was false? Try it. It can’t be done. If you try it and begin to see that it’s impossible, you are on your way to understanding deductive validity.

Here’s a second example:

1. If Wilbur won the race, he would have called to brag about it.
2. But Wilbur hasn’t called.

What follows from this? Is there any possible way that sentences 1 and 2 could both be true while at the same time the following sentence is false?

3. Wilbur did not win.

What about:

1. If a set is recursive, then it is recursively enumerable.
2. The set you mention is recursive.

What follows from this? Note that you don’t even need to understand the words ‘recursive’ or ‘enumerable’, much less know whether these two sentences are true, to see that sentence 3 follows logically from them.

3. The set you mention is recursively enumerable.

By way of contrast, consider the following argument:

1. If Wilbur won the race, he would have called to brag about it.
2. Wilbur did call to brag.

If we know that 1 and 2 are both true, can we be sure the following sentence is true?

3. Wilbur won the race.

No, we can’t be sure. In this case, it is possible for the two premises to be true while the conclusion is false. For example, Wilbur might be a compulsive liar who called to brag even though he came in last.

Validity: Less is More

As noted above, it is a very common mistake to think that the definition of deductive validity says more than it actually does. It only says what must be the case if all of the premises are true.

1. The definition does not require that either the premises or the conclusion of a valid argument be true.
2. The definition does not say anything about what happens if one or more of the premises are false. In particular, it does not say that if any of the premises are false, then the conclusion must be false.
3. The definition does not say anything about what happens if the conclusion is true. In particular, it does not say that if the conclusion is true, then the premises must be true.

The definition of validity only requires that the premises and conclusion be related in such a way that if the premises are (or had been) true, then the conclusion is (or would have been) true as well.

The definition of deductive validity is hypothetical; if all its premises are true, then the conclusion must be true as well. The ‘if’ here is a big one. It’s like the ‘if’ on the postcard you get that announces: “You will receive ten million dollars from the Publishers’ Clearing House—if you hold the winning ticket.” This does not mean that you do have the winning entry. And similarly, it does not follow from the definition of deductive validity that all the premises of each deductively valid argument are true.

There can be deductively valid arguments with:

1. False premises and a false conclusion.
2. False premises and a true conclusion.
3. All true premises and a true conclusion.

The only combination that cannot occur in a deductively valid argument is all true premises and a false conclusion. This can never happen, because, by definition, a deductively valid argument is one whose form makes it impossible for all its premises to be true and its conclusion false.

Invalid arguments can have any of these three combinations, plus the combination of all true premises and a false conclusion (which is the one combination that valid arguments cannot have).

Further Features of Deductive Validity

1. Deductive validity does not come in degrees. It is all or none.
2. In a deductively valid argument the conclusion contains no new information; there is no information in the conclusion that was not already contained in the premises. We won’t worry about the second feature now, but it will become important when we turn to inductively strong arguments.

Soundness

An argument is sound just in case:

1. It is deductively valid, and
2. It has all true premises.

Once you master the concept of validity (which is tricky), soundness will be easy. The conclusion of a sound argument must be true. Why?

A Note on Terminolgy

• Only arguments can be valid or invalid; sentences or statements cannot.
• On the other hand, only statements or sentences can be true or false (have a truth value); arguments can be neither.