25 Arguments IX – Deductive Arguments

I. Validity and Invalidity

An argument  as we’re using the term is a series of claims, in which some (the premises) are given as reasons that are supposed to establish the truth or probable truth of another (the conclusion). So far we’ve focused on inductive arguments, which aim to offer reasons (premises) to think a conclusion is probably or likely to be true. How well they succeed in this determines their strength or weakness. Lots of our reasoning about the world, whether in our ordinary lives or in science, is inductive. Whenever we generalize about a large group based on a small one (generalization), compare two things and make a prediction (analogical argument), or make a prediction about the future based on our past experience, we are reasoning inductively.

We can also reason deductively, however. When we do that, we aim to provide premises from which the conclusion follows with logical necessity. In other words, if we start with the given premises, then logically we have to accept the conclusion. (You actually did a bit of practice on this without knowing it in the hidden premises exercises.)

That means that:

• A deductive argument is an argument whose conclusion is intended to follow necessarily from the given premises.

As with the definition of an inductive argument, here the definition is about the intent of the argument. That means there is a question as to whether the intention succeeds or not. The basic kind of success (and we’ll see there’s another) we describe in terms of validity.

• A valid argument is a deductive argument that succeeds in having its conclusion necessarily follow from its premises.
• An invalid argument is a deductive argument that fails in having its conclusion necessarily follow from its premises.

Now, think back to our earlier chapter that talked about the importance of being clear on the definitions of words we use when we are reasoning carefully. Here it’s important to see that  we are defining “valid” and “invalid” in a very precise, technical way, one which does not exactly correspond to how we ordinarily use these terms. Often we use “valid” to mean “true” or “legitimate” or “makes sense.” But, as we’ll see, “validity” as we’re using the term here implies nothing about the truth or legitimacy of the claims that show up as either premises or conclusion.

Rather, validity (and invalidity) is strictly about the logical relation between premises and conclusions.  That means that you can have valid arguments that have false premises and invalid arguments that have true premises. I put all that in bold because it’s the one thing students stumble over more than any other.

Now, psychologically we don’t like to do this — i.e., disregard truth to focus on logical structure — but it’s important to do so. For one thing, we often don’t know what is true, but we need to explore “what if?” situations. So, we will often, as the expression has it, suppose for the sake of argument that something is true, and then see what would follow if it is. It’s possible that what we supposed to be true will turn out to be false, but we can still explore it and see what logic would also commit us to if it were true. This is really important in science, medicine, and also in our daily lives. We need to be able to reason well about unknown situations, or explore the logic of views we don’t know about or even believe to be false. (This also applies to situations where we are reasoning inductively.)

To see that the first of these is valid, we have to ask: if those premises are true, could the conclusion still turn out not to be? In other words, could there be a poodle that isn’t a mammal, given what the premises say about dogs, mammals, and poodles? The answer is no. Given the premises as stated, there couldn’t be a non-mammalian Poodle.

To see that the second argument is valid, first think about its first premise, which is a conditional (an “if…then…” claim). Conditionals tell you that the first part (the “if…”) is a necessary condition of the latter (the “then…”). It asserts that, if you know Mel is a dog, then you also know she likes to chew on bones. Then the second premise tells you that you do in fact know Mel is a dog. And so you, taken together, that means you know she likes to chew on bones. Given those premises, that conclusion has to follow, even if one or both of the premises are false.

So, in both cases, you can think through these and see that they are valid. As it happens, they also are instances of patterns of arguments that are always valid, no matter what is being talked about. More on the patterns below.

In both cases, what makes these invalid is that, given the stated premises, there’s not sufficient information to justify drawing the conclusion.

In the first example, the premises only tell you that those mammals which are dogs are not ugly; but, you aren’t told anything about any of the other kinds of mammals besides dogs. So, given those premises, there could be other mammals that are ugly, even if dogs aren’t.

In the second example, you know that one possible explanation of Mel liking to chew on bones is that she’s a dog (that’s what the first premise gives you).  But the premises don’t rule out all other possibilities that would account for her liking to chew on bones. Maybe, for instance, Mel’s a rat — rats also like to chew on bones. So, given the information in the premises, you don’t know enough to conclude that she definitely is a dog. (So, you could mount a reasonably strong inductive  argument  based on these premises, one that concluded “Mel might  be a dog;” but if you were going for a valid deductive argument, it wouldn’t work.)

• In general, then, if you can find a way to explain how the conclusion could be false even if the premises are true, then you’ve shown the argument is invalid.

II. Argument Patterns

Each of the four arguments in the two example boxes above fits a pattern of reasoning. There are, in fact, many other such patterns you can learn to identify that tell you if an argument is valid or not. You can take a more advanced logic class (we offer Phil-P 250) to learn some of these — which is a good idea of you hope to go to law school, as this sort of logical reasoning is central to what you have to do on the LSAT test. But, because of time, we’re not going to get into any besides these four. We could have picked other ones, but these are fairly common, and they give you the idea of what it means to have a logical pattern in an argument, which is the main goal.

So, here are the patterns. Next to each are two examples (one with true premises, one with false premises) but the idea is that, for each, no matter what you substitute for the variables (the capitalized letters), you have a logically valid argument (even if the premises themselves are false).

III. Soundness and Unsoundness

To assess an argument’s validity requires that you abstract from the truth of the claims in it. (Or, equivalently, it requires you to talk about their truth as only hypothetical: if the premises are true…  This you can do even if you know they are false). To repeat a point already made a couple of times, this means that arguments can be logically good – valid – even with false premises.

But it also means that there is a further way in which a deductive argument can be good: if it is not only valid but also has all true premises. When this happens, it guarantees that the conclusion is true too, since valid arguments are precisely those whose conclusions follow necessarily from their premises. True premises and a conclusion that necessarily follows from them means the conclusion inherits the truth of the premises.

Logical goodness we describe in terms of validity; when we get validity and truth both, we talk about soundness.

• A sound argument is a valid argument with all true premises. Such an argument guarantees that its conclusion is true as well.
• An unsound argument is a deductive argument with at least one false premise.

Crucial Point: not all the premises have to be false for an argument to be unsound. Just as it only takes one thing to break for your car not to run, so too an argument need only have one faulty part (premise) for it to fail.

So, how can you tell if a particular argument is sound or unsound?

You should ask about validity first, since you need to know if an argument is valid in order to see if it is sound (an invalid argument can’t, by definition, be sound). Plus, if a person isn’t reasoning logically, then it doesn’t matter what conclusion they draw from their premises, they won’t be entitled to it (at least not without more work).

If an argument is valid (logically successful), then you can go on to ask if it is sound, i.e., whether it has true premises. When you do this, ignore the conclusion: if you’ve already established validity, then what matters is whether the premises are each separately true. If they are, then the conclusion will be as well. If they are not, then even if the conclusion is true it will have to be because of reasons other than those stated by the premises.

What it takes to establish the truth of the premises depends entirely on the specific claims they make. If the claims are (non-evaluative) scientific ones, then you need science to establish them. If they are moral or religious (evaluative) claims, then it might not be so easy to establish truth in a way that all can accept  — but that doesn’t make them mere opinion! It is still possible to reason about such claims and do better or worse in finding arguments to support them.

Because what it takes to establish truth varies so much, most of your exercises will only ask you to evaluate validity, not soundness. There are surprisingly few things that everyone can be counted on to know are true or false, so generating examples isn’t so easy.