4.5: Logical Form

Since Aristotle, the first major logician, it’s been recognized that deductive validity is a matter of an argument’s logical form. We can display an argument’s logical form by replacing all but the logically operative vocabulary with symbols (we’ll use capital letters for this). So, consider the logical form of a few of our examples so far.

1. All planets are stars.
2. All stars are bodies that shine steadily.
3. All planets are bodies that shine steadily.

This argument has the following form:

1. All P are S
2. All S are B
3. All P are B

Any argument that has this logical form will be valid. Here is one example:

1. All fish are vertebrates.
2. All vertebrates are animals.
3. So, all fish are animals.

Remember, validity is just a standard of support. Validity does not assume true premises or a true conclusion. So even though it sounds a bit “off,” this argument is also valid:

1. All red things are bricks,
2. All bricks are rocket ships.
3. So, all red things are rocket ships.

Of course, this argument sounds silly. Both premises are ridiculously false. But then any possible world where both premises are true would be a possible world where all red things are rocket ships. The argument is valid in virtue of its valid logical form. Now consider this familiar argument:

1. If Socrates is human, then Socrates is mortal
2. Socrates is a human.
3. Therefore, Socrates is mortal

This argument has the following logical form:

1. If H, then M
2. H
3. M

Similarly, any argument that has this logical form will be valid. Plug any declarative sentences you like in for H and M and you will have a valid argument. The premises might be false, or even absurd, but it will remain the case that any way the world could be that makes both premises true will also make the conclusion true. Once you appreciate how deductive validity is a function of the logical form of an argument, it soon becomes clear that a valid argument can be constructed for any possible conclusion, true, false, or completely absurd. So, for instance:

1. If pigs fly, then the oceans will dry up.
2. Pigs fly
3. Therefore, the oceans will dry up.

So, you might be wondering what the point of all this silliness is. It’s partly to limber up your logical sense and help you recognize that logical validity is only about what follows from what, not about what is in fact true or false. Of course, the oceans aren’t going to dry up. But if both premises were true, then the conclusion would follow logically and also be true. But there is a further point to the hypothetical silliness. The fact that the conclusion of the “pigs fly” argument is absurdly false is a good indicator that at least some of the premises of this valid argument are also false. And this is a very useful thing to recognize. To see this, let’s look at another valid argument pattern that captures what we’ve just said about the pigs fly argument:

1. If P, then C
2. Not C
3. So, not P

This is a valid pattern of reasoning that we use routinely. For instance:

1. If I have milk, then it will be in the fridge
2. There’s no milk in the fridge
3. So, I am out of milk.

Now notice how we used this pattern of reasoning in our analysis of the “pigs fly” argument. It is valid, which means that if its premises are all true, then its conclusion is true. But obviously, its conclusion isn’t true. So now we can confidently infer that its premises are not all true.