Logic: A Quick Remedial Lesson

Editor's Note: Matt Teichman, our guest on the Frege episode, has been good enough to provide this primer on logic for our listeners who've not already had to sit through a class on it or who might need a refresher:

One of the things we said at the beginning the episode on Frege was that he is the father of modern logic.  But what is logic, anyway?  For those of you who never heard of logic before, here is a quick primer.

Logic is the study of what philosophers call valid reasoning.  In everyday conversation, valid just means 'good' or 'appropriate.'  But in logic, valid has a special technical meaning.  To see what it means, let's look at an example of an argument:

Argument 1:

All people drink water.
Matt is a person.
-----------------
Therefore, Matt drinks water.

Argument is also a term we're going to use in a special technical sense.  Here, rather than referring to e.g. a dispute between me and my wife about whether to get a new car, it refers to a line of reasoning meant to convince someone of something.  More specifically, we'll think of an argument as a sequence of sentences broken down into two parts: first come the premises, then comes the conclusion.  (You can also have an argument with just one premise, or even an argument with no premises!  But all arguments need to have a conclusion.)  In the above example, the premises are All people drink water and Matt is a person, and the conclusion is Matt drinks water.

Anyway, let's get back to validity.  An argument is valid just in case it is impossible for its premises to be true and for its conclusion to be simultaneously false.  Consider the above example: if all people drink water, and Matt is a person, then there's simply no way that Matt could fail to drink water.  It's absolutely, positively guaranteed.

Note that an argument can be valid even though it has false premises.  Here is an example of such an argument:

Argument 2:

All giraffes have short necks.
Barack Obama is a giraffe.
-----------------
Therefore, Barack Obama has a short neck.

This argument is valid, even though its premises are obviously false.  Why?  Because it is impossible for all giraffes to have short necks, for Barack Obama to be a giraffe, and yet for Barack Obama not to have a short neck.  If the first two sentences are true, the conclusion has to be true.  As we just saw, however, that doesn't mean the premises actually are true.

But don't we usually want our arguments to have true premises?  Isn't the whole point of making an argument to say something true?  Well, don't despair--logicians have another word for arguments which, in addition to being valid, also have true premises.  A valid argument with true premises is called a sound argument.

One reason that logicians make a distinction between arguments that are sound and arguments that are merely valid is that this distinction corresponds to two ways of criticizing an argument.  You could criticize an argument by saying that the conclusion doesn't follow from the premises, or you could criticize it by saying that some (or all) of the premises are false.  Argument 1 ( given above) is both sound and valid, and thus immune to both of these criticisms.  Argument 2 (also above) is valid but not sound, which means that you could reasonably criticize it for having false premises, even though the conclusion follows from the premises.  Here's a third argument that has true premises but isn't valid:

Argument 3:

All people drink water.
Matt drinks water.
-----------------
Therefore, Matt is a person.

Argument 3 has true premises: all people drink water, and Matt is a water drinker.  But since it is theoretically possible for all people to drink water, for Matt to drink water, and yet for Matt not to be a person, Argument 3 is invalid.

If you want to show that it is invalid, a simple method for doing so is to give a counterexample.  Think of a counterexample as an imaginary situation in which the premises are true but the conclusion is false.  If you show that such a situation can be imagined, then you've shown that the argument you're talking about is invalid.

Here's what a counterexample to Argument 3 might look like.  Imagine that Matt, rather than being a human being, was a lemur.  A lemur who drank lots of water.  In that imaginary situation, it would still be true that people drank water, and that Matt drank water, but it wouldn't be true that Matt was a person.  So Argument 3 is invalid.

Now, you've probably heard through the grapevine that logicians like to write things using funny symbols.  At first, this is likely to seem odd or perverse.  But to understand why logicians like to write things using symbols, it helps to consider what it takes for an argument to be valid.

Anyone who thinks about Argument 1 a little can see that it's valid.  What's remarkable about this fact is that you don't have to know anything about human physiology or have any intimate knowledge of Matt to know that it's valid.  You can tell that the argument is valid simply because the words in it are arranged a certain way.  The argument is valid because it falls into a certain abstract pattern.  The same goes for Argument 2.  You don't need to know anything about giraffes or Barack Obama to be able to see that it is valid.

As a further way of driving the point home, consider the following nonsense argument:

Argument 4:

All squinks are skwonks.
Stitchy is a squink.
-----------------
Therefore, Stitchy is a skwonk.

We obviously don't have any special knowledge of squinks or skwonks, because those are words I just made up.  Stitchy, likewise, is a character I just made up.  Nonetheless, we can tell from this argument that whatever squink and skwonk mean, the argument is still valid.  Whoever Stitchy turns out to be, the argument is still valid.  We can tell this just by looking at the shape of the argument.

That's kinda nuts, isn't it?

The big lesson to draw here is: the principles that determine whether an argument is valid often have to do with the abstract pattern the argument falls under.  If it falls under one abstract pattern, it will automatically be valid, no matter whether it's talking about people or giraffes.  And if it falls under another, it will automatically be invalid, no matter whether it's talking about water drinking or long necks.

So the reason logicians write arguments using funny symbols rather than in English (like we just did) is that the symbols are a way of representing these abstract patterns.  Here's how a logician would write Argument 1 down, using a symbolic language called the predicate calculus:

∀x(Person(x) --> DrinksWater(x))
Person(Matt)
-----------------
DrinksWater(Matt)

Never mind what these symbols mean for now.  Basically, the point of writing Argument 1 in the predicate calculus is to say that any argument of the following form will turn out valid, no matter what you stick in the blank spaces:

All ____ s drink water.
Matt is a ____.
-----------------
Therefore, Matt drinks water.

...and that any argument of this form will also turn out valid, no matter what you stick in the blank spaces:

All people ____.
Matt is a person.
-----------------
Therefore, Matt ____s.

So when people say that Frege was the father of modern symbolic logic, they mean that he invented a symbolic language--the predicate calculus--which essentially allows us to look at abstract argument patterns and consider whether they are valid without worrying about whether they're discussing giraffes, people, or anything else.

There are lots of other cool things you can do with the predicate calculus, but that's a topic for another post.

-Matt Teichman