Some basic information about logic

A redated post.

Argumentation about controversial issues is at the heart of philosophy, and logic is the science of studying arguments. The lecture material (which I didn’t write, just so you know), says some things about logic. The fundamental idea that founded the science of logic was discovered by Aristotle, who realized that arguments could you could distinguish the question of the internal logic of an argument from the question of the truth of the premises. He discovered that some arguments had the characteristic of being such that, if the premises are true, the conclusion must be true, while others do not. The key idea is the idea of validity. There is a common use of the term “valid,” which just means legitimate. So, for example, the you ask me “If I told you the dog ate my homework, would that be a valid excuse” I might well say no that is not valid. But when logicians talk about the concept of validity, what they mean is that the argument is structured in such a way that if the premises are true, the conclusion must be true. Conversely, if an argument is invalid, that means that it is possible that the premises are true and the conclusion is false.

Let’s take this argument:
1) If I say my dog ate my homework, then I have a valid excuse.
2) I say that my dog ate my homework.
3) Therefore, I have a valid excuse.

Even though the excuse isn’t valid in the informal, common-sense sense, the argument is valid in the logic al sense.

Aristotle worked with the logic of classes, so the arguments he considered were arguments like:

1. All dogs are animals.
2. All beagles are dogs.
3. Therefore, all beagles are animals.

This argument is valid, in that if the premises are true, the conclusion must be true. Now the premises are true, so the argument is sound. A sound argument is a valid argument with true premises, and when we have a valid argument with true premises, then the conclusion must be true.

But an argument can be valid if it has false premises. For example:

1. All angels are mortal beings.
2. All persons identical to Bill Clinton are angels.
3. Therefore, Bill Clinton is a mortal being.

In this case the premises are both false and the conclusion is true. But if the argument had had the same structure, you could never have and argument with the premises true and the conclusion false. That is the possibility that validity leaves out. One the other hand, some arguments are invalid, in that the premises can be true and the conclusion false. You have encountered this is what you have heard. There are some statements made, and then there is a leap to a conclusion that is not warranted by the prior statements.
1. All beagles are mammals.
2. All dogs are mammals.
3. Therefore, all dogs are beagles.

This argument has a true conclusion, but the same argument form could be used to prove a very different conclusion;

1. All cats are mammals
2. All dogs are mammals.
3. Therefore, all dog are cats.

Obviously a false conclusion. The conclusion doesn’t follow from the premises.

Now let’s look at the logic of conditional statements. That is what modus ponens and modus tollens are all about. A conditional statement is a statement of the type “If A then B.” That statements means that if A is true, B must also be true.

Modus ponens is:

1. If A then B.
2. A
3. Therefore B.

Let’s take a look at this:

1. If you study, you will pass.
2. You study.
3. Therefore, you will pass.

This arguments is a valid argument. You can’t accept both of these premises and reject the conclusion. The statements is about what will happen if you do study, namely, that you will pass.

But let’s try this argument.

1. If you study, you will pass.
2. You passed.
3. Therefore, you studied.

Uh, no. Just because studying means that you pass doesn’t mean that if you don’t study, you still won’t pass. Remember the irritating kid from high school who used to ace all his classes without studying? This is an argument form called affirming the consequent. It’s not valid. The premises can be true, and nevertheless the conclusion could be false. The structure of the argument is as follows:

1. If A then B
2. B
3. Therefore A.

Here’s another argument form:
1. If you study, you will pass.
2. You didn’t study.
3. Therefore, you didn’t pass.
Again, no. You could be that annoying kid from high school, in which case the premises of the argument are true but the conclusion is false. The structure here is

1. If A then B
2. Not A
3. Therefore not B.

This is denying the antecedent, and it’s invalid.

The last argument form is modus tollens, and it is valid.

1. If A then B
2. Not B
3. Therefore A.

Let’s try the same concepts again.
1. If you study, you pass.
2. You didn’t pass.
3. Therefore, you didn’t study.

This does follow. If studying is sufficient to make you pass, and you don’t pass, the only conclusion we can draw is that you just didn’t study. The structure is:

So two of these argument-forms are valid, and the conclusion follows from the premises. Two of these argument-forms are not valid. These argument forms involve a leap in logic that isn’t justified. The premise could be true and the conclusion not be true. The conclusion doesn’t follow from the premises.

Most arguments can be presented in a valid argument form. In fact, you can present a valid argument for an insane conclusion.

If the moon is made of green cheese, the moon is made of yellow cheese.
The moon is made of green cheese.
Therefore, the moon is made of yellow cheese.

That’s a valid argument, but the conclusion is, well, a little cheesy.

Now let’s take a look a controversial issue.

1. If fetuses are persons, then it is wrong in virtually all cases to abort them.
2. Fetuses are persons.
3. Therefore, it is wrong in virtually all cases to abort them.

The argument is perfectly valid. If the premises are true, then the conclusion must be true. However, both of the premises are controversial. Defenders of abortion often argue that fetuses are nonpersons, so they are abortable. They also maintain that even if fetuses are persons, there are a significant number of cases in which abortion is still justified. However, the valid argument clarifies the debate. An abortion defender has to attack one or both of the premises, in order to defend their position.