1.8: Patterns of Valid Arguments

Obviously, valid arguments play a very important role in reasoning, because if we start with true assumptions, and use only valid arguments to establish new conclusions, then our conclusions must also be true. But how do we determine whether an argument is valid? This is where formal logic comes in. By using special symbols we can describe patterns of valid argument, and formulate rules for evaluating the validity of an argument. Below we introduce a few patterns of valid arguments. You should make sure that you can recognize these patterns and make use of them in reasoning.

Modus ponens

Consider this argument:

• If copper is a metal, then it conducts electricity.
• Copper is a metal.
• So, copper conducts electricity.

Notice that it has a similar structure compared with this one:

• If there is a storm tomorrow, the park will close.
• There will be a storm tomorrow.
• So the park will close.

Both arguments are of course valid. What is common between them is that they have the same structure or form:

• If P then Q.
• P.
• Therefore Q.

Here, the letters P and Q are sentence letters. They are used to translate or represent statements.

By replacing P and Q with appropriate sentences, we can generate the original valid arguments.

This shows that the two arguments have a common form. It is also in virtue of this form that the arguments are valid, for we can see that any argument of the same form is a valid argument.

Because this particular pattern of argument is quite common, it has been given a name. It is known as modus ponens.

However, don’t confuse modus ponens with the following form of argument, affirming the consequent, which is not valid!

• If P then Q.
• Q.
• Therefore, P.

It is often a mistake to reason with an argument of this form. This is not valid:

• If Jane lives in London, then Jane lives in England.
• Jane lives in England.
• Therefore Jane lives in London.

Here are some other patterns of valid argument.

Modus tollens

• If P then Q.
• Not-Q.
• Therefore, not-P.

Here, not-Q simply means the denial of Q. So if Q means Today is hot., then not-Q can be used to translate It is not the case that today is hot, or Today is not hot.

• If there was a major earthquake, we would have felt it.
• We did not feel anything.
• So, there was no major earthquake.

But do distinguish modus tollens from the following fallacious pattern of argument, Denying the antecedent:

• If P then Q.
• Not-P.
• Therefore, not-Q.

An example:

• If Elsie is competent, she will get a promotion.
• But Elsie is not competent.
• So she will not get a promotion.

Hypothetical syllogism

• If P then Q.
• If Q then R.
• Therefore, if P then R.

Example: If God created the universe, then the universe will be perfect. If the universe is perfect, then there will be no evil. So if God created the universe, there will be no evil.

Disjunctive syllogism

• P or Q.
• Not-P.
• Therefore, Q.

Example: Either the government brings about more sensible educational reforms, or the only good schools left will be private ones for rich kids. The government is not going to carry out sensible educational reforms. So the only good schools left will be private ones for rich kids.

Dilemma

• P or Q.
• If P then R.
• If Q then S.
• Therefore, R or S.

When R is the same as S, we have a simpler form:

• P or Q.
• If P then R.
• If Q then R.
• Therefore, R.

Example: Either we increase the tax rate or we don’t. If we do, the people will be unhappy.If we don’t, the people will also be unhappy. (Because the government will not have enough money to provide for public services.) So the people are going to be unhappy anyway.

Arguing by Reductio ad Absurdum

The Latin name here simply means reduced to absurdity. It is actually an application of modus tollens. It is a method to prove that a certain statement S is false:

• First assume that S is true.
• From the assumption that it is true, prove that it would lead to a contradiction or some other claim that is false or absurd.
• Conclude that S must be false.

For example, suppose someone claims that the right to life is absolute and it is always wrong to kill a life, no matter what. Assume that this is true. We would then have to conclude that killing for self-defence is also wrong. But surely this is not acceptable. If killing an attacker is the only way to save your own life, then most people would agree that this is morally permissible. Since the original claim leads to an unacceptable consequence, we should conclude that the right to life is not absolute. This kind of reductio method is used in many famous mathematical proofs.

Other Patterns

There are of course many other patterns of deductively valid arguments. It is understandable that you might not remember the names of all these patterns. What is important is that you can distinguish the valid ones from the invalid ones, and construct examples of your own.