Validity, Soundness, and Valid Patterns

With valid arguments, it is impossible to have a false conclusion if the premises are all true. 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 which are the rules we should use to decide whether an argument is valid or not? 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.

§1. Modus ponens

Consider the following arguments :

• If this object is made of copper, it will conduct electricity. This object is made of copper, so it will conduct electricity.
• If there is no largest prime number, then 510511 is not the largest prime number. There is no largest prime number. Therefore 510511 is not the largest prime number.
• If Lam is a Buddhist then he should not eat pork. Lam is a Buddhist. Therefore Lam should not eat pork.

These three arguments are of course valid. Furthermore you probably notice that they are very similar to each other. What is common between them is that they have the same structure or form:

Here, the letters P and Q are called sentence letters. They are used to translate or represent statements. By replacing P and Q with appropriate sentences, we can generate the original three valid arguments. This shows that the three 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, which is not valid!

Giving arguments of this form is a fallacy - making a mistake of reasoning. This particular mistake is known as affirming the consequent.

If Jane lives in Beijing, then Jane lives in China. Jane lives in China. Therefore Jane lives in Beijing. (Not valid. Perhaps Jane lives in Shanghai.)

There are of course many other patterns of valid argument. Now we shall introduce a few more patterns which are often used in reasoning.

§2. Modus tollens

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 Betty is on the plane, she will be in the A1 seat. But Betty is not in the A1 seat. So she is not on the plane.

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

If Elsie is competent, she will get an important job. But Elsie is not competent. So she will not get an important job. Not valid. Perhaps Elsie is not very competent, but her boss couldn't find anyone else to do the job.

§3. Hypothetical syllogism

Hypothetical syllogism - If P then Q, If Q then R. Therefore, if P then R.

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.

§4. Disjunctive syllogism

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.

§5. Dilemma

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

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.

§6. Arguing by Reductio ad Absurdum

The Latin name here simply means "reduced to absurdity". Here is the method of argument if you want to prove that a certain statement S is false:

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

Those of you who can spot connections quickly might notice that this is none other than an application of modus tollens. A famous application of this pattern of argument is Euclid's proof that there is no largest prime number. A prime number is any positive integer greater than 1 that is wholly divisible only by 1 and by itself, e.g. 2, 3, 5, 7, 11, 13, 17, etc.

1. Assume that there are only n prime numbers, where n is a finite number : P₁ < P₂ < ... < P_n.
2. Define a number Q that is 1 plus the product of all primes, i.e. Q = 1 + ( P₁ x P₂ x ... x P_n).
3. Q is of course larger than P_n.
4. But Q has to be a prime number also, because (a) when it is divided by any prime number it always leave a remainder of 1, and (b) if it is not divisible by an prime number it cannot be divisible by any non-prime numbers either.
5. So Q is a prime number larger than the largest prime number.
6. But this is a contradiction, so the original assumption that there is a finite number of prime numbers must be wrong.
7. So there must be infinitely many primes.

Let us look at two more examples of reductio:

• Suppose someone were to claim that nothing is true or false. We can show that this must be false as follows : If this person's claim is indeed correct, then there is at least one thing that is true, namely the claim that the person is making. So it can't be that nothing is true or false. So his statement must be false.
• One theory of how the universe came about is that it developed from a vacuum state in the infinite past. Stephen Hawking thinks that this is false. Here is his argument : in order for the universe to develop from a vacuum state, the vacuum state must have been unstable. (If the vacuum state were a stable one, nothing would come out of it.) But if it was unstable, it would not be a vacuum state, and it would not have lasted an infinite time before becoming unstable.

§7. Other Patterns

There are of course many other patterns of deductively valid arguments. One way to construct more patterns is to combine the ones that we have looked at earlier. For example, we can combine two cases of hypothetical syllogism to obtain the following argument:

There are also a few other simple but also valid patterns which we have not mentioned:

Some of you might be surprised to find out that "P. Therefore P." is valid. But think about it carefully - if the conclusion is also a premise, then the conclusion obviously follows from the premise! Of course, this tells us that not all valid arguments are good arguments. How these two concepts are connected is a topic we shall discuss later on.

We shall look at a few more complicated patterns of valid arguments in another tutorial. It is understandable that you might not remember all the names of these patterns. But what is important is that you can recognize these argument patterns when you come across them in everyday life, and would not confuse them with patterns of invalid arguments that look similar.

Exercise #1

Consider the following arguments. Identify the forms of all valid arguments. Here are your choices: modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, dilemma, reductio ad absurdum, valid but not one of the above patterns, invalid.