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
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
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.
When R is the same as S, we have a simpler form :
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:
- 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.
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.
- Assume that there are only n prime numbers, where n is a finite number : P1 < P2 < ... < Pn.
- Define a number Q that is 1 plus the product of all primes, i.e. Q = 1 + ( P1 x P2 x ... x Pn).
- Q is of course larger than Pn.
- 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.
- So Q is a prime number larger than the largest prime number.
- But this is a contradiction, so the original assumption that there is a finite number of prime numbers must be wrong.
- 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:
- P and Q. Therefore Q.
- P. Therefore P.
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.
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.
- If Jesus loves me, then I love Jesus. I do not love Jesus. Therefore, Jesus does not love me. answer
- Either Jimmy is walking the dog or Cathy is feeding the cat (or both). Cathy is feeding the cat. Therefore Jimmy is not walking the dog. answer
- Either Jimmy is walking the dog or Cathy is feeding the cat. Cathy is not feeding the cat. Therefore Jimmy is walking the dog. answer
- If X is a man, then X is a human being. If X is a human being, then X is an animal. Therefore, if X is a man, then X is an animal. answer
- If I do not have Yellow Tail sashimi, then I will have scallop sushi instead. Now, I am having Yellow Tail sashimi. So I will not have scallop sushi. answer
- If some sheep are black, then some ducks are pink. It is not true that some ducks are pink. Therefore, it is not true that some sheep are black. answer
- Either she is right or she is wrong. If she is right, then he is wrong. If she is wrong, then he is also wrong. Therefore, he is wrong either way. answer
- Paul is a bachelor. Paul is poor. So at least one bachelor is poor. answer
- Either she is in Beijing or she is in Europe. If she is in Beijing, then she is eating lunch. If she is in Europe, then she is sleeping. Hence, either she is eating lunch or she is sleeping. answer
Identify the conclusions that can be drawn from these assumptions. Which basic patterns of valid arguments should be used to derive the conclusion?
- If God is perfect, then God knows what people intend to do in the future. If God knows what people intend to do in the future, then God can stop people from bringing about evil. answer
- If he is dead, then there will be no pulse. If there is no pulse, then the red light will turn on. There is no red light. answer
- Either Krypto is hot or Pluto is hot. If Krypto is hot, then there is no ice on its surface. But there is. answer
- Either you speak justly or unjustly. If you speak justly then men will hate you. But if you speak unjustly the gods will hate you. answer
- Johannes is either in Hong Kong or in Thailand. He is not at home. If he is in Thailand he is staying at the Peninsula. If he is in Hong Kong he is at home. answer
If the following statements are all true, who killed Pam and where was Jones in 1997? Which piece of information is not needed? answer
- Jones was either in HK or in London in 1997.
- If Jones did not kill Pam, then Peter did.
- If Pam died of suffocation, then either Jones killed her, or Pam committed suicide.
- If Jones was in HK in 1997, then Jones did not kill Pam.
- Pam died of suffocation but she did not kill herself.
Is "P or Q, P, Therefore not-Q" a valid argument form?
Suppose someone thinks that there is only a finite number of integers. How would he criticize the proof that there are infinitely many primes? Which step would he reject? answer
Here is a very nice example taken from the philosopher James Pryor :
A computer scientist announces that he's constructed a computer program that can play the perfect game of chess: he claims that this program is guaranteed to win every game it plays, whether it plays black or white, with never a loss or a draw, and against any opponent whatsoever. The computer scientist claims to have a mathematical proof that his program will always win, but the proof runs to 500 pages of dense mathematical symbols, and no one has yet been able to verify it. Still, the program has just played 20 games against Gary Kasparov and it won every game, 10 as white and 10 as black. Should you believe the computer scientist's claim that the program is so designed that it will always win against every opponent?
How would you use the reductio method to argue against the computer scientist? hint
Are arguments of the form "denying the antecedent” (or "affirming the consequent") necessarily invalid?