1.3 Deduction and Induction
Arguments are classified as deductive or inductive based on the inferential claim—the claim about how the premises support the conclusion. Deductive arguments claim to have conclusive support while inductive arguments do not.
1.3.1 Deductive Arguments
An argument is said to be deductive if its conclusion is claimed to necessarily follow from its premises. That is, if it is claimed that since the premises are true or acceptable, the conclusion must also be true or acceptable, then the argument is deductive. We can also define deduction by saying that in a deductive argument, the logical relation between the premises and the conclusion is claimed to be 100% supporting.
Notice that as long as the supporting relation between the premises and the conclusion is claimed to be a matter of logical necessity, the argument is treated as deductive. It is up to us to scrutinize carefully whether the conclusion indeed necessarily follows from the premises. The following are examples of deductive arguments:
| Workers would lose job security if more jobs go overseas. | ||
| More jobs would go overseas if globalization continues. | ||
| Workers would lose job security if globalization continues. | 1.3a |
| God does not exist in space-time | ||
| If something does not exist in space-time, then it cannot have a shape or form. | ||
| Things without any shape or form cannot be either male or female. | ||
| God cannot be either male or female. | 1.3b |
1.3.2 Inductive Arguments
Inductive arguments are more modest when it comes to the inferential claim. It claims only that its conclusion probably follows from its premises. That is, the inferential claim is that since the premises are true or acceptable, the conclusion is likely to be true or acceptable. Put differently, the logical relation between the premises and the conclusion is claimed to be less than 100% supporting.
As with deduction, the inferential claim in an inductive argument should be examined to see if the premises indeed makes the conclusion more likely to be true or acceptable. Here are some examples of induction:
| The windows are broken. | ||
| There are footprints with mud on the floor. | ||
| Some jewels and electronics are missing. | ||
| Some intruders entered the house and burglarized it. | 1.3c |
| It’s been observed that the farther galaxies are from the Earth, the faster they are moving away. | ||
| The universe is expanding. | 1.3d |
| Many people believe that crop circles are created by space aliens. | ||
| Space aliens have visited the earth. | 1.3e |
1.3.3 Proof vs. Confirmation
It’s important to distinguish between deductive proofs and inductive confirmations. The deductive proofs that we are most familiar with are derivations of mathematic theorems from axioms or postulates. On the other hand, scientific theories are accepted as true when they are well confirmed by evidence. Since such a confirmation is inductive, and not deductive, it would be unreasonable to demand a deductive proof for a scientific theory.
We often hear the charge that there is no proof for the theory of evolution. But using the term “proof” in this way is confusing and misleading. If by “proof” one means deductive proof, then it is trivially true that there is no deductive proof for the evolution theory. Nor should there be one. Scientific theories are confirmed by evidence. To demand a deductive proof for the theory of evolution reflects a lack of understanding of how science works.
If by “proof” one means evidence, then it is obviously false that there is no evidence for the theory of evolution. On the contrary, the theory of evolution has been one of the best-confirmed theories in the history of science. It has been accepted as one of our best scientific theories given the wide scope of evidence that confirms and supports it.
To gain a better understanding of how science works and how the theory of evolution has been confirmed, read the article “The Fossil fallacy (pdf)” by Michael Shermer, published in the March 2005 issue of Scientific America.
1.3.4 Validity and Soundness
As emphasized early on, it is very important to critically check the inferential claim made in arguments to assess how well the premises support the conclusion. For deductive arguments, we use the concept validity to do such an evaluation. Validity can be defined in three ways, even though they basically say the same thing:
- A deductive argument is valid if its conclusion indeed necessarily follows from its premises.
- If the premises are true/acceptable, then the conclusion must also be true/acceptable.
- 100% supporting
The third definition seems to be the simplest. A deductive argument is valid if its premises support its conclusion one hundred percent.
| Valid = 100% supporting |
Do not confuse the term “deductive” with “valid.” An argument is deductive and valid when its conclusion is claimed to necessarily follows from the premises, and indeed the supporting relation is airtight. However, an argument can be deductive but invalid. That is, its conclusion is claimed to necessarily follows from the premises, but as a matter of fact the supporting relation is found to be wanting.
Both of the argument (1.3a) and (1.3b) are valid. We will learn how to determine the validity of some deductive arguments in the next two chapters.
The second concept we use to evaluate deduction is soundness. For a deductive argument to be sound, it has to meet two conditions. First, it has to be valid. Second, each and every one of its premises has to be either true or acceptable.
| Sound = valid + true/acceptable premises |
Here is an example of sound argument:
| Platypuses are mammals. | ||
| Platypuses lay eggs. | ||
| Some mammals lay eggs. | 1.3f |
Given that arguments (1.3a) and (1.3b) are valid, they would be sound arguments if all of their premises are true. For each argument, go over its premises one by one and see if each of them is true. If you agree that each and every one of its premises is true, then you would have to accept its conclusion. A sound argument is the most compelling reason one can come up with to convince others to agree with her belief or position. A sound argumet is a proof. A logical person has to accept the conclusion of a sound argument as either true or acceptable. Refusing to accept the conclusion of a sound argument would be illogical, and thus unreasonable and irrational.
As long as a deductive argument fails to meet one of these two conditions, then it is unsound. So if a deductive argument is invalid, then it is unsound. The argument
| If John F. Kennedy was assassinated, then he is dead. | ||
| John F. Kennedy is dead. | ||
| John F. Kennedy was assassinated. | 1.3g |
is invalid despite all of its premises being true. The best way to recognize an invalid deductive argument is to identify its argument form. We will study how to do so in two deductive systems. Another way to see that (1.3g) is invalid is to compare it to the next argument:
| If John F. Kennedy was killed in a plane crash, then he is dead. | ||
| John F. Kennedy is dead. | ||
| John F. Kennedy was killed in a plane crash. | 1.3h |
We notice there is something wrong with (1.3h) because its two premises are true, but the conclusion is false. If it were valid, then according to the definition of soundness it would be sound. But it cannot be sound because its conclusion is false. This shows that (1.3h) cannot be valid. Since (1.3g) shares the same logical structure (form) with (1.3h), it is also invalid. An argument such as (1.3h) is called a counterexample. Using a counterexample is a clear and effective way of showing that any other argument with the same argument form is also invalid.
A deductive argument is also unsound if one or more of its premises are false/unacceptable. For example, the argument
| All Republicans are social conservatives. | ||
| All social conservatives are against gay-marriage. | ||
| All Republicans are against gay-marriage. | 1.3i |
is a valid categorical syllogism (we will learn how to determine the validity of categorical syllogisms in Chapter 2). But it is unsound because its first premise is not true.
If a deductive argument has false premises and a false conclusion, people tend to think that it must be invalid. But this is incorrect. Whether a deductive argument is valid or not is determined by its argument form, and not by whether its sentences are true or acceptable. Compare (1.3i) with (1.3j).
| All Democrats are conservatives. | ||
| All conservatives are against stem cell researches. | ||
| All Democrats are against stem cell researches. | 1.3j |
Notice that the premises and the conclusion of (1.3j) are false; yet it is valid because it has the same argument form as (1.3i).
1.3.5 Strength and Reliability
Recall that the inferential claim in inductive arguments is more modest. It claims only that its premises make its conclusion more likely to be true/acceptable. The degree of support its premises lend on its conclusion is call the strength. It can also be defined in three ways:
- An inductive argument is strong if its conclusion indeed probably follows from its premises.
- If the premises are true/acceptable, then the conclusion is likely to be true/acceptable.
- more than 50% supporting
When an argument is claimed to be less than 100% supporting, then it is inductive. If the supporting relation is in fact greater than 50%, then it is strong. Otherwise it is weak.
| Strong = greater than 50% supporting |
Here is a simple example of strong induction.
| All the ravens observed so far are black. | ||
| The next raven you see will be black. | 1.3k |
Since the premise of (1.3k) is true, the agrument is said to be reliable. A reliable argument is a strong inductive argument with true/acceptable premises.
| Reliable = strong + true/acceptable premises |
If an inductive argument fails to meet one of these two conditions, then it is unreliable. The argument (1.3e) we saw early on is unreliable because it is weak. So is the next example.
| Preliminary studies show that patients treated with Xanafin recover without serious side effects. | ||
| The new drug Xanafin is safe and effective. | 1.3l |
A strong argument may nevertheless be unrealible if it has one or more false/unacceptable premises. The next argument is strong but unreliable because its premise is false.
| Most swans observed so far are black. | ||
| The next swan you see will be black. | 1.3m |
Inductive reasoning by its very own nature can never give us one hundread percent certainty. However, one cannot dismiss inductive reasoning simply because of its lack of absolute certainty. It would still be illogical and irrational to reject the conclusion of a reliable argument. Even in our criminal justice system, the best we can demand is the benchmark of “beyond reasonable doubt.” By the same token, it would be unreasonable to refuse to accept a well-confirmed scientific theory by insisting on the lack of absolute certainty.
The chart below shows how arguments are classified based on the concepts introduced so far:
Exercise 1.3
- Determine whether each of the following arguments is deductive or inductive. If it is deductive, assess its validity and soundness. If it is inductive, estimate its strength and reliability.
- The lights were all out and the car was gone, so we conclude that nobody was home.
- Since many people believe that there are other life forms in the universe, extraterrestrial beings must have visited the earth.
- Women office workers work just as hard as men office workers, and are just as productive. Women office workers should receive the same pay as men in comparable positions.
- Marijuana should not be legalized because it is potentially dangerous and not enough is known about its long-term effects.
- Thinking is taken as the main function of the human mind. So if computers can think, that would strongly suggest that the human mind is a material thing.
- Most Toyota cars are very reliable, so the Toyota you just bought will be reliable too.
- If we had state sponsored religion, then religious freedom would be threatened. To make sure we do not have state sponsored religion, we need to uphold the separation of church and state. This means that religious freedom requires the separation of church and state.
- Given that there are billions of galaxies in the universe and billions stars in each galaxy, there must be other life forms beyond the earth.
- Many people believe that near death experiences are evidence for life after death. Since there are so many reports about the near death experience. There must be life after death.
- To be a sentient being one needs to exist in time. This means that if consciousness and memory are retained in afterlife, then afterlife cannot be a timeless existence.
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