10.1: Implying with Certainty vs. with Probability

From the supposition that all real moons are made of green cheese, does it follow that the Earth's real moon is made of green cheese?

Yes, it follows with certainty. That is, you can be certain of it on the basis of the reason given. However, the reason is an odd one, isn't it? As you can see from that odd argument, the notion of following-from is about what would be true if something else were true, regardless of whether that “something else” is true or not.

In a successful argument, the conclusion follows from the reasons given. The reasons used in an argument are called its premises. The basic premises are those used to establish the conclusion directly rather than by being a premise justifying another premise that supports the conclusion.

Sometimes the inference can make an argument's conclusion follow with certainty, in which case we call the argument deductively valid, but in many other arguments the premises are intended only to make the conclusion significantly more probable, in which case we say the argument is inductively strong.

The conception of deductive validity was introduced back in chapter 2. Here is how it was defined there:

• An argument is valid if the truth of its basic premises force the conclusion to be true.
• An argument is valid if it would be inconsistent for its basic premises to be true and its conclusion to be false.
• An argument is valid if its conclusion follows with certainty from its basic premises.
• An argument is valid if the conclusion would be true whenever the basic premises were true.
• An argument is valid if it has no counterexample, that is, a possible situation that makes the premises true and the conclusion false.

Suppose you have talked to ten baseball fans, and nine out of ten of them said the Baltimore Orioles won't win the next pennant. Suppose that on the basis of this information you conclude that the Orioles won't win the next pennant. You’ve just created an argument whose conclusion is probable, given the premises. Just how probable will be difficult to pin down precisely. If you talked only to fans who have biased views about the Orioles, your information makes your conclusion less probable than if you had talked to a wider variety of baseball fans, including perhaps some baseball experts. But even the experts can’t know for sure.

We have just been considering an inductive argument. Now, let's consider a deductively valid argument, an argument whose reasons, if true, force the conclusion to hold. Suppose all vice-presidents of the United States since Martin Van Buren have secretly been the coordinators of U.S. intelligence operations during their terms of office. Also suppose Andrew Johnson was the vice-president of the United States under President Ulysses S. Grant, a president who served after Van Buren. What would this information imply with certainty? It would imply that Andrew Johnson was once the coordinator of U.S. intelligence operations. Another way of saying this is that if you wanted someone to conclude that Andrew Johnson was once the coordinator of U.S. Intelligence operations, you might consider giving the person the following two reasons:

1. All vice-presidents of the United States since Van Buren have secretly been the coordinators of U.S. intelligence operations during their terms of office.
2. Andrew Johnson was the vice-president of the United States under President Ulysses Grant, a president who served after Van Buren.

Reasons 1 and 2 imply with certainty that Johnson once coordinated U.S. intelligence operations.

These notions of implying with certainty and implying with probability can be defined in terms of the notions of inconsistency and improbability:

Definition

A statement, or group of statements, P implies a statement Q with certainty if Q would have to be true if P were true. More formally, P implies Q with certainty if these two conditions hold: (1) it is logically inconsistent for P to be true without Q also being true.

Definition

A statement, or group of statements, P implies a statement Q with probability if Q would probably be true if P were true. More formally, P implies Q if these two conditions hold: (1) it is improbable for P to be true without Q also being true.

Implying with probability is the vaguer notion. Implying with probability admits of degrees. The probability usually cannot be measured with a number (the fancy phrase is “cannot be quantified”) but instead can only be measured as high, low, very high, and so forth. However, in those rare cases when the probability can be measured with a number, the probability is always less than 100 percent. If it were equal to 100 percent, then we’d say we have a case of implying with certainty instead of implying with probability.

When someone presents an argument intending to convince us of the conclusion, he or she tries to get us to see that an inconsistency or improbability is involved in accepting the premises but not the conclusion. If the arguer can show this, then the argument's premises really do imply their conclusion. The two definitions above try to codify these ideas. In criticizing an argument whose conclusion is intended to follow with certainty, the critic might try to show that there is no inconsistency in accepting the premises while rejecting the conclusion. If the critic shows this, then the argument's premises do not really imply their conclusion with certainty.

It is much easier to show that an invalid argument is invalid than it is to show that a valid argument is valid. To show that an invalid argument is invalid, you can show that there is a counterexample. That is, you can show that there could be a situation in which the premises are all true while the conclusion is false. To show that an argument is valid, you have to show that there couldn’t any counterexample, and that is harder to show.

OK, let’s go back over some of these points from a different perspective. To ask whether a statement implies another statement is to ask an ambiguous question. This question could mean implies with certainty or implies with probability. When implies occurs alone, it is best to assume that both senses might be intended. For example, suppose a person is arrested by the county sheriff. Does this action imply that the person is guilty of the crime he or she is arrested for? The best answer is "yes and no," depending on what is meant by the word imply. It doesn't follow with certainty that the arrested person is guilty, but it does follow with significant probability. This is because it is much more probable that the arrested person is the guilty one than that a typical non-arrested person is.

Even if it is probable that the person is guilty, the mere fact that the person is arrested does not make the probability so high that you as a juror should vote for conviction on the basis of just this fact. You shouldn't vote for conviction until you have been shown that the probability of guilt is high enough to be called "beyond a reasonable doubt." District attorneys try to show that from the evidence of the case it follows with very high probability that the defendant is guilty. The D.A. could never hope to show that the guilt follows with certainty. That would be too high a standard, and no one would ever get convicted.

The following are different, but equivalent, ways of saying that a statement P implies a statement Q with certainty:

• The argument from P to Q has no counterexamples.
• The argument from P to Q is deductively valid.
• Q can be deduced validly from P.
• Q follows with certainty from P.
• Q follows necessarily from P.
• Q is logically implied by P.
• Q follows logically from P.
• Q can be deduced from P.
• P logically implies Q.
• P necessitates Q
• P entails Q.

Let's look at some more examples of "following from" to get a better understanding of what follows from what and whether it certainly follows or just probably follows.

Does the statement "Everybody admires the first lady" imply with certainty that the current secretary of state admires the first lady? Yes, it does. Suppose you then learn that not everybody admires the first lady. Does the conclusion about the secretary of state admiring the first lady still follow with certainty? Yes, it still does. The point can be stated more generally:

Implying is a matter of "what if." We all do a lot of logical reasoning about false situations. Politicians, for example, will say, "This bill isn't law, but what if it were law? If it were law, then it would imply that things would be improved. So let's vote for the bill." The ability to do this kind of "what if" reasoning is valuable. At the beginning of this discussion, we did some "what if” reasoning from two statements: that Andrew Johnson was vice-president under Grant, who served after Van Buren, and that all vice-presidents of the United States since Van Buren have secretly been the coordinators of U.S. intelligence activities. Then we investigated what these statements implied. However, both statements are false. Vice-presidents have not actually been the coordinators of U.S. intelligence activities. Also, Andrew Johnson was not Grant's vice-president; he was Lincoln’s vice-president. Many unstated premises are like that: They're controversial and should not be left unexamined.

Consider this argument: All happy people are rich, all beautiful people are happy, so all beautiful people are rich. Although the key terms happy, beautiful, and rich are vague, the conclusion nevertheless follows from the two premises with absolute certainty.

If I feel certain that a particular person wouldn't have been arrested if the person weren't guilty, that doesn't make it certain that the person is guilty. My psychological state of feeling certain does not make it be certain; it doesn't make the person's being guilty follow with certainty from the fact about being arrested.

This point is worth repeating. The certainty mentioned in the definition of follows with certainty is not a psychological notion; it is a logical notion. That is, certainty is about the logical relationship of support among statements. Someone's feeling certain that Q follows from P is not what makes Q follow from P with certainty. Q follows from P with certainty only if it would be impossible for P to be true while Q is false. Another way of saying this is that Q follows from P with certainty only if P fully supports Q—that is, only if P entails Q.

Julius Caesar did conquer Rome. If this claim were in doubt, some historian might point out that it could be concluded with certainty from these two pieces of information:

The general of the Roman Legions of Gaul crossed the Rubicon River and conquered Rome.

Caesar was the general of the Roman Legions in Gaul at that time.

Notice that if "at that time" were missing from the second piece of information, then the conclusion would not follow with certainty. Here is why. Maybe Caesar was the general at one time, but Tiberius was the general at the time of the river crossing. The more doubt you have that "at that time" is intended if it is not stated explicitly, the less sure you can be in concluding that Caesar conquered Rome.

Exercise \(\PageIndex{1}\)

If an advertisement promotes a sale of clothes that are 100 percent genuine simulated cotton, then it

1. follows with certainty
2. follows with probability
3. doesn't follow

that this is an offer to sell clothes that are essentially all cotton.

Answer

Answer (c). Simulated cotton is not cotton.

Let’s try another concept check about the concept of following with certainty.

Exercise \(\PageIndex{2}\)

What follows with certainty from these three sentences?

Only bears sleep in this house. Goldilocks is not a bear. Smokey is a bear.

1. Smokey does not sleep in this house.
2. Smokey does sleep in this house.
3. None of the above.

Answer

Answer (c). Answer (b) would be the answer if the first sentence had said "all bears" instead of "only bears."