Cogency

I. What is a cogent argument?

Some arguments are not valid, but their logical structures are good in another way.

Consider:

1.Most of the people in the crowd are wearing coats.

2. Smith is a person in the crowd.

_________________________________

3. Therefore, Smith is wearing a crowd.

or,

1. Most of the substances that are carcinogenic for mice are carcinogenic for humans.

2. Benzene is carcinogenic for mice.

_______________________________________________

3. Therefore, benzene is carcinogenic for humans.

What can we say about these arguments? In the first, it seems like it's a pretty good bet that Smith is wearing a coat. And in the second, if the premises are true, then I wouldn’t want to be exposed to benzene.

But the arguments are invalid. The premises, if they were true, would not guarantee the truth of the conclusion.

So how should we characterize such an argument? It has some logical virtue; it is not ill-formed, as we will learn in a few weeks. So we need another category. We will call these arguments cogent:

An argument is cogent when these two conditions apply:

1. It is invalid.

AND

2. The premises, if they were true, would make the conclusion likely to be true.

The idea that cogent or inductively valid arguments have a logical structure such that the premises provide probabilistic grounds in support the conclusion. Valid arguments have this property: the premises, if they were true, would guarantee the truth of the conclusion. But we need to be able to identify arguments that are not deductively valid, but their premises still support their conclusion by making it probably true. Notice in the examples below that there is a general premise in every argument that has some language like, "most," "in almost every case," "vast majority," "probably," and so on. These premises make a claim about the majority of a class or group of objects having a particular property, like "Most Americans are female." "Most" means greater than 50%. If these premises said "some," or "several," they would not provide support for their conclusions. From the fact that "Some Americans are billionaires," for example, and the fact that "Matt McCormick is an American," we cannot reliably conclude that "McCormick is a billionaire. But this argument is cogent and it does provide probabilistic support for its conclusion:

1. Most Americans believe that the Earth is round.

2. McCormick is an American.

__________________________

3. Therefore, McCormick believes the Earth is round.

The first premise says that the majority of a group of things (Americans) possess a property (believing that the Earth is round.). So if we know that McCormick is in that group, as premise 2 asserts, then we can conclude that he probably believes the Earth is round. It's not a guarantee. He might be in the minority of people who do not believe it. But if these premises were true, and we knew nothing else, we could conclude that 3 is probably true. So that's a cogent or inductively valid argument.

II. More Examples: see if you can figure out which of these are cogent and which are not.

A. 1. In almost every case, increases in average global temperature greater than 3 degrees F have resulted in a rise of sea levels of more than 10 feet.

2. The average global temperature has increased more than 3 degrees F.

_____________________________

3. Therefore, the sea levels will rise more than 10 feet.

B. 1. The vast majority of murders of women are committed by an estranged husband or boyfriend.

2. Ann is a woman who was murdered.

_______________________________

3. Therefore, Ann was murdered by an estranged husband or boyfriend.

C. 1. The majority of the members of the House of Representatives are opposed to the President's job plan.

2. Nancy Pelosi is a member of the House of Representatives.

____________________________

3. Therefore, Nancy Pelosi is opposed to the President's job plan.

D. 1. Most Americans are atheists.

2. Dawkins is an atheist.

______________________

3. Dawkins is an American.

E. 1. Most Americans are atheists.

2. Billy Graham is an American.

___________________________

3. Therefore, Billy Graham is an atheist.

What about "Most of a most" cases, where there are more than one probability claim?

F. 1. Most of the people who live in rural parts of the country are conservative and oppose the President's foreign policies.

2. And most of the people who are conservative and oppose the President's foreign policies also oppose the President's domestic policies.

3. Smith lives in a rural part of the country.

________________________________

4. Therefore, Smith opposes the President's domestic policies.

G. 1. Most students at CSUS are women.

2. Most women live in Asia or Africa.

3. Smith is a student at CSUS.

_______________________

4. Therefore, Joan lives in Asia or Africa.

H. 1. Most students at CSUS are women.

2. Most CSUS students who are women are transfer students.

3. Jones is a student at CSUS.

________________________

4. Therefore, Jones is a transfer student.

I. 1. Humans with the abnormality in the G4S3 gene sequence have a greater than 50% chance of being over 6 feet tall.

2. Susan has the abnormality in the G4S3 gene sequence.

____________________________

3. Therefore, Susan has a greater than 50% chance of being over 6 feet tall. (or, Probably, Susan will be over 6 feet tall.)

III. When is an argument not cogent?

Notice argument D and argument F above.

Argument D, premise 1 asserts that some percentage greater than half of a group A has property B. “75% of dogs in the pound have had their rabies shots” has the same structure.

But a claim like “Most As are Bs,” or “Most A things have property B,” do not assert that most or all things with B property are also A. Consider the true sentence: “Most of the students at CSUS are women (54%)”. This is not equivalent to “Most of the women (in the world) are students at CSUS.” (There are over 6 billion people on the planet, and about 28,000 people at CSUS.

So in general, the sentence “Most As are Bs,” does not mean and is not equivalent to “Most Bs are As.”

So if I also know that some individual X is a member of the As--Roscoe is a dog in the pound, or Elizabeth is a student at CSUS—then, assuming that the “Most As are Bs,” sentence is true, then the conclusion, “Roscoe has had his rabies shot.” And “Elizabeth is a woman,” is likely to be true.

G. 1. 75% of the dogs in the pound have had their rabies shots.

2. Roscoe is a dog in the pound.

__________________________

3. Therefore, Roscoe has had his rabies shot. [COGENT]

H. 1. Most of the students at CSUS are women.

2. Elizabeth is a student at CSUS.

_____________________________

3. Therefore, Elizabeth is a woman. [COGENT]

But, these arguments are not cogent. Their premises, even if they were true, would not make their conclusions likely to be true:

I. 1. 75% of the dogs in the pound have had their rabies shots.

2. Roscoe has had his rabies shot.

__________________________

3. Therefore, Roscoe is a dog in the pound. [NOT COGENT]

J. 1. Most of the students at CSUS are women.

2. Elizabeth is a woman.

_____________________________

3. Therefore, Elizabeth is a student at CSUS. [NOT COGENT]

So argument D above is not cogent.

What about argument F:

F. 1. Most of the people who live in rural parts of the country are conservative and oppose the President's foreign policies.

2. And most of the people who are conservative and oppose the President's foreign policies also oppose the President's domestic policies.

3. Smith lives in a rural part of the country.

________________________________

4. Therefore, Smith opposes the President's domestic policies.

The pattern here is:

1. Most As are Bs.

2. Most Bs are Cs.

3. X is an A.

____________________

4. Therefore, X is a B.

There could be cases where if these premises were true, they would make the conclusion likely to be true. If the “Most” in premise 1 was in fact 99%. And then of those rural people who are conservative and oppose the President's foreign policies, 99% of them also oppose the President's domestic policies.

But consider the case where a mere 51% of rural people are conservative and oppose the President's foreign policies. And then of that 51% of the general population, only 51% (most) of them oppose the President's domestic policies.

In that case, of all the rural people in the country, this argument actually asserts that a relatively small percentage, a minority of them, oppose the domestic policies. So if all I know about Smith is that she lives in a rural area, the odds are not good that she also opposes the President's domestic policies.

So these “Most of a most,” arguments are not cogent. From these premises alone, we cannot infer that the conclusion is likely to be true when the premises are assumed to be true.

In general, when the conclusion of an argument is built on the compounding on probabilities, it is not going to be cogent. Here's why:

The multiplication rule in probability theory says that if the odds of an individual's having one property are, say, 60%, and the odds of having some other independent property are 60%, then you multiply the two probabilities to determine the likelihood that the individual has both: .6 x .6 = .36 or 36%

So it is not probable that an individual has both properties, even it is probable that the individual has each of the properties separately.

So examples F, G, and H are NOT cogent.

What about this argument?

K. 1. If a person inherits a huge fortune from a dead relative, then they will be rich.

2. Smith is rich.

____________________

3. Therefore, Smith inherited a huge fortune from a dead relative.

This argument is invalid. There are many ways to get rich, so if all we know about Smith is that she is rich, we can’t infer that she got it through inheritance.

But is it cogent? That is, if these premises were true, would they make the conclusion likely to be true? If you know that inheritance produces wealth and that Smith is wealthy, can you infer that Smith probably got her money from inheritance?

No, we can’t. Think about it this way. How many other ways are there to get rich? Hundreds? Thousands? Of all the rich people in the world, what percentage of them got that way by inheritance? Maybe a lot of them. But this argument, especially premise one, does not give us that information. It doesn’t tell us the rate at which rich people inherited their wealth. If it said, “Most rich people get their money through inheritance.” And then we added that Smith is rich, then we could infer cogently that Smith probably got her money through inheritance. But the argument, as it stands, doesn’t give us that information.

The mere fact that an argument is invalid is not sufficient to indicate that it is cogent.

And the arguments that we've been considering with conditionals (If P then Q) claims in them, or universal generalizations (All As are Bs) generally won't be cogent arguments unless there are other premises that give us the probabilistic grounds, or "most," or "majority" language that we have seen is necessary for a cogent argument. So this argument:

1. If you are a politician, then you are a liar.

2. Trump is a liar.

_________________

3. Therefore, Trump is a politician.

is just ill-formed. It is neither valid nor cogent. It follows an invalid, fallacious argument pattern that we have seen before. And it also does not provide probabilistic grounds to support its conclusion. These premises don't guarantee this conclusion, nor do they make the conclusion likely. (Also notice that this argument is ill formed independent of the actual truth of the premises and conclusion.)

A useful video explanation about inductive reasoning: