Cogency

I.  What is a cogent argument? 

 

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

 

Consider:

 

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

2.  Benzene in amounts above 10 parts per million is carcinogenic for mice.

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3.  Therefore, benzene in amounts above 10 parts per million is carcinogenic for humans. 

 

What can we say about this sort of argument?  If these premises are true, then I wouldn’t want to be exposed to benzene. 

 

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

 

So how should we characterize such an argument:

 

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. 

 

 

II.  More Examples:

 

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.  In the last 10 years, the average global temperature has increased more than 3 degrees F. 

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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.  Anna Nicole Smith is a woman who was murdered. 

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3.  Therefore, Anna Nicole Smith was murdered by an estranged husband or boyfriend. 

 

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

2.  Joe Biden is a member of the House of Representatives. 

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3.  Therefore,  Joe Biden is opposed to President Obama's job plan.   

 

D.  1.  Most Americans are atheists. 

2.  Dawkins is an atheist. 

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3.  Dawkins is an American. 

 

E.  1.  Most Americans are atheists.

2.  Billy Graham is an American.

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3.  Therefore, Billy Graham is an atheist. 

 

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

2.  And most of the people who are conservative and oppose Obama's foreign policies also oppose Obama's policy concerning Don't Ask Don't Tell. 

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

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4.  Therefore, Smith opposes Obama's policy concerning Don't Ask Don't Tell.  


G.  1.  Most students at CSUS are women.  

2.  Most women live in Asia or Africa.  

3.  Smith is a student at CSUS.

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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.  

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4.  Therefore, Jones is a transfer student.  


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 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.  


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.  
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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 Eizabeth 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. 

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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.

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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.

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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 Obama's foreign policies.  

2.  And most of the people who are conservative and oppose Obama's foreign policies also oppose Obama's policy concerning Don't Ask Don't Tell. 

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

________________________________

4.  Therefore, Smith opposes Obama's policy concerning Don't Ask Don't Tell.  


 

1.  Most As are Bs. 

2.  Most Bs are Cs.

3.  X is an A.

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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 Obama's foreign policies, 99% of them also oppose Obama's policy concerning Don't Ask Don't Tell.  

 

But consider the case where a mere 51% of rural people are conservative and oppose Obama's foreign policies.  And then of that 51% of the general population, only 51% (most) of them oppose Obama's policy concerning Don't Ask Don't Tell.   

 

In that case, of all the rural people in the country, this argument actually asserts that a relatively small percentage, a minority of them, support NCLB.  So if all I know about Smith is that she lives in a rural area, the odds are not good that she also supports NCLB. 

 

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.



What about this argument: 


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

2.  Smith is rich.  

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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.  

Generally, the mere fact that an argument is invalid is not sufficient to indicate that it is cogent.  


A useful video explanation about inductive reasoning:


Induction in Science