Strong, Cogent and Weak Arguments
Let's turn now to Emily from example 2.2.1, who has left the role of mathematician and now assumes the role of biologist.
Emily, who lives in the United States, is classifying swans and notes that every swan she has seen is white. She has seen many swans, as they are her main subject of study. So she forms an argument that goes something like this:
Premise 1: The first swan I observed was white.
Premise 2: The second swan I observed was also white.
Premise 1003: The 1003th swan I observed was also white.
Conclusion: Hence all swans are white.
Now let P be the conjunction of all 1003 premises, and C be the conclusion as noted above. What is the inference type between P and C?
Upon reflection the inference between P and C is not implication, since it is possible that when she sees more swans she will discover a swan which is not white. This means that it is possible that all of the premises to her argument are true, but the conclusion is false. Hence in our classification the argument is invalid. However, it does seem that the fact that all 1003 swans observed as being white is strongly relevant to the conclusion that all swans are white, especially if we also assume the swans she has observed are were randomly selected of the total population. Whenever this is the case, we classify the argument as being strong.
So similar to the classifications of valid - sound - valid but not sound we have the classification of strong - cogent - strong but not cogent. Note this is a similarity in classification, but the characteristics of each classification have important differences. The following example illustrates this.
According to the 2010 census, the population of the 5 largest cities is over 1.5 million.
Phoenix Arizona is the 6th largest city.
Hence Phoenix has a population at least over 1 million.
If one deems the above argument to be strong, then it is an example of a cogent argument since the premises are true according to the 2010 census. While it turns out that Phoenix has a population of 1,445,632 according to the 2010 census, this fact is not important - it could have been otherwise. Cogent arguments are just strong arguments with all true premises. Strong arguments are just arguments whose inference between the premises and conclusion is considered more probable than not. Be careful to keep this in mind, cogent arguments, unlike sound arguments, can have a false conclusion!
To create an example of a strong argument that is not a cogent argument, just change the first premise of the above argument to, "According to the 2010 census, the population of the 5 largest cities is over 3 million". Since this sentence is false, the resulting argument becomes strong but not cogent, since it was created from a cogent argument with one premise changed to be false.
Not all relevant inferences are strong. For example, suppose that Emily, now a budding biologist, has only observed two swans and both were white and from this data concludes that all swans are white. As above, her conclusion could be false, so the inference between the premises and conclusion is only not that of implication, and since she is basing the conclusion on only a few observed cases the inference between the premise and conclusion is only weak.
I once ate at that restaurant on the corner, and become ill the next day.
Today Sally told me she ate there and did not feel good either.
Hence the restaurant at the corner has cooks who do not wash their hands.
Even if we assumed that 2 cases of people eating at a restaurant and getting sick were good evidence that the restaurant prepared bad food, it surely does not give good evidence to come to the more specific conclusion that the cooks do not wash their hands, so Example 2.4.3 is an example of a weak argument.
We now turn to our last classification of arguments based on inference type.
Suppose John is trying to prove some conclusion - in other words John is trying to create an argument whose inference is valid. However he makes a mistake in that proof and his proof turns out to be erroneous (perhaps John is in an advanced mathematics class or logic class where he is asked to prove a theorem and his proof is wrong). These cases will be called erroneous proofs in our classification system, but it should be noted in the traditional classification system based on induction vs deduction these arguments are the only arguments classified as being 'invalid'.
We will examine such arguments in more detail in our next section but for now we will just define the concept and give one example.
Suppose x2 = 25, prove x = 5.
x2 is mathematical shorthand for the product of any real number x times itself.
We are given that x2= 25.
But (5)(5) = 52 = 25.
Therefore x = 5.
Since the above is a mathematical argument, we can easily say it is intended to prove its conclusion. However all the premises can be true but the conclusion false, since the value of x could be -5 rather than 5, so the above is an example of an erroneous proof.
We can now list a diagram that shows all of the possible argument classifications based on the inference of an argument.
Exercise 2.4.5 Classify the following arguments using the most specific classification possible (go as far to the right in the diagram as you can). For example if the inference of an argument is valid and the premises can be shown to be true, then classify the argument as sound rather than just valid.
The Empire State Building is taller than the Eiffel Tower.
The Eiffel Tower is taller than the Statue of Liberty.
Therefore the Empire State Building is taller than the Statue of Liberty.
This is a large hat.
Someone is the owner of this hat.
The owners of large hats are people with large heads.
People with large heads have large brains.
Hence the owner of this large hat has a large brain.
No one was at the front desk of the office.
Over 30 people work at the office.
Therefore everyone was probably on break..
For argument (a), we first need to discover whether the inference is valid or not. We try to show it is not by coming up with some possibility where all premises are true and the conclusion is false - but find that is impossible (think it through)- so the inference is valid.
Next we check to see whether the premises are actually true (if this check is impossible we stop at the classification of valid). A quick check gives the following heights:
- Empire State building: 1,250 feet
- Statue of Liberty: 305 feet
- Eiffel Tower: 984 feet
So all premises are true, hence the argument is sound.
For (b) we check to see if the inference is invalid the same way we do for (a), and discover that if all of the premises were true, the conclusion would have to be true, so again the inference is valid, since it is impossible to have all true premises and a false conclusion. Can we go further? Yes! In general it is not true that owners of large hats are people with large heads, or that people with large heads have large brains. Hence at least one premise is false, so this argument is valid but not sound.
For (c) the check of whether it is possible to have all true premises and a false conclusion turns out to be true. So we know the argument's inference is invalid. The argument is not mathematical or logical in nature and does not show any evidence of being a proof of the conclusion. Experience tells us that there are many reasons why no one might be at the front desk of an office, everyone being on break being just one possibility, so we classify the argument as being weak.
Check your understanding now by taking the following quiz.
Test your understanding: argument classification 2