Are these arguments invalid?

Question

In a book on statistics, the following were presented as examples of deductively invalid arguments:

A1: No addictive things are inexpensive (Premise 1)
Some cigarettes are inexpensive (Premise 2)
Therefore, some addictive things are not cigarettes (Conclusion)

A2: No cigarettes are inexpensive (Premise 1)
Some addictive things are inexpensive (Premise 2)
Therefore, some cigarettes are not addictive (Conclusion).

For the life of me I cannot see how these arguments are invalid. Am I missing something here?

Answer 1

For the first one, you see that addictive things and inexpensive things are disjoint sets. Making a statement about cigarettes having a nontrivial intersection with inexpensive things doesn't give you any information about addictive things. Addictive things could for example only consist of expensive cigarettes. This possibility is not excluded by the premises.

Answer 2

In A1 it is possible that every addictive thing is an expensive cigarette, as this has not been ruled out by the premises. It is likely that there are some addictive things that aren't cigarettes, but this hasn't been proven, so you can't conclude it.

For A2, it is possible that all cigarettes are expensive and addictive, and that there were some inexpensive and addictive chewing gum that caused the second premise. So it is possible that all cigarettes are addictive, so the conclusion hasn't been proven.

Answer 3

Compare A1 to the following, which is (hopefully) more obviously wrong.

No addictive things are inexpensive (Premise 1)

No sunlight comes at night.

Some cigarettes are inexpensive (Premise 2)

Some light comes at night.

Therefore, some addictive things are not cigarettes (Conclusion)

Therefore, some sunlight is not light.

Answer 4

A nice tool for analyzing these kinds of categorical syllogisms are Venn Diagrams. Let's do this for the first argument. First, draw a Venn diagram for the 3 sets of things involved in the argument:

Now we are going to put the information from the premises into the diagram. First:

No addictive things are inexpensive (Premise 1)

OK, so this says that there is nothing that is both addictive and inexpensive, i.e. the intersection between Inexpensive Things and Addictive Things is empty. This is how we will show that in the Venn Diagram:

(so when an area is blacked out, it means there cannot be anything there)

OK, second premise:

Some cigarettes are inexpensive (Premise 2)

So this premise is saying that there is something that is both a cigarette as well as inexpensive, i.e. there is something in the intersection of Cigarettes and Inexpensive Things. We will indicate that by putting an 'X' in that intersection:

OK, so now we have a diagram that reflects all the information contained in the premises. Now, if the argument were valid, then that would mean that the conclusion should be forced to be true as well:

Therefore, some addictive things are not cigarettes (Conclusion)

Well, on the basis of the diagram, are there indeed addictive Things that are not Cigarettes?

Well, we don't know! The relevant area is not blacked out, but there is not an 'X' either. So, there could be something there, but it is also possible for nothing to be there. In other words, the conclusion could be false, meaning that it is not forced to be true, meaning that the argument is invalid. Indeed, the counterexample is easy to produce:

Notice that in the scenario that this last Venn diagram represents, the premises are both true, but the conclusion is false. So: invalid argument!

Can you analyze A2 using this tool?