Part II: Examples

The definitions above are great, but it’s much easier to see what they really mean when we look at some examples. These examples also help us to see the different types of propositions that can be involved in each of the relations. 

1. Contradictory Relation: 

• This relation exists between Universal Affirmative and Particular Negative propositions. If “all gas stations are dirty” (UA) is true, then “some gas stations are not dirty” (PN) must be false. Conversely, if “some gas stations are not dirty” (PN) is true, then “all gas stations are dirty” (UA) must be false. Euler Circles will help us intuitively visualize this:

It is evident that one of these being true implies that the other is false. Suppose that what’s represented by the right diagram (UA) is true. If (UA) is true, then all members of “gas stations” are also members of the category “dirty” (so no members of “gas stations” are not members of the category “dirty.”) But then the left (PN) is explicitly false, since it claims there are members of the category “Gas stations” which are ‘outside’ the category “dirty”. Now, suppose that (UA) here is false. Then it is not the case that all members of “gas stations” are also members of the category “dirty.” In other words, some members of “gas stations” must be ‘outside’ of “dirty”. This is explicitly what the left diagram states, so (PN) would be true.

The contradictory relation also holds between Universal Negative and Particular Affirmative propositions. If “no T-shirts are stylish” (UN) is true, then “some T-shirts are stylish” (PA) must be false. Conversely, if “some T-shirts are stylish”(PA) is true, then “no T-shirts are stylish” (UN) must be false.

Suppose the right diagram (UN) is true. Then there is no overlap between the categories “T-shirt” and “stylish.” Then the left diagram (PA) is explicitly false, since it says there is such an overlap. Now, suppose that (UN) is false. Then the two categories are not completely separate (there are some objects that are members of both of them). But then (PA) is explicitly true.

2. Contrary Relation: 

• This relation exists between Universal Affirmative and Universal Negative propositions. “All children are cute” (UA) and “no children are cute” (UN) cannot both be true. That said, both propositions can be false—it may be the case that some, but not all, children are cute! 

Now it is clear that the left (UA) and the right (UN) cannot both be true at the same time. It can’t both be the case that all of the category “children” is contained within “cute” and that the two categories are completely separate at the same time. However, both (UA) and (UN) can be false at the same time. There could be overlap between the categories, which would make both (UA) and (UN) false. 

3. Subcontrary Relation: 

• This relation exists between Particular Affirmative and Particular Negative propositions. “Some candy bars are free” (PA) and “some candy bars are not free” (PN) cannot both be false. However, both propositions have the potential to be true (if we only lived in a more generous world than the one we actually live in!). 

We won’t need diagrams for this one, since we can simply use the ones already given. Intuitively, both (PA) and (PN) can be true at the same time. This would just mean that the diagram would look different from either given in (2) above, even after all information is known. By what we learned in (1), if “some candy bars are free” (PA) is false, then “No candy bars are free” (UN) is true. Likewise, if “some candy bars are not free” (PN) is false, then “All candy bars are free” (UA) is true. But, by what we learned in (2), (UN) and (UA) cannot both be true! Therefore, (PA) and (PN) cannot both be false!

4. Subalternation:

• This relation exists between Universal Affirmative and Particular Affirmative propositions. “All chocolate desserts are delicious” (UA) being true implies that “some chocolate desserts are delicious”(PA) is also true. But be careful—if “some chocolate desserts are delicious” (PA) is true, it does not necessarily mean that “all chocolate desserts are delicious” (UA) is also true! This means that (PA) propositions are the subalterns of (UA) propositions, but not vice versa. In other words, (PA) propositions do not have subalterns.  

This relation also exists between Universal Negative and Particular Negative propositions. “No teenagers are early risers” (UN) being true implies that “some teenagers are not early risers” (PN) is also true. However, “some teenagers are not early risers” (UN) being true does not imply that “no teenagers are early risers” (PN) is also true. This means that (PN) propositions are the subalterns of (UN) propositions, but not vice versa. In other words, (PN) propositions do not have subalterns. We therefore have the following:

• (PN) is the subaltern of (UN)
• (PA) is the subaltern of (UA)
• Nothing is the subaltern of (PN)
• Nothing is the subaltern of (PA)