Categories, Necessity, and Sufficiency

To see how these hierarchical relationships are established, let’s look at our prototypical Barbara syllogism again.

All dogs are mammals
all mammals are warm-blooded
therefore all dogs are warm-blooded

This syllogism works only because there is a relationship between these three categories that can be pictured like this.

The easiest way to understand what this picture means is to think of it as a view from an airplane, from which we see inter-nested large territories occupied by the members of the categories, and fenced off into different regions. Imagine that all of the warm blooded creatures in the world have been herded together by an obsessive-compulsive psychopath into the large circular pen. Imagine further that all the mammals have been herded into a smaller pen completely inside the larger pen, and all of the dogs have been herded together inside an even smaller pen inside that one. The section of the mammal pen outside of the dog pen, is filled with roaming kangaroos, aardvarks, wildebeests, bandicoots etc., but is an area from which the dogs are forever banned. In other words, although all dogs are inside the mammal space, it is not true that all mammals are inside the dog space. The outermost ring contains…