Categorical Logic

Categorical Logic was invented by Aristotle, and presupposes his system of categories. To understand how it works, we have to know the method Aristotle used to classify things into categories. For Aristotle, an individual substance is not just an individual. It is always a member of a kind or category. Fido is not just Fido, he is also a dog. Socrates is not just Socrates, he is also a man. By itself, this principle was probably not very innovative even in Aristotle’s time. In order to talk about the world at all, we need words that refer to the kinds of things that we encounter in it: words like dog, chair, football, kangaroo, asparagus etc. Words like these (called nouns, as you may remember from English class) are the simplest kinds of categories. Aristotle, however, turned this process back onto itself, and developed a system for classifying the kinds themselves into kinds. Thus a dog is a mammal, a chair is a piece of furniture, asparagus is a vegetable, a kangaroo is a marsupial. And when we make judgments about those kinds, we classify them into kinds of kinds. Mammals are animals, furniture is an artifact, vegetables are living things. This may seem like an obvious thing to do today, but that is only because we have grown up in a society which is so thoroughly permeated by Aristotle’s ideas that they have become common sense[1]

If you study biology you will see a very sophisticated example of this way of thinking. Biology not only has names for animals and for kinds of animals, but even has specific names for kinds of kinds of kinds, kinds of kinds of kinds of kinds, and…