1. Categorical versus Propositional Logic

1. Categorical vs Propositional Logic

In any standard logic textbook you’ll see separate chapters on both propositional logic and categorical logic. Sometimes categorical logic is called “Aristotelian” logic, since the key concepts in this branch of logic were first developed by the Greek philosopher Aristotle.

I’m not planning on doing a whole course on categorical logic at this stage, but there are a few concepts from this tradition that are important to have under your belt when doing very basic argument analysis, so in this next series of tutorials I’m going to introduce some of these basic concepts.

In this introduction I’m going to say a few a words about what the basic difference is between categorical logic and propositional logic.

In the course on “Basic Concepts in Logic and Argumentation” we saw a lot of arguments and argument forms that are basically categorical arguments and that use the formalism of categorical logic.

Here’s a classic example.

1. All humans are mortal.
2. Simon is human.
Therefore, Simon is mortal.

This argument is valid. When you extract the form of this argument it looks like this:

1. All H are M
2. x is an H
Therefore, x is an M

The letters are arbitrary, but it’s usually a good idea to pick them so they can help us remember what they represent.

Now, the thing I want to direct your attention to is how different this symbolization is from the symbolization in propositional logic.

When we use the expression “All H are M”, the “H” and the “M” DO NOT represent PROPOSITIONS, they don’t represent complete claims. In propositional logic each letter symbolizes a complete proposition, a bit of language that can be true or false. Here, the H and the M aren’t propositions.

So what are they?

They’re categories, or classes. H stands for the category of human beings, M stands for the category of all things that are mortal, that don’t live forever.

These categories or classes are like buckets that contain all the things that satisfy the description of the category.

What they don’t represent is a complete claim that can be true or false. This is a fundamental difference in how you interpret symbolizations in categorical logic compared to how you interpret them in propositional logic.

In categorical logic, you get a complete claim by stating that there is a particular relationship between different categories of things.

In this case, when we say that all humans are mortal, you can visualize the relationship between the categories like this:

We’re saying that the category of mortals CONTAINS the category of humans. Humans are a SUBSET of the category of things that die. The category of mortals is larger than the category of humans because lots of other things can die besides human beings. This category includes all living things on earth.

Now, when you assert this relationship between these two categories, you have a complete proposition, a claim that makes an assertion that can be true or false.

This is the fundamental difference between symbolizations in propositional logic and categorical logic.

In propositional logic you use a single letter to represent a complete proposition.

In categorical logic the analysis is more fine-grained. You’re looking INSIDE a proposition and symbolizing the categories that represent the subject and predicate terms in the proposition, and you construct a proposition by showing how these categories relate to one another.

Now, what does that small “x” represent, in “x is an H”?

It represents an INDIVIDUAL human being, Simon.

In categorical logic you use capital letters to represent categories or classes of things, and you use lower-case letters to represent individual members of any particular category.

On a diagram like this you’d normally us a little x to represent Simon, like this:

So, putting the X for Simon inside the category of humans is a way of representing the whole proposition, “Simon is human”.

Notice, also, that from this diagram you can see at a glance why the argument is VALID. This diagram represents the first two premises of the argument. When judging validity you ask yourself, if these premises are true, could the conclusion possibly be false?

And you can see that it can’t be false. If x is inside the category of humans, then it HAS to be inside the category of mortals, since humans are subset of mortals.

In a full course in categorical logic you would learn a whole set of diagramming techniques for representing and evaluating categorical arguments, but that’s not something we’re going to get into here.

What we’re going to talk about is what sorts of claims lend themselves to categorical analysis. These are claims with the following forms:

  • All A are B
  • Some A are B
  • No A are B
  • All A are not-B
  • Some A are not-B
  • No A are not-B

Claims like “All humans are mortal”, “Some men have brown hair”, “No US President has been female”, “All mammals do not have gills” and so on.

The Greek philosopher Aristotle worked out a general scheme for analyzing arguments that use premises of this form. In the tutorial course on “Common Valid and Invalid Argument Forms” we look at a few of the most common valid and invalid categorical argument forms.

In the remaining lectures in this section all I really want to do is look at the semantics of categorical claims, what they actually assert, and how to write the contradictory of these categorical claims.