Chapter 6, Tutorial 4

Categorical Syllogisms

Traditional logic, going back to Aristotle and before, spent much time and energy working out an intricate systematization of the logic of categorical statements and "syllogisms", a.k.a. arguments. In this tutorial, we will address these syllogisms but use Venn Diagrams to circumvent the intricacies.

Let's begin with an example of a categorical syllogism:

No mammals are fish.

__All whales are mammals.__

So, no whales are fish.

Makes sense, yes? Try this one...

All whales are mammals.

__No fish are whales.__

So, no fish are mammals.

OK? Well anyway, these are two examples of categorical syllogisms. But only one of the two is valid...how do we tell? Venn Diagrams. Let's get to work on it...

1. Let's see categorical syllogisms how are defined. Each consists of two premises, both of categorical form, and one categorical statement as conclusion.

The conclusion is of the form: "All/No/Some S are P" (choose one quantity word from the three types). Each premise contains one of the category terms, S or P, from the conclusion. In addition, there is a "middle term", M, in both premises. So, in the first argument above...

No mammals are fish.

__All whales are mammals.__

So, no whales are fish.

S is "whales" and P is "fish" (see the conclusion) while the middle term, M, is mammals. Note that M is the one category term in both premises.

2. Because there are three terms in our syllogisms, our diagrams will be a little more complicated. We'll need three circles to represent the three groups.

Notice that the possibilities of overlap are increased. We could continue to talk about these in terms of their colors. Now there's an orange area that is inside the M and P circles but outside S. Right? This means that any objects in orange are in category M and P but they are non-S.

But it's better to have numbers:

Now, instead of "orange" we can just say 6.

The important thing is what it means to be in an area. Anything in area 3 is just P, it is neither M nor S.

Now let's see how we put these to work.

Here's that first categorical syllogism again:

No mammals are fish.

__All whales are mammals.__

So, no whales are fish.

Instead of writing out the whole category description, sometimes they get quite long!, we'll just do letters. So, let's rewrite this as

No M are F.

__ All W are M. __

So, no W are F.

Now we just do the Venn Diagram by diagramming what the premises mean together...the argument will be valid if the premises mean that the conclusion is true.

...way of diagramming. The only difference is that now we have a third category in the way.

Here goes, let's first take our Venn Diagram with numbers...

and diagram the first premise.

The first premise is "No M are F". To say what it means for this to be true, we just exclude everything in the area of overlap between M and F.

This is just what we did before. Here it means that areas 5 and 6 are completely empty. We use our stakeout to show this. Do it now.

Next we diagram the second premise right over the top of the first diagram. "All W are M" means that there is nothing in W outside of M: something can't be a whale without also being a mammal. OK? So, diagram this in the normal way, just strike-out the areas 1 and 2 because both of these areas would, if non-empty, contain whales that are not mammals. Do it now.

What do all these stripe mean? Well, the are of overlap of W and F is marked off. So, our premises together mean that there is *nothing* that is both W and F: No Whales are Fish. This is to say that the premises are enough. If they are true, then the conclusion "No W are F" must be. So, this argument is valid.

Here's the test of validity:

- Diagram both premises.
- Stop diagramming and see if premises together indicate that the conclusion is true.
- If they do make the conclusion true, then the argument is valid. Otherwise, it's invalid.

Now which of the following diagrams is right for our second argument?

All whales are mammals.

__No fish are whales.__

So, no fish are mammals.

Click on the Venn Diagram below for this argument. Feedback is shown below the diagrams.

Yes, this is the right one. Time to move on...

Nope. The lines through 4 and 7 are correct for the first premise. But the lines through 5 and 2 do not diagram the second premise.

Not this one. The first premise needs to exclude any whales that aren't mammals. So you need to strike-out 4 and 7 with lines.