Chapter 6, Tutorial 2

Categorical Logic and Venn diagrams

In 6.1 we introduced the logic of *quantities*. The new symbols here are the existential ('%') and universal ('^') quantifiers. Still, we restricted all discussion to *a single * subject matter: our "universe of discourse" included a single subject, for example *Cats*. We need to loosen that restriction and gradually work up to the full logic of quantities. We begin with categorical logic later in this tutorial. But there are important lessons to learn from the restricted case.

The last exercise in 6.1 is very important. If you've not yet done this one, this would be the time: 6.1d.

This exercise deals
with *cats alone*:

Universe of Discourse: Cats (and only cats! for this symbolization **felines are our only subject**)

Mx: x is a mammal, Rx: x is a reptile; Wx: x is wild.

f: Felix the Cat (pretend he's real)

t: Tony the Tiger (of course, he's real, he was on TV)

So, to say in PL "**all** cats are mammals", we just wrote

(^x)Mx

With the subject restricted to cats alone, we need just say "all are mammals" or "everything x is such that it is a mammal".

...we can symbolize (the obviously false) statement "**some** cats are reptiles" as

(%x)Rx

This just means that there is at least one thing in our universe of discourse, cats, that is also a reptile. Is there such an animal? Well, apparently not! No cats are reptiles! We see then how to symbolize something else: "**no** cats are reptiles" is a negation:

~(%x)Rx

Think about this a bit more. This statement in the last blue box means that *it is not true that there is a reptile* (from among the cats). Well, this is one way to say "No cats are reptiles".
But there is another way:

There is another way to say "**no** cats are reptiles"; which is it?