Chapter Six, Tutorial Five

PL for Categorical Logic

PL for Categorical Logic

Natural language provides many ways to quantify over objects. Often these
involve very complicated constructions.
Think about the complicated English quantification you needed to symbolize in the exercises of 6.4.

English uses "every", "all", "only", "none but", sometimes "any", "a" *and other means* to make universal categorical statements. And we had numerous ways to represent existential and universal categorical statements as well.

... we treated general quantification with universal and existential quantifiers (in tutorial one: (^x) and (%x)). But these quantifiers are hard when there are *many subjects*. (Remember, we could symbolize while we were talking about all cats alone, 6.1d, but *not* when the subject changed between cats, dogs, mammals, etc.) Categorical logic helps remedy that shortcoming. But categorical logic is just
a stepping stone to the full treatment: PL.

Fortunately, symbolizing in PL often boils down to our three categorical forms. And because obversion allows us to define a negative statement in terms of a universal one, we can take just two forms as basic. In this tutorial we focus on these two forms, the "existential" and the "universal".

The Existential Form

The existential form is pretty easy to symbolize in PL.

For anyone skipping over 6.2 to 6.4: Any English sentence like "Some prime number is even" or "Some students are freshmen" are simple examples of the form. An English sentence is of the form if it fits this mold:

Some S are P.

where 'S' (the subject)
and 'P' (the predicate
of the expression) range over English predicates. The predicates stand
for categories (like *students* and *freshmen*).

Now, it's pretty easy to see how to symbolize sentences of this form. For instance,

Some whales are living in Ohio

means that there is some thing or things that are whales and that are living in Ohio. This may be symbolized as

(%y)(Wy&Oy)

(Read this as "there is a y such that y is W and y
is O".)

assuming that 'Wx': "x is a whale", 'Ox': "x is living in Ohio".

Now, many English sentences are not *exactly* of this existential
form but are close enough. An example we used in the last chapter was:

There is an even number less than three

This has the same meaning as "Some even number is less than three" so might be symbolizes as

(%x)(Ex&Lx)

Another familiar example

Someone will attend law school and need a loan.

is a simple stylistic variant of "Some persons who will attend law school are persons who will need a loan" and can be symbolized as

(%x)(Wx&Nx)

Once again, notice that all the English sentences highlighted on this
page can be paraphrased as of the *existential form*:

Some S are P.

Each can be symbolized as of the form

(%x)(Sx & Px)

One may usefully force many English sentences into this form!

Now, given the symbolization used above, which of the following might symbolize "Some even number is prime"?